Here are a number of boxes packed by congruent polycubes.
Box Collection
Michael Reid (reid@math.ucf.edu)
September 15, 2007
view in fixed-width font!
.............................................................................
Tricubes
.............................................................................
*
**
1x2x3 minimal
+--+--+--+
| | |
+ +--+ +
| | |
+--+--+--+
also: 1x5x9, 3x3x3
.............................................................................
Tetracubes
.............................................................................
*
***
1x2x4 minimal
+--+--+--+--+
| | |
+ +--+--+ +
| | |
+--+--+--+--+
also: 1x3x8, 2x2x6, 2x3x6, 3x3x4
.............................................................................
***
*
1x4x4 minimal
+--+--+--+--+
| | |
+--+ +--+ +
| | | |
+ +--+--+ +
| | | |
+ +--+ +--+
| | |
+--+--+--+--+
also: 3x3x8, 3x5x8, 5x5x8
.............................................................................
**
**
2x3x4 minimal
+--+--+--+ +--+--+--+
| |u | | d| |
+ +--+--+ +--+--+ +
|u | |u | | d| | d|
+--+ + + + + +--+
| | | | | |
+ +--+--+ +--+--+ +
| |u | | d| |
+--+--+--+ +--+--+--+
level 1 level 2
.............................................................................
2*
*
2x2x2 minimal
+--+--+ +--+--+
|u | | d| |
+ +--+ +--+ +
| |u | | d|
+--+--+ +--+--+
level 1 level 2
also: 2x5x8, 2x5x12, 4x4x5 by Andrew Clarke
also: 4x5x6 by Andris Cibulis
.............................................................................
**
2
2x2x2 minimal
+--+--+ +--+--+
| | | |
+ +--+ +--+ +
|u |u | | d| d|
+--+--+ +--+--+
level 1 level 2
.............................................................................
Pentacubes
.............................................................................
*
****
1x2x5 minimal
+--+--+--+--+--+
| | |
+ +--+--+--+ +
| | |
+--+--+--+--+--+
also: 1x7x15, 3x5x5
.............................................................................
**
***
1x2x5 minimal
+--+--+--+--+--+
| | |
+ +--+ +
| | |
+--+--+--+--+--+
also: 1x7x15, 3x3x5
.............................................................................
*
****
1x5x10 (minimal) by David Klarner
+--+--+--+--+--+--+--+--+--+--+
| | | | |
+ +--+--+ +--+ +--+--+ +--+
| | | | | | |
+ +--+ +--+--+--+--+--+--+ +
| | | | |
+ +--+--+--+--+--+--+ +--+ +
| | | | | | |
+--+ +--+--+ +--+ +--+--+ +
| | | | |
+--+--+--+--+--+--+--+--+--+--+
also: 1x9x20, 1x9x30, ... (many others)
.............................................................................
* *
***
+--+ +--+
| | | |
+--+--+--+ +--+ +--+
| u | | | d| |
+--+--+--+ +--+ +--+
| | | |
+--+ +--+
level 1 level 2
2x3x5 minimal
+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+
(2 levels deep)
also: 2x10x10, 2x10x14, 2x11x20, 2x11x25, 3x3x10, 3x5x7, 4x4x5, 4x5x5, 5x5x11
.............................................................................
***
**
2x4x5 minimal
+--+--+--+--+--+ +--+--+--+--+--+
| u | | | | d | |
+--+--+ +--+--+ + +--+--+--+ +
|u | | | | d| | |
+ +--+--+ +--+ +--+ +--+--+ +
| | |u | | | | d|
+ +--+--+--+ + +--+--+ +--+--+
| | u | | | | d |
+--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2
.............................................................................
*
*
***
+--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+
|u |u | |ud|ud| | d| d| |u |u | |ud|ud| | d| d|
+ + + +--+--+ +--+--+ +--+--+ +--+--+ + + +
| | | |u |u | |ud|ud| | d| d| | | | | | |
+ + +--+ + + +--+ +--+--+--+ +--+--+--+ + + +--+ + + +--+
| | |u | | | |ud| | d| | u | | | |ud| | | | d|
+--+--+--+ + + +--+ +--+--+--+ +--+--+--+ + + +--+ +--+--+--+
| u | | | |ud| |u |u | d| |ud|ud|u | | d| d|ud| | d|
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
level 1 level 2 level 3 level 4 level 5 level 6
3x5x6 minimal
+--+--+--+
| |
+--+--+ +
| | |
+ + +
| | |
+ +--+
| |
+ +
| |
+--+--+--+
(6 levels deep)
also: 4x5x6, others
.............................................................................
***
*
*
3x10x10 (co-minimal) by Frits Go"bel
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
|x |x |x | | |x | |x | |x |x |x | x | | x |
+--+--+--+ + +--+--+ +--+--+ + + + +--+--+--+ +--+--+--+
| | | | | |x | | | | | | | x |
+ +--+--+ + +--+--+ + +--+ + + + +--+ +--+ +--+--+--+
| | | | |x | | |x | | | | | | | |x | x |
+ +--+--+--+--+ +--+--+ +--+ +--+--+--+ + + +--+--+--+--+
| | |x | | | | | x | | | | | |
+--+ +--+--+--+ + +--+--+--+ +--+--+--+ +--+ +--+--+ + +
|x | | | | | | |x | | | | | |
+--+ +--+ +--+--+ +--+ +--+ + + +--+--+--+--+--+--+ + +
| | | | | |x | | | | | | |x |
+--+--+--+ + +--+--+--+ +--+ + + +--+--+ +--+ +--+--+--+
| | | |x | | | | | | | | |x |
+--+ +--+--+ +--+--+--+--+ + +--+--+--+--+ + + +--+--+--+
|x | | |x | | | | | |x |x | | | | | | |
+--+ + +--+--+ + +--+--+ + +--+--+--+ +--+ +--+ + + +
|x | | | | | | |x | | | | | |
+--+--+ +--+--+ + +--+--+--+ +--+--+--+ +--+--+--+ + + +
|x | |x | | |x |x |x | |x | |x |x |x |x |
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
levels 1, 3 level 2
5x5x12 (co-minimal) by Michael Beeler
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| u | |u | | |ud| ud| |u | d| |u | d| |ud |ud| | | d| | d |
+--+--+--+ +--+ + +--+--+--+--+ +--+--+ +--+--+ +--+--+--+--+ + +--+ +--+--+--+
| u | |u | | |ud| ud| |u | d| |u | d| |ud |ud| | | d| | d |
+--+--+--+ +--+ + +--+--+--+--+ +--+--+ +--+--+ +--+--+--+--+ + +--+ +--+--+--+
|u | | | |ud|u |u |u | | | d|ud|ud|ud|u | | | d| d| d|ud| | | | d|
+--+ +--+--+--+ +--+--+--+--+ + +--+ +--+ +--+ + +--+--+--+--+ +--+--+--+ +--+
|u | | | |ud | | | | d| | | |u | | | | ud| | | | d|
+--+ +--+ +--+ +--+--+--+ + + +--+ + + +--+ + + +--+--+--+ +--+ +--+ +--+
| | |u | | | |ud| |u | | | | d| |ud| | | | d| | |
+--+--+--+ +--+ + +--+--+ +--+ +--+--+ +--+--+ +--+ +--+--+ + +--+ +--+--+--+
| u | |u | | |ud|u | |ud| |u | d|ud|u | d| |ud| | d|ud| | | d| | d |
+--+--+--+--+--+ + +--+--+--+ + +--+--+--+--+--+ + +--+--+--+ + +--+--+--+--+--+
|u | | u | |ud| |u |ud| | | d|u |ud| d|u | | |ud| d| |ud| | d | | d|
+--+ +--+--+--+ +--+ +--+--+ + +--+--+ +--+--+ + +--+--+ +--+ +--+--+--+ +--+
|u | | | |ud| | | | d| | | |u | | | |ud| | | | d|
+--+ +--+ +--+ + + +--+--+--+ +--+ + + +--+ +--+--+--+ + + +--+ +--+ +--+
| | |u | | | | ud| |u | | | | d| |ud | | | | d| | |
+--+--+--+ +--+ + +--+--+--+--+ +--+ +--+ +--+ +--+--+--+--+ + +--+ +--+--+--+
| | |u | | |u |u |u |ud| |u |ud|ud|ud| d| |ud| d| d| d| | | d| | |
+--+ +--+--+--+ +--+--+--+--+ + +--+--+ +--+--+ + +--+--+--+--+ +--+--+--+ +--+
|u | | u | |ud |ud| | | d|u | | d|u | | |ud| ud| | d | | d|
+--+ +--+--+--+ +--+--+--+--+ + +--+--+ +--+--+ + +--+--+--+--+ +--+--+--+ +--+
|u | | u | |ud |ud| | | d|u | | d|u | | |ud| ud| | d | | d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3 level 4 level 5
.............................................................................
**
**
*
3x6x10 (co-minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u | u | | | ud| |ud |u | | | d | d| | d|
+ +--+--+--+ +--+ +--+--+ +--+--+ + + +--+--+--+ +--+
| | |u | | | | | d| | | |
+--+ +--+--+ +--+ + +--+--+ + + + +--+ +--+--+--+ +
|u | |u | | | |ud| ud| |u | | | d| | d| | d| |
+ +--+--+--+--+ + +--+--+--+--+ +--+ +--+--+--+ +--+--+
| |u | | |u |u | | |ud | | d|ud| | d| | d|
+--+--+ + +--+--+ + +--+--+ +--+ + +--+--+--+--+ +--+
|u | | | |ud| | | | | d| | | | |
+ +--+ +--+ +--+ +--+ +--+--+ +--+ +--+ +--+ +--+ +
| | | | |u | | | | |ud| | | | d|
+--+ +--+--+--+--+ + +--+ +--+--+ + +--+--+ + +--+--+
|u | |u | |ud|u | | ud| | | d| d| | | d| |
+--+--+ +--+--+--+ +--+ +--+--+--+--+ + +--+--+--+--+ +
| |u | |u | |u | | | d| |ud |ud| | | | d| | d|
+ +--+--+--+ +--+ + + + +--+--+ + +--+ +--+--+ +--+
| | | |u | | | | | d| | |
+--+ +--+--+--+ + + +--+--+ +--+--+ +--+ +--+--+--+ +
|u | |u | u | | | d| ud| |ud | | | d | d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3
5x6x6 (co-minimal) by Torsten Sillke and Ishino Keiichiro (independently)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u | u | | | d u |ud | | |u |u | d| d|u |u | | d|ud | ud| d| | d| | d |
+ +--+--+--+ +--+ +--+--+--+--+--+ + +--+--+--+--+--+--+ + +--+--+--+--+ + +--+--+ +--+--+--+
|u | |u | |u | |ud| | d u |ud| | d|u | | d|u | d| |u | d| u d|u | | d| | d| | d|
+--+ +--+--+ + + + + +--+--+--+--+ +--+--+ +--+--+--+ + + +--+--+--+ + +--+--+ +--+ +--+
| | | | | | |u |u | |u | | | d|ud| ud| | | |u | d| | d| | | | d| | |
+ +--+--+ +--+--+ +--+ +--+ +--+--+ + +--+--+--+--+ + +--+ + + +--+--+ + +--+--+ +--+--+
| |u | | | | | d| | | | | | | | | | | | | | | |
+--+--+ +--+ +--+ + +--+--+--+ + + +--+ +--+--+ +--+ + +--+--+--+ +--+ +--+ +--+--+ + +
|u | |u | |u | |ud| u d| | d| | d| | d|u | |u | |u | u d| |ud| | d| | d| | d|
+ +--+ +--+--+--+ +--+--+--+--+--+ + +--+--+--+--+--+--+ + +--+--+--+--+ + +--+--+--+--+ +--+
| |u | |u | | d u | ud|u | |u |u | d|u | d| d| | d|ud |ud | | | d| d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4 level 5
.............................................................................
*
**
**
5x6x6 by Torsten Sillke (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| u | u | | | | d u | d u | | | |u | d|u | d| | | | d u | d u | | | | d | d |
+--+--+--+--+ +--+ + +--+--+--+--+ + + +--+--+--+--+ + + +--+--+--+--+ + +--+ +--+--+--+--+
| | u | | | | | d u | | | |u | | | | | d u | | | | | d | |
+ +--+--+ +--+ + +--+ +--+--+ +--+ +--+ +--+--+ +--+ +--+ +--+--+ +--+ + +--+ +--+--+ +
|u | | | | | d| | |u | |u | | | d| | d| | |u | | | | | d|
+--+ +--+--+ +--+ + +--+--+--+--+ + +--+--+--+--+--+--+ + +--+--+--+--+ + +--+ +--+--+ +--+
| | | |u | |u | | | d| | d| | |u | |u | | | d| | d| | | |
+ +--+ +--+--+ + +--+ +--+--+ +--+ +--+ +--+--+ +--+ +--+ +--+--+ +--+ + +--+--+ +--+ +
| | |u | | | |u d| | | | d|u | | | |u d| | | | d| | |
+--+ +--+--+--+--+ + +--+--+--+--+ + + +--+--+--+--+ + + +--+--+--+--+ + +--+--+--+--+ +--+
| |u |u | | |u d|u d| | | | |u | d|u | | | |u d|u d| | | d| d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4 level 5
also: 5x7x9 by Patrick Hamlyn
.............................................................................
**
*
**
6x10x10 by Yoshiya "Wolf" Shindo
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
| | | |u | | |u | | |u d|u |u |ud| | | | d|u | d|ud| d| |
+ + +--+ +--+--+ + +--+--+ +--+ +--+--+--+--+--+--+--+ + +--+ +--+--+--+ +--+--+--+ +
|u | |u | | | | |ud| | d u | u | | | d| | u | d| |ud| |
+--+ +--+--+--+ + +--+--+ + +--+ +--+--+--+--+--+ +--+--+ + + +--+--+--+ +--+ +--+--+
| | | |u | | |u | | | | | | d u | | d|u | | | | |u |u | d| | | d|
+--+--+ + +--+--+--+--+--+--+ + +--+--+ +--+--+--+--+ + + +--+--+--+ +--+--+--+--+ +--+
| | | | | |u |u |u | | | | | |ud|ud| d| | |u |u |
+ +--+--+--+--+ + +--+ +--+ +--+ + +--+--+ + +--+ + + +--+--+--+--+--+ + +--+--+ +
| | | | | |u | |u | |u | | | | | | d|u | d| | d u | | | |u | d| |
+--+ +--+ +--+--+ +--+ +--+ +--+--+ + +--+--+ + +--+--+ +--+--+--+ +--+--+ +--+--+--+
|u | |u | | | | | | | d|u | d| | | | | |u | | | d|u | | | |u d|
+--+ +--+ + +--+--+--+--+ + + + +--+ + +--+--+ + +--+ + +--+--+ + +--+--+--+--+--+
| | | | | | | | | |u |u |u | |u |u | | | d|ud|ud|
+--+--+--+--+--+--+ + +--+--+ + + +--+--+--+--+ +--+--+ + +--+ +--+--+--+--+ +--+--+--+
| |u | | |u | | | | |u | d| |u d| | | | | d| | | d|u |u | | | |
+ +--+--+ + +--+--+--+ +--+ +--+--+ +--+--+--+--+--+ +--+ +--+--+ +--+ +--+--+--+ + +
| | | |u | |u | | |u |u d| |ud| | |ud| | d|u | | d|
+--+--+ + +--+--+ +--+ + + + +--+--+--+--+--+--+--+ +--+ + +--+--+--+ +--+--+--+ +--+
| u | | |u | | | | | |ud| u |u | d u | | | | d |ud| d| u | d| |
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
level 1 level 2 level 3
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
| |u |ud|u | d|u | | | |ud| d| d|u d| | | d| | | d| | | |
+ +--+--+--+ +--+--+--+ +--+ + +--+--+--+--+--+--+--+ +--+ +--+--+ + +--+--+ +--+ + +
| |ud| |u | d | |u | | | d | d u | |ud| | | | | d| | d|
+--+--+ +--+ +--+--+--+ + + +--+--+ +--+--+--+--+--+ +--+ + +--+--+ + +--+--+--+ +--+
|u | | |u | d| d| | | | | d|u | | d u | | | | | | d| | | d| | | |
+--+ +--+--+--+--+ +--+--+--+ + + +--+--+--+--+ +--+--+ + +--+--+--+--+--+--+ + +--+--+
| d| d| | |u |ud|ud| | | | | | d| d| d| | | | | |
+ +--+--+ + +--+--+--+--+--+ + + +--+ + +--+--+ + +--+ +--+ +--+ + +--+--+--+--+ +
| |u | d| | | | d u | |u | d|u | | | | | | d| | d| | d| | | | | |
+--+--+--+ +--+--+ +--+--+--+ +--+--+ + +--+--+ + +--+--+ +--+ +--+ +--+--+ +--+ +--+
|u d| | | | d|u | | | d| | | | | |u | d|u | | | | | | | d| | d|
+--+--+--+--+--+ + +--+--+ + +--+ + +--+--+ + +--+ + + + +--+--+--+--+ + +--+ +--+
|ud|ud|u | | | d| d| | d| d| d| | | | | | | | | |
+--+--+--+ +--+--+--+--+ +--+ + +--+--+ +--+--+--+--+ + + +--+--+ + +--+--+--+--+--+--+
| | | | d| d|u | | |u | | | | |u d| |u | d| | | | | d| | | d| |
+ + +--+--+--+ +--+ +--+--+ +--+ +--+--+--+--+--+ +--+--+ +--+ +--+--+--+ + +--+--+ +
|u | | d|u | |ud| | |ud| |u d| d| | | d| | d| | | |
+--+ +--+--+--+ +--+--+--+ + +--+ +--+--+--+--+--+--+--+ + + + +--+ +--+--+ + +--+--+
| |u | d |u |ud| u | | | | d u | d| d |ud| | | | | | d| | | d |
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
level 4 level 5 level 6
also 6x6x25 by Yoshiya "Wolf" Shindo
.............................................................................
**
2*
2x2x5 minimal
+--+--+--+--+--+ +--+--+--+--+--+
|u | | u | | d | | d|
+--+ +--+ +--+ +--+ +--+ +--+
|u | |u | | d| | d|
+--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2
.............................................................................
2**
*
2x2x5 minimal
+--+--+--+--+--+ +--+--+--+--+--+
|u | |u | | d| | d|
+--+ +--+ +--+ +--+ +--+ +--+
|u | | u | | d | | d|
+--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2
.............................................................................
2*
**
+--+--+ +--+--+
|u | | d| |
+ +--+--+ + + +--+
|u |u u | | d| d d|
+--+--+--+ +--+--+--+
level 1 level 2
2x2x5 minimal
+--+--+--+--+--+
| | |
+ +--+ +
| | |
+--+--+--+--+--+
(2 levels deep)
also: 2x3x5, others
.............................................................................
2
***
+--+--+--+ +--+--+--+
| | | |
+ +--+--+ +--+--+ +
|u | |u | | d| | d|
+--+ +--+ +--+ +--+
level 1 level 2
+--+ +--+ +--+ +--+
|u | |u | | d| | d|
+--+--+--+--+ +--+ +--+ +--+--+--+--+
|u | |u | | d| | d|
+--+ +--+--+--+--+ +--+--+--+--+ +--+
|u | | | u | | d | | | d|
+ + + +--+--+--+ +--+--+--+ + + +
| | |u |u | | d| d| | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
2x5x6 co-minimal
+--+--+--+--+--+--+
| | |
+ +--+ + +--+ +
| | | | | | |
+--+ +--+--+ +--+
| |
+ +
| |
+ +
| |
+--+--+--+--+--+--+
(2 levels deep)
3x4x5 co-minimal
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| | | | u | u | | | d| | d|
+ +--+ +--+--+ +--+--+--+--+--+ +--+ + + + +
|u |u |u |u | | |ud|ud| d| d | | d| d| | | |
+--+--+--+ + + +--+--+--+--+--+ +--+--+--+ + +
|u |u | | | | |ud|ud|u |u | | d| d| d| d| |
+--+ + + + + +--+--+--+--+--+ + +--+ +--+--+
| |u | |u | | d| d| | | |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3
same pentacube, allowing reflections
+--+--+ +--+--+ +--+--+
|u |u | | d|ud| | | d|
+ +--+--+ +--+--+--+ + + +--+
| | |u | | d| |
+--+--+--+ +--+--+--+ +--+--+--+
level 1 level 2 level 3
2x3x5 minimal
+--+--+--+--+--+
| | |
+ +--+ +
| | |
+--+--+--+--+--+
(3 levels deep)
.............................................................................
2
***
2x5x5 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+ +--+--+--+--+--+
| u |u | | | | d| d |
+--+--+--+ + + + + +--+--+--+
|u | | |u | | d| | | d|
+--+--+ +--+ + + +--+ +--+--+
| |u | u | | | | d | d| |
+ +--+ +--+--+ +--+--+ +--+ +
|u | | | u | | d | | | d|
+ + +--+--+--+ +--+--+--+ + +
| |u | u | | d | d| |
+--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2
.............................................................................
2**
*
3x5x6 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u | |ud| | d|u | |ud| | d| d| d|
+ +--+ +--+--+--+ +--+ +--+--+ +--+ +--+--+--+ +--+ +
| | | | u | |u | | u | d| | d | | d| |
+--+ +--+--+--+ + +--+ +--+--+--+--+ + +--+--+--+--+--+
| | | u | | | | u | | d|u | | | | d| d| |
+ + +--+ + +--+ +--+--+ +--+--+ + +--+--+--+--+ + +
| |u | | |u | |u | d| | d| | d | | | |
+ +--+--+ + +--+ +--+--+ +--+--+--+ + +--+--+ +--+ +
|u |u | |u | | d| d|u | d|u | | | d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3
.............................................................................
*
**2
4x5x6 (minimal) by Patrick Hamlyn
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
|u |u |u | |ud| d| d| u | | d |u |u |ud| | d| d| d|
+ + +--+--+ + +--+--+--+--+--+ +--+--+--+--+--+ + +--+--+ + +
| | | |u | | |u | d|u | | d|u | d| | | d| | | |
+--+ + +--+--+ +--+--+ + +--+ +--+ + +--+--+ +--+--+ + +--+
| | u | | | | | d| |u | | | | | d | |
+--+--+--+--+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+--+--+--+--+
|u | | | d| | | | | | | |u | | | d|
+--+--+ + +--+ +--+ + +--+--+ +--+--+ + +--+ +--+ + +--+--+
| |u | | | | |u | d| u | | d |u | d| | | | | d| |
+ +--+--+ + + +--+--+--+--+--+ +--+--+--+--+--+ + + +--+--+ +
| u |u |u | |u | d| d|ud| |ud|u |u | d| | d| d| d |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3 level 4
same pentacube, allowing reflections
3x5x6 (minimal), by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u | | u | | d|ud|u | |ud| | d| d | d|
+ + + +--+--+--+ +--+--+--+--+ +--+ +--+--+--+--+ + +
| | | |u |u | | |u |ud| | d| | d| d| | |
+ +--+ +--+--+ + + +--+--+--+ +--+ + +--+--+ +--+--+
| |u |u |u | | | |u | d| d| d| | | | d| | | |
+--+--+--+ +--+ + + + +--+--+--+ + +--+--+ +--+--+ +
|u | u | | | | d| |ud|u | | | | d| d| | |
+--+--+--+ +--+--+ +--+ +--+--+--+--+ + + +--+--+ + +
|u | |u | |ud| |u |ud|u | | d| d| d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3
.............................................................................
**
*2
5x8x8 by Helmut Postl
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|u | |u | | |u | | d |u d| u |u d| |u | | d|u | | d| d|u |
+--+ +--+--+ +--+ +--+ + +--+ +--+--+ +--+ + +--+ +--+--+ +--+--+--+
| u | | u |u |u | |u | d| |u | d| d| d| | | d| u | d |u | |
+--+--+ +--+--+--+--+ + +--+--+--+--+--+--+--+--+ +--+--+ + +--+--+ +--+
| |u | u |u | | | |u d| | d| d | u | | d|u | | |u | u | d|
+ +--+--+ +--+ +--+--+ + +--+ +--+ +--+--+ + +--+--+--+--+--+--+--+--+
| u |u | | u | |u | | | d| d u |u | d|u | d| | |u | d| d| d|u |
+--+ +--+--+--+--+ +--+ +--+--+--+--+--+--+--+--+ +--+ +--+--+--+--+ +--+
|u | |u | |u |u | | d|u | d|u |u d| d| | |u | d | d| d|u | |
+--+--+ +--+ +--+--+ + + +--+--+ +--+ +--+ + +--+--+--+--+--+--+--+--+
| | |u |u |u | | |u | d| d| | d u | | d|u |u | | |u | d|
+ +--+--+--+--+ +--+--+ +--+--+--+--+--+--+--+--+ +--+ +--+--+ + +--+--+
| u |u |u | |u | | | d| d| d|u | | d|u | | |u | d|u | d|
+--+ +--+ +--+--+ +--+ + +--+ +--+--+ +--+ + +--+--+--+ +--+--+ +--+
|u | | |u | |u | | d u |u | d u | d| |u | d| d| |u | d| |u |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 level 2 level 3
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| d|u |u d| u |u d| | d| d| | d| d| |
+ +--+--+ +--+ + +--+ +--+ +--+--+ +--+--+ +
|u | d| |u | d| |u | | d| | | d | d|
+--+--+--+--+--+--+--+--+ +--+--+--+ +--+--+ +--+
|u d| |u | d | d|u | | d| | d| | | d|
+ +--+ +--+ +--+--+ + +--+ + +--+--+ +--+--+
| |u | d u |u |u | d| | d| | d| d| d | |
+--+--+--+--+--+--+--+--+ + +--+--+--+--+--+--+ +
| d |u |u |u d|u | | | | d| d| d| | d |
+ +--+--+ +--+ +--+ + +--+--+ +--+--+ + +--+
|u | d| d|u | | d u | | d| | | d | | d|
+--+--+--+--+--+--+--+--+ +--+ +--+--+ +--+--+--+
|u | | d|u | | d| u | | d | d| | | |
+--+ + +--+ +--+--+ + + +--+--+ +--+--+ +--+
| d u |u | d u |u | d| | | d| d| | d| d |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 4 level 5
same pentacube, allowing reflections
4x5x7 (minimal) by Patrick Hamlyn
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| u | u | | |u | d|u | d | | d|u d|u | | | d| d| |
+--+ +--+ + + + +--+--+--+--+ +--+--+ + +--+ + +--+--+--+ +
|u | | |u | | d | | d| |u | | |u | | d| | | d|
+--+--+--+--+--+ +--+--+ +--+ + + +--+--+--+ + +--+ + +--+--+
|u | |u | | | d | d u | | | |u |u | d| | | d| d | |
+--+ +--+ +--+ + +--+--+--+--+ +--+--+ +--+--+ + +--+--+--+ +
| u | u | | |u | d| | d|u | | d|u | |u | d| | | d | d |
+ +--+--+--+ + +--+--+ +--+--+ +--+--+--+--+ + +--+ +--+ +--+
| | u |u | | |u | d| d| | | | d u | u | | | d| | d|
+--+--+ + +--+ + +--+--+--+ + + +--+ +--+--+ +--+--+--+--+--+
|u | | |u | | d| | | d| |u | | u | | d| | | d|
+ +--+--+--+ + +--+ + +--+--+ +--+--+--+--+ + + + +--+ +--+
| |u |u | | | d|u d|u | | u | d|u | d| | | d | d |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*2
2
4x9x20 by Helmut Postl
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u |u |u | u|u |u |u | u |u |u |u | u |u |
+ +--+ + +--+--+--+--+ + +--+--+--+--+ +--+ + +--+ +
| u |u | | | | | | | | |u |u |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+ +--+--+--+--+
|u |u |u |u | u |u |u |u | u |u | u | |
+--+--+--+ + +--+--+--+--+--+ +--+ +--+--+--+ +--+--+ +
| |u |u | u | |u | |u |u |u |u |u |u |u |
+ +--+ +--+--+--+--+ +--+--+--+ +--+ +--+ +--+--+ +--+
|u | | u |u |u |u | | |u | u |u |u | |u |
+--+ +--+--+ +--+ +--+ +--+--+--+ +--+--+--+--+ +--+ +
|u | u |u | u |u |u |u | | u | |u |u |u | |
+ +--+--+ +--+--+--+ +--+ +--+--+--+--+--+ + +--+--+--+
| |u | u |u |u |u | u |u |u |u | u | u |
+--+--+--+--+ +--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| u |u | | | | | | | | |u |u |
+ +--+ + +--+ +--+--+--+--+ + +--+--+--+--+ + +--+ +
|u |u | u |u | u |u |u |u | u |u | | u |u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 1
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d| d|u | d | d| d| d| d | d| d| d| d| u | d| d|
+--+--+--+--+--+ + +--+--+--+--+ + +--+--+ + +--+--+--+
|u | d| d|u | | | | | | | |u | d| d|u |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ +
| d|u | d|u | d| d | d| d| d| d | d| d| u | d|u |
+ +--+--+--+--+--+--+ +--+ +--+--+--+--+--+ +--+--+--+--+
| | d| d|u | d| | d| | d| d| d| d|u | d| d| d|
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+
| d | | d| d| d| d| | | d | d| d| d| | d|
+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| d | d| d|u | d| d| d| d | | d| | d|u | d| d| |
+--+--+--+--+ +--+--+--+--+--+ +--+ +--+--+--+--+--+--+ +
| u | d|u | d| d| d| d | d| d| d| d|u | d|u | d|
+ +--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|u | d| d|u | | | | | | | |u | d| d|u |
+--+--+--+ + +--+--+ + +--+--+--+--+ + +--+--+--+--+--+
| d| d|u | d| d | d| d| d| d | d| d| |u | d| d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 2
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u |u | d |u |u |u |u | u |u |u |u | u | d|u |u |
+--+--+--+ + +--+--+ + +--+--+--+--+ + +--+--+--+--+--+
| d|u |u | d| | | | | | | | d|u |u | d|
+ +--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| d|u | d |u |u | u |u |u |u | u |u | d|u | d|u |
+--+--+--+--+ +--+--+--+--+--+ +--+ +--+--+--+--+--+--+ +
|u |u |u | d|u |u |u |u | |u | |u | d|u |u | |
+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|u | |u |u |u | u | | |u |u |u |u | | u |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+
| |u |u | d|u | |u | | u |u |u |u | d|u |u | u |
+ +--+--+--+--+--+--+ +--+ +--+--+--+--+--+ +--+--+--+--+
|u | d|u | d|u |u |u |u | u |u |u |u | d|u | d |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ +
| d|u |u | d| | | | | | | | d|u |u | d|
+--+--+--+--+--+ + +--+--+--+--+ + +--+--+ + +--+--+--+
|u |u | d|u |u |u | u |u |u |u | u |u | d|u |u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 3
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d| d | d| d| d| d | d| d| d| d | d| d| d|
+ +--+ + +--+ +--+--+--+--+ + +--+--+--+--+ + +--+ +
| d| d| | | | | | | | | d| d |
+--+--+--+--+ +--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| | d | d| d | d| d| d| d | d| d| d| d|
+ +--+--+ +--+--+--+ +--+ +--+--+--+--+--+ + +--+--+--+
| d| d| d| d| d| d| d| | d| | d | d| d| |
+--+ +--+--+ +--+ +--+ +--+--+--+ +--+--+--+--+ +--+ +
| d| | d| d| d | d| | | d| d| d| d | | d|
+ +--+ +--+--+--+--+ +--+--+--+ +--+ +--+ +--+--+ +--+
| | d| d| d| | d | | d| d| d| d | d| d | d|
+--+--+--+ + +--+--+--+--+--+ +--+ +--+--+--+ +--+--+ +
| d | d | d| d| d| d | d| d| d| d | d| |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+ +--+--+--+--+
| d| d| | | | | | | | | d| d |
+ +--+ + +--+--+--+--+ + +--+--+--+--+ +--+ + +--+ +
| d| d | d| d| d | d| d| d| d | d| d | d| d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 4
.............................................................................
*2*
*
7x8x20 by Helmut Postl
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | u | u | |u | u |u |u | | u | |u |
+ +--+ +--+--+ +--+ +--+--+ +--+--+--+ +--+ +--+ +--+
|u | |u | | |u |u |u | | | | u | |u |u |
+ +--+--+--+ +--+--+ +--+--+--+ + +--+--+--+--+--+ +--+
| | u |u | | |u | | u |u | | u | |u |
+--+--+ +--+ +--+ +--+--+ +--+ + +--+ +--+ +--+--+--+
|u | | |u | | |u | u | | | | |u | | u |
+--+ +--+--+--+ + +--+--+ + +--+--+ + +--+--+--+ +--+
| u | | u | | | | |u | u | | |u | | |u |
+--+--+--+ +--+ +--+ + +--+ +--+--+ +--+ +--+ +--+--+
|u | | u | | u |u | |u | | | u | u | |
+--+ +--+--+--+--+--+ + +--+--+--+ +--+--+ +--+--+--+ +
| u |u | | u | | | |u | u |u | | |u | | u |
+--+ +--+ +--+ +--+--+--+ +--+--+ +--+ +--+--+ +--+ +
|u | | u | |u |u | u |u | | u | u | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 1
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u | | d| |u | d| | ud| | d| ud|ud |u | d| | ud|
+--+ +--+ +--+--+ +--+ + +--+--+ + +--+--+--+ +--+ +
| d|u | d | d| d| |ud |u | | | d| | d| d| |
+--+ +--+ +--+--+ +--+--+--+--+--+--+--+--+ +--+ +--+--+
| | | d| | d|u | | d |u | d| d|u | u | d| | ud|
+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+ +
|ud | d |u | d| | | d|u | u | d| | |u | d| |
+--+--+--+ +--+--+--+ +--+--+--+--+ +--+--+--+ +--+--+--+
| | d|u | | | d| u |u | d| | | d|u | d | ud|
+ +--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +
|ud | | d|u |u | d| d|u | d | |u | d| | d| | |
+--+--+ +--+ +--+--+--+--+--+--+--+--+ +--+--+ +--+ +--+
| | d| d | | d| | |u | ud| | d| d | d| u | d|
+ +--+ +--+--+--+ + +--+--+ + +--+ +--+--+ +--+ +--+
|ud | | d|u | ud|ud | d| |ud | | d|u | | d| |u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 2
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|ud |u | d |u | | d|u |u | | d| d| |ud |u | | d|
+ +--+--+--+ +--+--+ +--+--+--+ +--+--+ + +--+--+ +--+
| | d| |u | | | u | | d| |ud | u | |u | u |
+--+--+ +--+--+ +--+ + +--+ +--+--+--+ +--+--+--+--+--+
|u | u |u | |ud | |u |ud | ud| | d|u | | | d|
+--+--+ +--+ +--+--+--+ +--+--+--+--+ +--+--+--+ + +--+
| d| | | u |ud |u | | |ud | d| | | | ud|u |
+--+ +--+--+ + +--+--+--+ + +--+--+--+ + +--+--+ +--+
| u |ud | | | | d| ud| | |u | ud|u | | | d|
+--+ + +--+--+--+ +--+--+--+--+ +--+--+--+ +--+ +--+--+
| d| | |u | d| |ud | ud| u | | ud| |u |u |u |
+--+--+--+--+--+ +--+--+--+ +--+ + +--+ +--+--+ +--+--+
| u |u | |u | ud| | d| |u | | |u | | d| |
+--+ +--+--+ + +--+--+ +--+--+--+ +--+--+ +--+--+--+ +
| d| |u | ud| | d| d| |u |u | d| |u | d |u | ud|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 3
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d| ud| |u | d | ud|ud | u | d| |ud |u |
+--+--+ + +--+--+ +--+--+ + +--+--+ +--+--+ + +--+--+
| |u | |ud |u | | d|u | | | d|u | | d| ud| | d| |
+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +
| d | d|ud | d|u | | d|u | d|u | | d|u | ud|u | u |
+ +--+--+ +--+--+--+ +--+--+--+--+ +--+--+--+ +--+--+ +
| |u |u | | d| d| d |u | d| u |u |u | | d| d| |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | d| d| |u |u | u | d|u | d | d| d| |u |u | |
+ +--+--+ +--+--+--+ +--+--+--+--+ +--+--+--+ +--+--+ +
|u |u |ud |u | d| |u | d|u | d| |u | d| ud| d| d|
+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +
| | d| |ud | d| |u | d| | |u | d| |u | ud| |u | |
+--+--+ + +--+--+ +--+--+ + +--+--+ +--+--+ + +--+--+
|u | ud| | d| u | ud|ud | d |u | |ud | d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 4
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u | | d| ud| |u |u | | d| d|u | | d| u |d | ud|
+--+ +--+--+ + +--+--+ +--+--+--+ +--+--+ +--+--+--+ +
| d | d| | d | ud| |u | | d | | | d| |u | |
+--+--+--+--+--+ +--+--+--+ +--+ + +--+ +--+--+ +--+--+
|u | | | d|u | |ud | ud| d| | ud| | d| d | d|
+--+ + +--+--+--+ +--+--+--+--+ +--+--+--+ +--+ +--+--+
| d|ud | | | |u | ud| | | d| ud| d | | |u |
+--+ +--+--+ + +--+--+--+ + +--+--+--+ + +--+--+ +--+
|u | | | d|ud | d| | |ud |u | | | | ud| d |
+--+--+ +--+ +--+--+--+ +--+--+--+--+ +--+--+--+ + +--+
| d| d| d| |ud | | d |ud | ud| |u | d| | |u |
+--+--+ +--+--+ +--+ + +--+ +--+--+--+ +--+--+--+--+--+
| |u | | d| | | d| |u | |ud | d| | d| d |
+ +--+--+--+ +--+--+ +--+--+--+ +--+--+ + +--+--+ +--+
|ud | d| u | d| |u | d| d| |u |u | |ud | d| |u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 5
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|ud | |u | d| ud|ud |u | |ud | |u | d| |u | | d|
+ +--+ +--+--+--+ + +--+--+ + +--+ +--+--+ +--+ +--+
| |u |u | |u | | | d| ud| |u |u | u | d|u |
+--+--+ +--+ +--+--+--+--+--+--+--+--+ +--+--+ +--+ +--+
|ud | |u | d | d|u |u | d| u | | d|u | |u | | |
+ +--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +
| |u | d| | |u | d | d|u | | |u | d| u | ud|
+--+--+--+ +--+--+--+ +--+--+--+--+ +--+--+--+ +--+--+--+
|ud | u | d|u | | |u | d| d |u | | | d|u | |
+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+ +
| | |u | |u | d| | u | d|u |u | d| d|u | | ud|
+--+ +--+ +--+--+ +--+--+--+--+--+--+--+--+ +--+ +--+--+
|u | d |u | u |u | |ud | d| | |u | | u |u | |
+--+ +--+ +--+--+ +--+ + +--+--+ + +--+--+--+ +--+ +
| d| |u | | d|u | | ud| |u | ud|ud | d|u | | ud|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 6
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d| | d | | d| d| d | d| | d | d | |
+--+ +--+ +--+ +--+--+--+ +--+--+ +--+ +--+--+ +--+ +
| d| d| | d | | | | d| d| d| | | d| | d|
+--+ +--+--+--+--+--+ + +--+--+--+ +--+--+ +--+--+--+ +
| d| | d | | d| d | | d| | | d| d | |
+--+--+--+ +--+ +--+ + +--+ +--+--+ +--+ +--+ +--+--+
| d | | d| | | | | d | d| | | d| | | d|
+--+ +--+--+--+ + +--+--+ + +--+--+ + +--+--+--+ +--+
| d| | | d| | | d | d| | | | | d | | d |
+--+--+ +--+ +--+ +--+--+ +--+ + +--+ +--+ +--+--+--+
| | d | d | | | d| | d| d | | d | | d|
+ +--+--+--+ +--+--+ +--+--+--+ + +--+--+--+--+--+ +--+
| d | | d| | | d| d | d| | | | d | | d| d |
+ +--+ +--+--+ +--+ +--+--+ +--+--+--+ +--+ +--+ +--+
| | d | d | | d| d | d| d| | d | | d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 7
.............................................................................
Hexacubes
.............................................................................
*
*****
1x2x6 minimal
+--+--+--+--+--+--+
| | |
+ +--+--+--+--+ +
| | |
+--+--+--+--+--+--+
.............................................................................
**
****
1x2x6 co-minimal
+--+--+--+--+--+--+
| | |
+ +--+--+ +
| | |
+--+--+--+--+--+--+
1x3x4 co-minimal
+--+--+--+--+
| |
+--+--+ +
| | |
+ +--+--+
| |
+--+--+--+--+
also: 1x5x18, 1x6x11, 1x9x10, 3x3x10, 3x5x6
.............................................................................
* *
****
1x3x4 minimal
+--+--+--+--+
| |
+--+ +--+ +
| | | | |
+ +--+ +--+
| |
+--+--+--+--+
also: 2x5x12
.............................................................................
*
**
***
1x3x4 minimal
+--+--+--+--+
| | |
+ +--+ +
| | |
+ +--+ +
| | |
+--+--+--+--+
.............................................................................
*
* *
***
1x3x4 minimal
+--+--+--+--+
| | |
+ +--+ + +
| | | | |
+ + +--+ +
| | |
+--+--+--+--+
also: 3x6x6 by Torsten Sillke
.............................................................................
**
****
1x4x6 minimal
+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ +--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+
also: 1x5x12, 3x4x4
.............................................................................
*
***
**
3x4x5 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| |u | u u | | |ud u |ud d| | d d| d| |
+ +--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ +
| | | | | | | | |
+ +--+ + + +--+ + + +--+ +
| | | | | | | | |
+--+--+--+--+ + +--+--+--+--+ + + +--+--+--+--+
|u u |u | | | d ud|u ud| | | | d| d d |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3
also: 1x9x12 (by David Klarner) and others
also: 3x6x6 by Torsten Sillke
.............................................................................
*
*****
2x7x12 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+
| | | | | | |u | u | | | d | d| |
+ + + + + + +--+--+--+--+--+--+ +--+--+--+ +--+--+--+--+--+--+--+ +
|u |u |u | |u | |u | u | | d| d| d| | d| d | d| |
+ + + + + + +--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+ +
| | | | | | | | | | | | | | | | | |
+ + + + + + + + + + + + + + +--+ +--+--+--+--+ +--+--+--+ +
| | | |u | |u |u | |u | | | | | | | d| | d| d| | d| | |
+ + + + + + + + + + + + + + +--+--+--+ +--+--+--+--+ +--+ +
| | | | | | | | | | | | | | | | | |
+--+--+--+--+--+--+ + + + + + + + +--+--+--+--+--+--+--+--+--+--+--+
| u |u | |u | |u |u |u | | | d| d | d| | d| d| d|
+--+--+--+--+--+--+ + + + + + + + +--+--+--+--+--+--+--+ +--+--+--+
| u |u | | | | | | | | | d| d | |
+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+
level 1 level 2
also: 1x23x24 (by T.W. Marlow and Karl A. Dahlke, independently)
also: 3x5x12 by Andrew Clarke
also: 5x5x12, 5x6x8, 5x6x14, 5x6x18, 5x7x12, 5x8x9
.............................................................................
*
***
* *
3x4x5 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| u | | |u |ud| | | d| d | |
+--+--+--+ +--+ + +--+--+ +--+ +--+--+--+ +--+
| | |u | | | | |ud| | | | d|
+--+ +--+ +--+ + + +--+ + + +--+ +--+ +--+
|u | | | |ud| | | | | d| | |
+--+ +--+--+--+ +--+ +--+--+ + +--+ +--+--+--+
| | u | | |ud|u | | | d| d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3
.............................................................................
* *
****
+--+--+ +--+--+
|u |u | | d| d|
+--+--+--+--+ +--+ + +--+
|u u | | d| | | d|
+--+--+--+--+ +--+ + +--+
|u u | | d| | | d|
+--+--+--+--+ +--+ + +--+
|u |u | | d| d|
+--+--+ +--+--+
level 1 level 2
2x4x6 (confirmed co-minimal by Patrick Hamlyn)
+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+
(2 levels deep)
+--+--+ +--+--+ +--+--+ +--+--+
|u |u | |ud|ud| |ud|ud| | d| d|
+--+ + +--+ +--+--+--+--+ +--+--+--+--+ +--+ + +--+
|u | | |u | | d d| |u u | | d| | | d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
3x4x4 (confirmed co-minimal by Patrick Hamlyn)
+--+--+--+--+
| |
+ +--+--+ +
| | | |
+--+ +--+
| |
+--+--+--+--+
(4 levels deep)
.............................................................................
* **
***
3x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u u | | | u | d|u | d | | | d d| |
+--+ +--+--+--+ + +--+--+--+--+--+--+ + +--+--+--+ +--+
|u | |u | | | d d|u u | | | d| | d|
+--+ +--+--+ +--+ +--+--+--+--+--+--+ +--+ +--+--+ +--+
| | u | |u | |u u | d d| | d| | d | |
+ +--+--+--+ +--+ +--+--+--+--+--+--+ +--+ +--+--+--+ +
| |u u | | | d|u | d|u | | | d d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3
also 6x6x6 by Torsten Sillke
.............................................................................
**
**
**
3x4x5 (minimal) by Helmut Postl
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
|u u |u | | | d ud| d u u | | | d | d d|
+--+--+--+ + +--+--+--+--+--+ + +--+--+--+--+
|u |u u | | |ud| d d u |u | | d| | d| d|
+ +--+--+ +--+ + +--+--+--+ + +--+ +--+--+ +
|u |u | |u | | d| d u u |ud| | | d d| d|
+--+--+--+--+ + +--+--+--+--+--+ + +--+--+--+
|u u | u | | | d d u |ud u | | | d| d d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3
also 3x4x6
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u | u | | | d| | d |
+ +--+--+--+--+ + +--+--+ +--+--+
|u | | | | d| | | |
+--+ + +--+ + + +--+--+ + +
| | |u | | | | | d|
+ +--+--+--+--+ + +--+--+ +--+--+
| |u | u | | | d| | d |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 levels 2, 3
also: 3x6x6 by Torsten Sillke
.............................................................................
*
*
****
+--+ +--+
|u | | d|
+--+ + +
| | | |
+ + + +
| | | |
+ +--+--+--+--+ + +--+--+--+--+
| |u | | | d|
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2 level 3
3x4x7 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+--+
| | | |
+ + +--+--+--+--+ +
| | | | |
+ + +--+--+--+ + +
| | | | |
+ +--+--+--+--+ + +
| | | |
+--+--+--+--+--+--+--+
(3 levels deep)
also: 3x5x20, 3x6x14, 3x6x17, 3x6x20, 3x7x10, 3x8x8, 3x8x9, 4x4x9, 4x5x21,
4x6x6, 4x6x8, 4x6x9, 4x6x11, 4x9x10, 5x7x12, 6x6x7
and 4x4x6 by Helmut Postl, 6x6x6 by Torsten Sillke
.............................................................................
*
***
* *
3x5x6 (minimal) by Andrew Clarke
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| x | |x | | |x | x | |
+--+--+ +--+--+--+ + +--+--+--+--+ +
|x | |x |x | |x | | x | |x |
+ +--+ +--+--+ + +--+ +--+--+ +--+
| |x | |x | | | |x | |x | |
+--+ +--+--+ +--+ + +--+ +--+--+ +
|x | | x | |x | |x | |x |x |
+ +--+--+--+--+ + +--+--+ +--+--+--+
| |x | x | | | x | |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 3 level 2
also: 3x6x6 by Torsten Sillke
.............................................................................
**
*
***
+--+--+ +--+--+ +--+--+ +--+--+
| | | | | | | | | |
+ + + + + + +--+ + +--+ +
| | | |u | | |ud| | | d| |
+ + +--+--+ +--+ +--+--+ +--+ +--+--+ + + +--+--+
|u | | |ud| | | d| | | | |
+--+--+--+ + +--+--+--+ + + +--+--+--+ + +--+--+--+
| u | | | u |ud| | | |ud| d | | | d |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
4x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+
(4 levels deep)
also: 4x7x24, 4x9x20, 4x10x18
.............................................................................
*
***
**
4x6x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u | u | | | | d u |ud| | | |ud| d u | | | | d | d| |
+ +--+--+--+ +--+ + +--+--+--+ +--+ +--+ +--+--+--+ + +--+ +--+--+--+ +
| | |u | | | | d| |u | | | | d| | |
+--+ + +--+--+--+ +--+ + +--+--+ + + +--+--+ + +--+ +--+--+--+ + +--+
|u | | | | | |ud| | | |u | | d| | | |ud| | | | | | d|
+ + +--+--+ + + +--+ +--+--+ +--+ +--+ +--+--+ +--+ + + +--+--+ + +
| | | | |u | |u | | | |ud| |ud| | | | d| | d| | | | |
+--+--+--+ + +--+ + +--+--+ + +--+ +--+ + +--+--+ + +--+ + +--+--+--+
|u | | | | d| | | | | |u | | | | d|
+--+ +--+--+--+ + +--+ +--+--+--+ + + +--+--+--+ +--+ + +--+--+--+ +--+
| |u |u | | | |ud|u d| | | |u d|ud| | | | d| d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
**
**
**
3x6x8 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u |u | | |u d|u d| | | d| d| |
+--+ +--+--+--+--+ + +--+--+--+--+ + +--+--+--+--+ +--+
| | |u | | | |u d| | | | d| | |
+ +--+ +--+--+ + +--+ +--+--+ +--+ + +--+--+ +--+ +
| | | |u | |u | | | d| | d| | | |
+--+ +--+--+ +--+ + +--+ + +--+ + +--+ +--+--+ +--+
|u | | |u | | | d|u | | | d|u | | | d| | | d|
+ +--+ + +--+ + +--+ +--+--+ +--+ + +--+ + +--+ +
| |u | | |u | |u | d| | |u | d| | d| | | d| |
+--+ +--+--+ +--+ + +--+ + +--+ + +--+ +--+--+ +--+
|u | | | | | d| | |u | | | | | d|
+ +--+--+ +--+ + +--+ +--+--+ +--+ + +--+ +--+--+ +
| | u | | | | | d u | | | | | d | |
+--+--+--+--+ +--+ + +--+--+--+--+ + +--+ +--+--+--+--+
| u | u | | | | d u | d u | | | | d | d |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3
also 6x6x6 by Torsten Sillke
.............................................................................
****
**
2x6x8 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| |u | |u | | d| | d| |
+ +--+--+--+--+ +--+--+ +--+--+ +--+--+--+--+ +
| | u | | |u | | | | | d | | d|
+ +--+--+--+--+ + + + + + +--+ +--+--+ +--+
| | u | | | | | | | | d | |
+ +--+--+--+--+ + + + + + + +--+--+--+--+ +
| | u | | | | | | | | d | |
+--+ +--+--+ +--+ + + + + + +--+--+--+--+ +
|u | | u | | | | | d| | | d | |
+ +--+--+--+--+ +--+--+ +--+--+ +--+--+--+--+ +
| |u | |u | | d| | d| |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 level 2
2x8x9
+--+--+--+--+--+--+--+--+--+
|u | |u |u | |
+--+ +--+--+--+--+--+ +--+
| |u |u | |u | levels 1, 2, 3
+--+--+--+--+--+--+--+--+--+
+--+--+--+--+--+--+--+--+--+
| d|u |u |u |u | d| d|u |u |
+ +--+ + +--+ + +--+ +
|u |u | d| d|u |u |u |u | d| level 4
+--+--+--+--+--+--+--+--+--+
+--+--+--+--+--+--+--+--+--+
|u d| d| d| d u |u d| d|
+--+--+--+--+--+--+--+--+--+
| d| d u |u d| d| d| d u | level 5
+--+--+--+--+--+--+--+--+--+
+--+--+--+--+--+--+--+--+--+
| d| | d| d| |
+--+ +--+--+--+--+--+ +--+
| | d| d| | d| levels 6, 7, 8
+--+--+--+--+--+--+--+--+--+
.............................................................................
*
*****
5x5x6 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | |x |x |
+ +--+ +--+ +--+ +--+--+ +--+--+ +
| | | | | | |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 4, 5 level 3
.............................................................................
***
***
4x6x7 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u u | |u | | d d|u | |ud| |u u |ud| | d| d d| | d|
+--+ +--+--+--+ + + +--+--+--+ + + + +--+--+--+ + +--+--+--+--+ +--+
|u | | |u | |ud| |u | |ud| | d| |ud| | d| | | d| |
+--+--+--+--+ +--+ +--+ + + +--+ +--+ + + +--+ +--+ +--+ +
| |u | | | | |ud| | | | | d| | | | | |
+--+ +--+ + + +--+ +--+ + + +--+ +--+ + + +--+--+--+--+ +
| | | | | | | | | | | | | | | | | |
+ +--+--+--+--+ + + +--+ +--+ + + +--+ +--+ + + +--+ +--+
| | | | | |u | | | | |ud| | | | | d| |
+ +--+ +--+ +--+ + + +--+ +--+ + + +--+ +--+ +--+--+--+--+
| |u | | |u | |ud| |u | |ud| | d| |ud| | d| | | d|
+--+ +--+--+--+--+ + +--+--+--+ + + + +--+--+--+ + + +--+--+--+ +--+
|u | |u u |u | |ud| d d| |ud| | d|u u | | d| | d d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*
****
*
3x8x8 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| |x |x |x |x | | | |x |x |x |x |
+ +--+--+--+--+--+ +--+ + + + + +--+--+--+--+
| | | | |x | | | | | | x |
+ +--+--+--+ + + +--+ + + + + +--+--+--+--+
| | | | |x | | | | | | x |
+--+--+--+--+ + + +--+ + + + + +--+--+--+--+
|x | | | | |x | |x | | | | x |
+--+--+ +--+--+ +--+--+ +--+--+--+--+--+--+--+--+
|x | | | | |x | |x | | | |x |
+--+ + + +--+--+--+--+ +--+--+--+--+ + + + +
|x | | | | | |x | | | | |
+--+ + + +--+--+--+ + +--+--+--+--+ + + + +
|x | | | | | |x | | | | |
+--+ +--+--+--+--+--+ + +--+--+--+--+ + + + +
| |x |x |x |x | | | x |x |x |x | |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
levels 1, 3 level 2
.............................................................................
*
*
****
+--+--+--+ +--+--+--+
|x |x |x | |x |x |x |
+--+--+--+--+--+--+ + + + +--+--+--+
| | |x | | | | | x |
+ +--+--+ +--+--+ + + + +--+--+--+
| | | |x | | | | | x |
+ +--+--+ + +--+ +--+--+--+--+--+--+
| | |x | | |x | | x | x |
+ + +--+--+ +--+ +--+--+--+--+--+--+
| | | | | | | | | |
+--+ +--+--+ + + + + + + +
| | | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2, 4 level 3
4x12x12
+--+--+--+--+--+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+ + +
| | | |
+ +--+--+--+--+--+--+ +
| | | |
+ + +--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
also: 6x6x8 (minimal) by Ishino Keiichiro
also: 4x8x18 by Helmut Postl
.............................................................................
*
**
***
6x6x8 by Ishino Keiichiro
also: 4x8x30 by Helmut Postl
.............................................................................
*
**
***
4x12x14 by Ishino Keiichiro
also: 4x8x36 by Helmut Postl
.............................................................................
*
***
**
6x12x35 by Helmut Postl
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u | |u u | |u u | | |u u | |u u | |u u | |u u | |u u | |u u |u |
+ + +--+--+ +--+--+ + +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +
|u | | | | | | |u | | | | | | |u | |u |
+--+--+ +--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+ +--+--+ +--+
| | | | | | | | | | |u | | | | | | | | | | |
+ +--+--+ +--+ +--+ +--+ +--+--+--+ +--+ +--+--+ +--+ +--+--+ +--+ +
| | | | | | | |u | | | | | | | | | |
+--+ +--+ +--+--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+--+ +--+--+ +--+ +--+--+
|u | | | | | | | | | | | | | | | | | |
+ +--+ +--+--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+ + +--+--+--+ +
|u | | |u |u | | | | | | | | | | | | |u |u | |
+--+ +--+--+ +--+--+--+--+ +--+--+ +--+ +--+ +--+ +--+ +--+ +--+--+ +--+--+--+--+ +--+--+ +--+
| |u |u | | | | | | | | | | | | |u |u | | |u |
+ +--+--+--+ + +--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+--+ +--+ +
| | | | | | | | | | | | | | | | | |u |
+--+--+ +--+ +--+--+ +--+--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+--+ +--+ +--+
| | | | | | | | | |u | | | | | | | |
+ +--+ +--+--+ +--+ +--+--+ +--+ +--+--+--+ +--+ +--+ +--+ +--+--+ +
| | | | | | | | | | |u | | | | | | | | | | |
+--+ +--+--+ +--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+ +--+--+
|u | |u | | | | | | |u | | | | | | |u |
+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ + +--+--+ +--+--+ + +
|u |u u | |u u | |u u | |u u | |u u | |u u | | |u u | |u u | |u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 1
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d|u | ud d| | d ud |u u | ud d|u | d ud | ud d|u u | d ud |u | d ud | ud d| d|
+ +--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +
|ud|u | | | |u u | |u u | |u ud |u u | |u u | |u u | | ud u |ud|
+ + + +--+ +--+ +--+--+ +--+--+ +--+--+--+--+--+ +--+--+ +--+--+ +--+--+--+ +
| |u | | | | | | |u ud |u | | |u | | | | | |
+--+--+--+ +--+--+ +--+ +--+ +--+--+ +--+--+--+--+--+--+ +--+ +--+--+ +--+ +--+ +--+ +--+
| | | |u | | | | | | | ud u |u u | | | | | | | | |u |
+ + +--+ +--+ +--+ +--+--+ +--+--+--+--+--+--+ +--+ +--+ +--+--+ +--+--+ +--+--+
|ud| |u |u | | | | | |u u | | | | | | |u | | |
+ +--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+--+--+ +--+--+ +--+ +--+--+ + +--+ +
| d| | | | d|ud| | | | | |u u | | | | | | |ud|ud| |
+--+ +--+ + + + +--+--+--+ +--+--+ +--+--+--+ +--+--+ +--+--+--+ + + + +--+ +--+
| |ud|ud| | | | | | |u u | | | | | |ud| d| | | | d|
+ +--+ + +--+--+ +--+ +--+--+ +--+--+--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+ +
| | |u | | | | | | |u u | | | | | |u |u | |ud|
+--+--+ +--+--+ +--+--+ +--+ +--+ +--+--+--+--+--+--+ +--+--+ +--+ +--+ +--+ + +
|u | | | | | | | | |u u |u ud | | | | | | |u | | | |
+--+ +--+ +--+ +--+ +--+--+ +--+ +--+--+--+--+--+--+ +--+--+ +--+ +--+ +--+--+ +--+--+--+
| | | | | |u | | |u | ud u | | | | | | |u | |
+ +--+--+--+ +--+--+ +--+--+ +--+--+--+--+--+ +--+--+ +--+--+ +--+ +--+ + + +
|ud|u ud | |u u | |u u | |u u | ud u | |u u | |u u | | | |u |ud|
+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+ +
| d| d ud | ud d|u | ud d|u u | d ud | ud d|u | d ud |u u | ud d| | d ud |u | d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 2
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u ud | d| |u u | d| ud d|u | d| ud u | d|u u | d| ud d|u | d| ud u | d| | d|u u |
+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+
| d| d| | |u | ud d|u | d ud |u | d d| ud d|u | ud d|u | ud d| | d d| d|
+--+ + +--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+
|u |ud| | | | |u u | |u u | d d| ud u | |u ud | | | | |u |u |
+ + +--+ +--+--+ + +--+--+ +--+--+--+--+--+--+--+ +--+--+--+ + +--+ +--+ +
|u | | | | d| | | | |u | d d| d ud | | | | | |ud|
+--+--+ +--+ +--+ +--+--+ +--+--+ +--+ +--+--+--+--+--+--+--+ +--+ +--+ +--+--+ +--+--+ +--+ +
| d| |ud| d| | | | | | |u | d ud | | | | | | |u | d| | | |
+--+--+ + + +--+ +--+ +--+--+ +--+--+--+--+ +--+--+ +--+ +--+--+ +--+ +--+
| | | | d| | | | | |u | d ud | | | | | | d| d| |
+ +--+--+--+--+ +--+ +--+--+ +--+ +--+--+--+--+--+--+--+ +--+ +--+--+ +--+ +--+--+--+--+ +
| | d| d| | | | | | ud d|u | | | | | | d| | | |
+--+ +--+ +--+--+ +--+ +--+--+ +--+--+--+--+ +--+--+ +--+ +--+ + + +--+--+
| | | | d|u | | | | | | | ud d|u | | | | | | | d|ud| | d|
+ +--+ +--+--+ +--+--+ +--+ +--+ +--+--+--+--+--+--+--+ +--+ +--+--+ +--+--+ +--+ +--+ +--+--+
|ud| | | | | | ud d| d d|u | | | | | d| | | |u |
+ +--+ +--+ + +--+--+--+ +--+--+--+--+--+--+--+ +--+--+ + +--+--+ +--+ + +
|u |u | | | | | ud u | |u ud | d d|u u | |u u | | | | |ud|u |
+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+ + +--+
| d| d d| | d ud |u | d ud |u | d ud | d d|u | ud d|u | d ud |u | | | d| d|
+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+
|u u | d| | d|u ud | d|u | d ud | d|u u | d|u ud | d|u | d ud | d|u u | | d| ud u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 3
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d d|u | |u | d ud |u | d|u ud |u | d d|u | d ud |u | d|u ud |u | d d| |u | ud d|
+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+
|u |u u | |u ud | d|u ud | d|u ud |u u | d| ud u | d|u ud | d| | |u |u |
+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+ + +--+
| d| d| | | | | ud d| | d ud |u u | d d| | d d| | | | |ud| d|
+ +--+ +--+ + +--+--+--+ +--+--+--+--+--+--+--+ +--+--+ + +--+--+ +--+ + +
|ud| | | | | | ud u |u u | d| | | | |u | | | | d|
+ +--+ +--+--+ +--+--+ +--+ +--+ +--+--+--+--+--+--+--+ +--+ +--+--+ +--+--+ +--+ +--+ +--+--+
| | | |u | d| | | | | | | ud u | d| | | | | | |u |ud| |u |
+--+ +--+ +--+--+ +--+ +--+--+ +--+--+--+--+ +--+--+ +--+ +--+ + + +--+--+
| |u |u | | | | | | ud u | d| | | | | |u | | | |
+ +--+--+--+--+ +--+ +--+--+ +--+ +--+--+--+--+--+--+--+ +--+ +--+--+ +--+ +--+--+--+--+ +
| | | |u | | | | | | d|u ud | | | | | |u |u | |
+--+--+ + + +--+ +--+ +--+--+ +--+--+--+--+ +--+--+ +--+ +--+--+ +--+ +--+
|u | |ud|u | | | | | | | d|u ud | | | | | | | d|u | | | |
+--+--+ +--+ +--+ +--+--+ +--+--+ +--+ +--+--+--+--+--+--+--+ +--+ +--+ +--+--+ +--+--+ +--+ +
| d| | | |u | | | | | d|u u |u ud | | | | | |ud|
+ + +--+ +--+--+ + +--+--+ +--+--+--+--+--+--+--+ +--+--+--+ + +--+ +--+ +
| d|ud| | | | | d d| | d d|u u | ud d| | d ud | | | | | d| d|
+--+ + +--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+
|u |u | | | d| ud u | d|u ud | d|u u | ud u | d| ud u | d| ud u | |u u |u |
+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+
| d ud |u | | d d|u | ud u | d|u | ud d|u | d d|u | ud u | d|u | ud d|u | |u | d d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 4
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|u |u ud | ud u | d| ud u | d d|u ud | ud u | d|u ud | d d| ud u | |u ud | d|u |
+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+ +
|ud| d ud | | d d| | d d| | d d| ud d| | d d| | d d| | | | d|ud|
+ +--+--+--+ +--+--+ +--+--+ +--+--+--+--+--+ +--+--+ +--+--+ +--+ +--+ + + +
| | | | | | d| | | d| ud d| | | | | | | d| |
+--+ +--+ +--+ +--+ +--+--+ +--+ +--+--+--+--+--+--+ +--+--+ +--+ +--+ +--+--+ +--+--+--+
| d| | | | | | | | | d d| d ud | | | | | | | d| | | |
+--+--+ +--+--+ +--+--+ +--+ +--+ +--+--+--+--+--+--+ +--+--+ +--+ +--+ +--+ + +
| | | d| | | | | | | d d| | | | | | d| d| |ud|
+ +--+ + +--+--+ +--+ +--+--+ +--+--+--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+ +
| |ud|ud| | | | | | | d d| | | | | |ud|u | | | |u |
+--+ +--+ + + + +--+--+--+ +--+--+ +--+--+--+ +--+--+ +--+--+--+ + + + +--+ +--+
|u | | | |u |ud| | | | | | d d| | | | | | |ud|ud| |
+ +--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+--+--+ +--+--+ +--+ +--+--+ + +--+ +
|ud| | d| d| | | | | | d d| | | | | | | d| | |
+ + +--+ +--+ +--+ +--+--+ +--+--+--+--+--+--+ +--+ +--+ +--+--+ +--+--+ +--+--+
| | | | d| | | | | | | ud d| d d| | | | | | | | | d|
+--+--+--+ +--+--+ +--+ +--+ +--+--+ +--+--+--+--+--+--+ +--+ +--+--+ +--+ +--+ +--+ +--+
| | d| | | | | | | d ud | d| | | d| | | | | |
+ + + +--+ +--+ +--+--+ +--+--+ +--+--+--+--+--+ +--+--+ +--+--+ +--+--+--+ +
|ud| d| | | | d d| | d d| | d ud | d d| | d d| | d d| | ud d|ud|
+ +--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+ +
|u | d| ud u | |u ud | d d| ud u | d|u ud | ud u | d d|u ud | d|u ud | ud u |u |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 5
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| d| d d| | d d| | d d| | d d| | d d| | d d| | | d d| | d d| | d|
+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ + +--+--+ +--+--+ + +
| d| | d| | | | | | | d| | | | | | | d|
+--+ +--+--+ +--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+ +--+--+
| | | | | | | | | | | d| | | | | | | | | | |
+ +--+ +--+--+ +--+ +--+--+ +--+ +--+--+--+ +--+ +--+ +--+ +--+--+ +
| | | | | | | | | | d| | | | | | | |
+--+--+ +--+ +--+--+ +--+--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+--+ +--+ +--+
| | | | | | | | | | | | | | | | | | d|
+ +--+--+--+ + +--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+--+ +--+ +
| | d| d| | | | | | | | | | | | | d| d| | | d|
+--+ +--+--+ +--+--+--+--+ +--+--+ +--+ +--+ +--+ +--+ +--+ +--+--+ +--+--+--+--+ +--+--+ +--+
| d| | | d| d| | | | | | | | | | | | | d| d| |
+ +--+ +--+--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+ + +--+--+--+ +
| d| | | | | | | | | | | | | | | | | |
+--+ +--+ +--+--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+--+ +--+--+ +--+ +--+--+
| | | | | | | | d| | | | | | | | | |
+ +--+--+ +--+ +--+ +--+ +--+--+--+ +--+ +--+--+ +--+ +--+--+ +--+ +
| | | | | | | | | | | d| | | | | | | | | | |
+--+--+ +--+ +--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+ +--+--+ +--+
| d| | | | | | | d| | | | | | | d| | d|
+ + +--+--+ +--+--+ + +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +
| d| | d d| | d d| | | d d| | d d| | d d| | d d| | d d| | d d| d|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
level 6
.............................................................................
*
***
**
6x8x60 by Helmut Postl
.............................................................................
*
***
**
6x8x60 by Helmut Postl
.............................................................................
*
****
*
6x10x56 by Helmut Postl
.............................................................................
2*
***
2x2x3 minimal
+--+--+--+ +--+--+--+
|u |u | | d| d|
+ +--+ +--+ +
| | | |
+--+--+--+ +--+--+--+
level 1 level 2
.............................................................................
**
**2
2x2x3 minimal
+--+--+--+ +--+--+--+
| |u | | d|
+ +--+ + +--+
| u | | | d|
+--+--+--+ +--+--+--+
level 1 level 2
also: 2x5x15, 2x7x9
allowing reflections: also 2x3x9, 3x3x8, 3x3x12, 3x3x14, 3x3x18, 3x4x7
.............................................................................
2*
*2
2x2x3 minimal
+--+--+--+ +--+--+--+
|u |u | | d| d|
+--+ +--+ +--+ +--+
|u | u | | d | d|
+--+--+--+ +--+--+--+
level 1 level 2
.............................................................................
*2*
* *
2x2x3 minimal
+--+--+--+ +--+--+--+
| u | | | d| |
+ +--+ + + +--+ +
| |u | | | d |
+--+--+--+ +--+--+--+
level 1 level 2
also: 3x5x8, 3x5x12
.............................................................................
22
**
2x2x3 minimal
+--+--+--+ +--+--+--+
|u |u | | d| d|
+ + + + + +
|u |u | | d| d|
+--+--+--+ +--+--+--+
level 1 level 2
also: 2x5x9
.............................................................................
2*
*2
2x2x3 minimal
+--+--+--+ +--+--+--+
|u | u | | d | d|
+ +--+ + + +--+ +
|u |u | | d| d|
+--+--+--+ +--+--+--+
level 1 level 2
.............................................................................
2
*3
2x2x3 minimal
+--+--+--+ +--+--+--+
| | | |
+ +--+--+ +--+--+ +
|u | u | | d | d|
+--+--+--+ +--+--+--+
level 1 level 2
.............................................................................
**
2**
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u | u | | d | d|
+--+--+ + + +--+--+
| | | | | |
+ +--+--+ +--+--+ +
|u |u | | d| d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
same hexacube, allowing reflections
2x2x3 minimal
+--+--+--+ +--+--+--+
| |u | | d|
+ +--+ +--+ +
|u | | d| |
+--+--+--+ +--+--+--+
level 1 level 2
.............................................................................
* *
**2
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u | u | | d | d|
+--+ +--+ + + +--+ +--+
| | | | | | | | | |
+ +--+ +--+ +--+ +--+ +
|u |u | | d| d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
.............................................................................
2
*2*
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u |u | | d| d|
+--+ +--+ + + +--+ +--+
| | | | | | | | | |
+ +--+ +--+ +--+ +--+ +
|u | u | | d | d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
.............................................................................
2
**2
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u |u | | d| d|
+--+--+ + + + + +--+--+
| | | | | | | | | |
+ + +--+--+ +--+--+ + +
|u | u | | d | d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
also: 2x5x12
.............................................................................
**
**2
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u |u | | d| d|
+--+--+ +--+ +--+ +--+--+
| | | | | |
+--+ +--+--+ +--+--+ +--+
|u | u | | d | d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
.............................................................................
2
**
**
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u | |u | | d| | d|
+ +--+ +--+ +--+ +--+ +
| | | | | |
+--+ +--+ + + +--+ +--+
|u | |u | | d| | d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
.............................................................................
*2
***
+--+ +--+
|u | | d|
+ +--+ + +--+
|u | | d| |
+--+--+--+ + + +--+
| u u | | | d d|
+--+--+--+ +--+--+--+
level 1 level 2
2x3x4 minimal
+--+--+--+--+
| | |
+ +--+ +
| | |
+ +--+ +
| | |
+--+--+--+--+
(2 levels deep)
.............................................................................
**
*2*
+--+ +--+
|u | | d|
+ +--+ + +--+
|u | | d| |
+ +--+--+ +--+ +--+
| |u u | | d d|
+--+--+--+ +--+--+--+
level 1 level 2
2x3x4 minimal
+--+--+--+--+
| | |
+ +--+ +
| | |
+ +--+ +
| | |
+--+--+--+--+
(2 levels deep)
.............................................................................
*
*2**
2x2x6 minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u | u | | d | d| |
+ +--+--+--+--+ + + +--+--+--+--+ +
| u |u | | | | d| d |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
2*
***
2x2x6 minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+
|u | |u | | d| | d|
+--+ +--+--+ +--+ +--+ +--+--+ +--+
|u | | u | | d | | d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
* 2
***
+--+ +--+ +--+ +--+
| | |u | |ud| | d|
+ +--+ + +--+ +--+--+ + +--+
| u | | |ud| | d| | |
+--+--+ +--+--+ +--+--+ +--+--+
level 1 level 2 level 3 level 4
2x3x4 minimal
+--+--+--+
| | |
+ +--+ +
| | |
+--+--+--+
(4 levels deep)
also: 4x5x9
.............................................................................
2
2**
2x3x4 minimal
+--+--+--+--+ +--+--+--+--+
|u |u | | d| | | d|
+ +--+--+ + + + + + +
|u | u | | d| | | d|
+--+--+--+--+ +--+ + +--+
|u u |u u | | d d| d d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
also: 2x6x6 by Torsten Sillke
.............................................................................
2
****
+--+--+ +--+--+
|u |u | | d| d|
+--+--+ +--+ +--+ +--+--+
| | | |
+--+--+--+--+ +--+--+--+--+
level 1 level 2
2x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ +--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+
(2 levels deep)
also: 2x5x12
.............................................................................
2
2**
+--+--+ +--+--+
|u |u | | d| d|
+--+ +--+--+ +--+--+ +--+
|u |u | | d| d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
2x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ +--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+
(2 levels deep)
also: 2x5x12
.............................................................................
**
22
+--+--+ +--+--+
| | | |
+--+ +--+--+ +--+--+ +--+
|u u |u u | | d d| d d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
2x4x6
+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ +--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+
(2 levels deep)
also: 2x5x12
.............................................................................
2
****
+--+ +--+ +--+ +--+
|u | |u | | d| | d|
+--+--+--+ + + +--+--+--+
| | | |
+--+--+--+--+ +--+--+--+--+
level 1 level 2
+--+--+ +--+--+
| u | | | d|
+--+--+--+--+ +--+ + +--+
| u | | | | | d|
+--+--+--+--+ + + + + +
|u | | d| | | |
+--+--+--+--+ +--+ + +--+
|u | | d| |
+--+--+ +--+--+
level 1 level 2
2x4x6
+--+--+--+--+
| |
+ +--+--+ +
| | | |
+--+ +--+
| |
+ +
| |
+--+ +--+
| | | |
+ +--+--+ +
| |
+--+--+--+--+
(2 levels deep)
also: 6x6x6 by Torsten Sillke
same hexacube, allowing reflections
2x3x6
+--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u | | d| d| d|
+--+--+--+--+ +--+ + +--+--+--+--+ +
| | | | | | | |
+ +--+--+--+--+ + +--+ +--+--+--+--+
|u |u |u | | d| d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
***
3
3x4x4 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u |u | | d| d| | | |
+--+--+--+ + + +--+--+--+ + + +--+--+
| |u |u | | | |ud|ud| | | | d| d |
+ +--+--+ + + +--+--+ + + +--+--+ +
| |u |u | | | |ud|ud| | | d| d| |
+ +--+--+--+ +--+--+--+ + +--+--+ + +
|u | u | | d | d| | | |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3
also: 6x6x6 (allowing reflections) by Torsten Sillke
.............................................................................
_________
/ /\
/________/ \
\ \ /
_______\________\/
/ / /
/___________/ /
\ \ /
\___________\/
2x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| u |u | | | | | | d| d |
+--+--+--+ + + + + + + +--+--+--+
|u | | |u | | d| | | d|
+--+ +--+--+ +--+ +--+ +--+--+ +--+
|u | | | u | | d | | | d|
+ + + +--+--+--+ +--+--+--+ + + +
| | |u |u | | d| d| | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
2**
**
2x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| u | u | | | | d | d |
+--+--+--+--+ + + + + +--+--+--+--+
| u | | |u | | | | d | d|
+--+--+ + +--+--+ + +--+--+--+--+ +
|u | | |u | | d | d| | |
+ + +--+--+--+--+ +--+--+--+--+ + +
| |u |u | | d| d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
***2
*
allowing reflections
2x6x6 (minimal) by Torsten Sillke
+--+--+--+--+--+--+ +--+--+--+--+--+--+
|u | u |u | | d | d| d|
+--+ +--+--+--+ + +--+--+--+ +--+--+
|u | |u | | | | d| | d| | |
+--+--+ + +--+ + + + +--+--+--+ +
| | | |u | | | | d| |
+ +--+ + +--+ + + +--+--+--+--+ +
| |u | | | | | | d | |
+ +--+ + +--+--+ + +--+--+--+ + +
| | |u | |u | | | | d| | d|
+ +--+--+--+ +--+ +--+--+ +--+--+--+
|u |u |u | | d| d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
2**
*
*
4x4x6 (minimal) by Helmut Postl
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u |u | | d| d| |u |u | | d| d|
+ +--+--+--+ +--+--+--+ + + +--+--+--+ +--+--+--+ +
| |u | | |ud|u | | | | d|ud| | | d| |
+ + +--+--+ + +--+--+ + + +--+--+ + +--+--+ + +
| | |u | | |u |ud| | | |ud| d| | | d| | |
+--+--+ +--+ + +--+--+--+ +--+--+--+ + +--+ +--+--+
|u | | | | | d | | | | u | | | | | d|
+--+ + + + +--+--+--+ + + +--+--+--+ + + + +--+
|u | | | | |ud| u | | |ud| | | | | d|
+--+ +--+ + +--+--+--+--+ +--+--+--+--+ + +--+ +--+
| u | u | |u |ud|u |ud| |ud| d|ud| d| | d | d |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
3*
**
4x4x6 by Helmut Postl
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u |u | |ud| d|u | | d|ud |ud| | d| d|
+--+ +--+ + +--+--+--+--+ +--+--+--+--+ + +--+ +--+
|u | | | | d u |u | | | | | d u | | | | d|
+--+--+--+ + +--+--+ + + + + +--+--+ + +--+--+--+
| |u | | | d| | | | | |u | | | d| |
+ +--+ +--+ +--+--+ + + + + +--+--+ +--+ +--+ +
| |u | | | |u d| |u | | d| |u d| | | | d| |
+ +--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+ +
| u | | | |u | d |u | | d| u | d| | | | d |
+--+--+--+ + + +--+--+--+ +--+--+--+ + + +--+--+--+
|u |u | | d|u |u d| |u d| d|u | | d| d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
**2*
*
+--+ +--+
| | | |
+--+--+ +--+--+ +--+--+ +--+--+
|u |u | | d| d|
+--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
| u | | | | d| |u | | | | d |
+--+ +--+--+ + + +--+--+ +--+--+ + + +--+--+ +--+
|u | | u | |ud| | |ud| |ud| | |ud| | d | | d|
+--+--+--+--+ +--+ + +--+ +--+ + +--+ +--+--+--+--+
|u |u | | d| d| |u |u | | d| d|
+--+--+--+--+ +--+ + +--+ +--+ + +--+ +--+--+--+--+
|u | |u | |ud| | |ud| |ud| | |ud| | d| | d|
+--+--+ +--+ +--+--+ + + + + +--+--+ +--+ +--+--+
|u | | d| | | | | |u | | d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
4x5x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+
(4 levels deep)
.............................................................................
*2**
*
4x6x6 (minimal) by Helmut Postl
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u | |u |u | |u | |ud |ud|ud| ud| |ud| ud|ud |ud| | d| | d| d| | d|
+--+ +--+--+ +--+ + +--+--+--+--+ + +--+--+ + +--+--+ +--+ +--+--+ +--+
| u |u | | | d| | | d| | | |u | | |u | | | d | d |
+--+--+--+--+ +--+ +--+--+ + +--+--+ + +--+--+--+--+ + +--+--+--+--+ +--+
|u | |u |u | |u | |ud| ud|ud |ud| |ud |ud|ud| ud| | d| | d| d| | d|
+--+ +--+--+ +--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+ +--+--+ +--+
|u | |u |u | |u | |ud| ud|ud |ud| |ud |ud|ud| ud| | d| | d| d| | d|
+--+ +--+--+--+--+ +--+--+ + +--+--+ + +--+--+--+--+ + +--+ +--+--+--+--+
| u | u | | | d| | | d| | | |u | | |u | | | d| d |
+--+ +--+--+ +--+ + +--+--+--+--+ + +--+--+ + +--+--+ +--+ +--+--+ +--+
|u | |u |u | |u | |ud |ud|ud| ud| |ud| ud|ud |ud| | d| | d| d| | d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
____
/ /\
____/___/ \
/ /\ \ \
/___/ \___\ \
\ \ / / /
\ \/___/ /
\ \ \
\____ \ \
\ \ /
\___\/
2x7x12 (minimal) by Helmut Postl
+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+
| |u |u | u | u | u | | | d| d| | d | d | d |
+ +--+--+ +--+--+--+--+ +--+--+ + +--+ +--+--+ +--+ +--+--+--+ +--+
|u | |u | u | | u | | d| | d| | d | | d|
+--+ +--+--+--+--+--+--+--+--+ +--+ + +--+--+--+--+--+--+ +--+--+--+ +
| | | |u |u | u | | | | | | | d| d| | d | | |
+ +--+ +--+--+--+--+--+--+ +--+ + +--+ +--+ +--+ +--+--+ +--+ +--+
|u |u | | | |u |u | | d| d| | | | d| d|
+--+ +--+ +--+--+ +--+ +--+ +--+ + +--+ +--+--+--+--+--+--+ +--+ +
| | | u | |u |u | | | | | | | d | d| d| | | |
+ +--+--+--+ +--+--+--+--+--+--+ + +--+ +--+--+--+--+--+--+--+--+ +--+
|u | | u | |u | |u | | d | | d | d| | d|
+--+ +--+--+--+ +--+ +--+--+ +--+ + +--+--+ +--+--+--+--+ +--+--+ +
| u | u | u | |u |u | | | d | d | d | d| d| |
+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
**
*2*
4x4x12 (minimal) by Patrick Hamlyn
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u | u | |ud | d|u | | d|u | ud| | d | d|
+--+ +--+--+ +--+--+--+ + + +--+--+--+ +--+--+ +--+
| | |u | | | | | d| |u | | | | | d| | |
+--+--+ +--+ + + + +--+ +--+ + + + +--+ +--+--+
|u |u | | | d| d| | | |u |u | | | d| d|
+ +--+--+ + +--+ + +--+ +--+ + +--+ + +--+--+ +
| |u |u | | | d| d| |u |u | | | d| d| |
+ +--+ + + + +--+--+--+ +--+--+--+ + + + +--+ +
| | | | | | | | | | | | | | | | | | | |
+--+ +--+--+ +--+ + + + + + + +--+ +--+--+ +--+
| |u |u | | | d| d| |u |u | | | d| d| |
+--+ + +--+ + +--+--+ + + +--+--+ + +--+ + +--+
|u |u | | | d| d| | | |u |u | | | d| d|
+--+--+ +--+ + + + +--+ +--+ +--+ + +--+ +--+--+
| | | | | | | | | | | | | | | | | | | |
+ + +--+ + +--+--+--+ + + +--+--+--+ + +--+ + +
|u |u | | | d| d| | | |u |u | | | d| d|
+ +--+--+ + +--+ + +--+ +--+ + +--+ + +--+--+ +
| |u |u | | | d| d| |u |u | | | d| d| |
+--+ +--+--+ +--+ + +--+ + + + +--+ +--+--+ +--+
|u | | | | d| | | | | | | |u | | | | d|
+--+--+ +--+ + +--+--+--+ +--+--+--+ + +--+ +--+--+
| u |u | |u | d| ud| |ud |u | d| | d| d |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
same hexacube, allowing reflections
4x4x6
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u | u | |ud | d|u | | d|u | ud| | d | d|
+--+ +--+--+ +--+--+--+ + + +--+--+--+ +--+--+ +--+
| | |u | | | | | d| |u | | | | | d| | |
+--+--+ +--+ + + + +--+ +--+ + + + +--+ +--+--+
|u |u | | | d| d| | | |u |u | | | d| d|
+--+ + +--+ + +--+--+ + + +--+--+ + +--+ + +--+
| |u |u | | | d| d| |u |u | | | d| d| |
+--+ +--+--+ +--+ + + + + + + +--+ +--+--+ +--+
|u | | | | d| | | | | | | |u | | | | d|
+--+--+ +--+ + +--+--+--+ +--+--+--+ + +--+ +--+--+
| u |u | |u | d| ud| |ud |u | d| | d| d |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
3
***
6x7x8 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u |u | | |u | | | d| |ud | ud| | u | d| |u | d|
+--+--+--+--+ + +--+ + + + +--+ + +--+ +--+--+--+--+ + +--+
|u |u | u | | d| | |ud| | |ud| |u | d|u | | d|
+--+--+--+--+ + +--+ + + + +--+ + +--+ +--+--+--+--+ + +--+
| | u | | |u | | | | d|u | | |ud| | |ud| | | d|
+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ + + +--+--+ +--+--+--+
| u |u | | | |u |ud|ud| | | |ud| d| d| |u |
+ +--+--+--+ +--+ + +--+--+--+ +--+--+--+ +--+--+ +--+--+--+--+
| | u |u |u | | | d|ud|ud|u | |u | | d| d|ud|
+--+--+--+--+--+--+ + + +--+--+ +--+--+--+ +--+--+--+--+--+--+ +
|u | | | u | | |ud| | | | d| | | | d| | |u | |
+--+ + +--+--+--+--+ +--+ + +--+ + + + +--+ + +--+--+--+--+
|u | u | u | |ud| | |ud| | | d| | d| |u | d| u |
+--+ + +--+--+--+--+ +--+ + +--+ + + + +--+ + +--+--+--+--+
|u | | |u | u | |ud | ud| | d| | | d|u | | d| u |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2 level 3
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| | d| |u | d | | |u | |ud |u | | | d | d| d |
+ + + +--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+
| d| | u | d | |u | | |ud| | |u | | d | d| d|
+ + + +--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+
| | | | d| | | | |u | | | | | | | d | |
+--+--+--+--+--+--+ + +--+--+--+ +--+--+--+ + +--+--+--+--+ +--+
| d| u |u |u |ud| |u | d|ud|ud| d| | d| | d| d| |
+--+--+--+--+--+--+--+ +--+--+--+ +--+--+--+ + +--+ +--+ +--+ +
|ud|u |u | | d| | d|ud|ud| | u | | | d| d| | d|
+ +--+--+ +--+--+--+ +--+--+--+--+--+--+--+ +--+ +--+--+--+--+ +
| |ud| | | | | | | | d|u | | | | | d | |
+--+--+--+--+ + + + + + + +--+ + + + +--+--+--+--+--+--+--+
| d|u | | d| |u | | |ud| | |u | | d | d| d|
+--+--+--+--+ + + + + + + +--+ + + + +--+--+--+--+--+--+--+
| d |u | | d| | | |u | ud| |u | | | d | d| d |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 4 level 5 level 6
also: 4x8x12 by Helmut Postl
.............................................................................
*2*
*
*
6x8x8 by Helmut Postl
same hexacube, allowing reflections
4x4x6
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u | | |ud| u | | d |ud| | | d|
+--+--+--+ + + +--+--+--+ +--+--+--+ + + +--+--+--+
| u |u | | |u |ud| d| |u |ud| d| | | d| d |
+--+ +--+ + +--+--+--+--+ +--+--+--+--+ + +--+ +--+
|u | | | | |ud| u | | d |ud| | | | | d|
+--+--+ +--+ + +--+--+--+ +--+--+--+ + +--+ +--+--+
| |u | | |ud|u | | | | d|ud| | | d| |
+ +--+--+--+ +--+--+--+ + + +--+--+--+ +--+--+--+ +
| | |u | |u |ud| |ud| d| | d| | |
+ +--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+ +
|u |u | |ud|ud|u |u | | d| d|ud|ud| | d| d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
2*
2*
4x10x42 by Helmut Postl
.............................................................................
*2*
*
*
10x10x36 by Helmut Postl
.............................................................................
2
***
*
10x10x48 by Helmut Postl
.............................................................................
2*
*
**
10x10x72 by Helmut Postl
.............................................................................
*2
*
**
10x10x72 by Helmut Postl
.............................................................................
**2
*
*
10x10x72 by Helmut Postl
same hexacube, allowing reflections
4x6x6
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| | |u | | |ud|u | | | | d|ud| | | | d| |
+ +--+ +--+--+ + +--+--+ +--+--+ + +--+--+ +--+--+ + +--+ + +--+--+ +
| |u | u | | |u |ud| |u |ud| | |ud| d| |ud| d| | | d| | | d |
+ + +--+--+--+--+ +--+--+ +--+--+ + +--+--+ +--+--+ + + + +--+--+--+--+
|u | |u |u | | |ud|u | d| d | | d|ud|u |u | | | d| d| d| |
+--+ +--+--+ +--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ + +--+--+--+--+ +
| |u |u | |u | | d| d|u |ud| | u |u |ud| d| | | d| d| d| |
+--+--+--+--+ + + + +--+--+ +--+--+ + +--+--+ +--+--+ +--+--+--+--+ + +
| |u |u | | | |ud|u | |ud|u | | | d|ud| | d|ud| | d| | | d|
+ +--+--+ +--+ + + +--+--+ +--+--+ + +--+--+ +--+--+ + +--+--+ + +--+
| u | | | | |u |ud| | | |ud| d| | | | d | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
2**
*
*
10x10x72 by Helmut Postl
.............................................................................
____
/ /\
/ / \
/ / /
____/___/ /
/ \ \ \
/ ___\ \ \
/ /\ \ /
/___/ \_______\/
\ \ /
\___\/
12x12x72 by Helmut Postl
.............................................................................
Heptacubes
.............................................................................
*
******
1x2x7 minimal
+--+--+--+--+--+--+--+
| | |
+ +--+--+--+--+--+ +
| | |
+--+--+--+--+--+--+--+
also: 1x9x21, 3x7x7
.............................................................................
**
*****
1x2x7 minimal
+--+--+--+--+--+--+--+
| | |
+ +--+--+--+ +
| | |
+--+--+--+--+--+--+--+
also: 3x5x7, others
.............................................................................
***
****
1x2x7 minimal
+--+--+--+--+--+--+--+
| | |
+ +--+ +
| | |
+--+--+--+--+--+--+--+
also: 3x5x7, others
.............................................................................
***
***
*
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u u u | |u u | | d d ud| |ud ud| | | | d| d d|
+--+--+--+ +--+--+ +--+--+--+ +--+--+ + + +--+--+--+--+
|u | | |ud| | | d| |
+ + + + + + + + +
|u | | |ud| | | d| |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3
3x3x7 minimal
+--+--+--+--+--+--+--+
| | |
+ +--+--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(3 levels deep)
also: 1x14x14 (by David Klarner), many others
.............................................................................
**
*****
+--+--+ +--+--+
|x | | |x |
+--+ +--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+ +--+ +--+--+--+--+
|x | | |x | |
+--+ +--+ +--+--+ +
|x | |x | |x |x |
+--+ +--+ +--+--+--+
|x | | |x |
+--+--+ +--+--+
levels 1, 4, 5 levels 2, 3
+--+--+--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+ +--+--+ +--+--+--+--+--+
| | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 4, 5 levels 2, 3
2x7x8 minimal
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| |u u | u u | | d d | d d| |
+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +
| |u u | u u | | d d | d d| |
+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +
| | | | | | | |
+ +--+--+ +--+ + + +--+ +--+--+ +
| | | | | | | | | |
+ +--+ +--+--+ + + +--+--+ +--+ +
| | | | | | | |
+--+--+--+--+--+--+--+ + + +--+--+--+--+--+--+--+
| u u |u u | | | | d d| d d |
+--+--+--+--+--+--+--+ + + +--+--+--+--+--+--+--+
| u u |u u | | | | d d| d d |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 level 2
5x5x7
+--+--+--+--+--+--+--+
| | | |
+ +--+--+ +--+ +
| | | | |
+ + +--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+
(5 levels deep)
3x5x14
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+--+ +
| | | | |
+ +--+--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
also: 1x19x28 (by T.W. Marlow and Karl A. Dahlke, independently), many others
.............................................................................
* *
*****
+--+--+--+ +--+--+--+
|u |u |u | | d| d| d|
+--+--+--+--+--+ +--+ + + +--+
|u u | | d| | | | d|
+--+--+--+--+--+ +--+ + + +--+
|u u | | d| | | | d|
+--+--+--+--+--+ +--+ + + +--+
|u u | | d| | | | d|
+--+--+--+--+--+ +--+ + + +--+
|u |u |u | | d| d| d|
+--+--+--+ +--+--+--+
level 1 level 2
2x5x7 (minimal) by Andrew Clarke
+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+
(2 levels deep)
.............................................................................
* **
****
+--+--+ +--+--+
|x |x | |x |x |
+--+--+ + + +--+--+--+--+
| x | | | | |x | |
+--+--+--+--+ + +--+ +
| | | | | | |
+ + + + + +--+--+--+--+
|x |x |x |x | |x |x |x |x |
+--+--+--+--+ +--+--+--+--+
levels 1, 2, 4 level 3
4x4x7 minimal
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(4 levels deep)
also: 4x6x14
.............................................................................
*
**
****
4x4x7 minimal
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
| u u | | |u |ud d| |u ud| d| | | d d |
+--+--+--+--+ + +--+--+--+ +--+--+--+ + +--+--+--+--+
| u u | | |u |ud d| |u ud| d| | | d d |
+--+--+--+--+ + +--+--+--+ +--+--+--+ + +--+--+--+--+
| | | | | | | | | |
+ +--+--+ + +--+ + + +--+ + +--+--+ +
| |u |u | | | d| d| |u |u | | | d| d| |
+--+ + + + +--+--+ + + + + +--+--+ + + + +--+
|u | |u |u | |ud|u |ud|ud| |ud|ud| d| d| | d| d| | |
+--+--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+--+
|u u | |u | |ud d |ud| |ud| u ud| | d| | d d|
+--+--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+--+
|u u | |u | |ud d |ud| |ud| u ud| | d| | d d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
* *
*****
+--+--+--+--+--+ +--+--+--+--+--+
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ + + + + + +
| | | | | | | |
+ +--+--+ +--+ +--+--+--+--+--+
| |x |x | | | x |x |
+--+--+--+--+ +--+--+--+--+
levels 1, 3, 4 levels 2, 5
+--+ +--+
|x | |x |
+--+--+--+--+--+ +--+--+--+--+ +
|x | |x |x | | |x | x |x | |
+--+ +--+--+ + + +--+--+ +--+
| | | | | | | |
+--+--+--+--+--+ +--+ + +--+ +
|x |x | |x | |x |x | |x |
+--+--+ +--+ +--+--+ +--+
levels 1, 3, 4 levels 2, 5
5x5x7 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+--+
| | | |
+ +--+ + +
| | | |
+ + + +
| | | |
+ +--+ + +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+
(5 levels deep)
also: 5x6x7, 5x8x14, 5x9x14
.............................................................................
* *
*****
+--+--+ +--+--+
|x |x | |x |x |
+--+--+--+--+--+--+ +--+--+ + +--+--+
| |x | | | | | | x |
+--+--+ +--+ +--+ + + +--+--+--+--+
|x |x | |x | |x | |x |x | x | x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 4 levels 3, 5
+--+--+--+--+ +--+--+--+--+
| |x | |x | | x | x |
+ +--+ +--+--+ +--+--+--+--+--+
| | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2, 4 levels 3, 5
3x5x14 co-minimal
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+--+ +--+ + +
| | | | | | |
+ + +--+ +--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
5x6x7 co-minimal
+--+--+--+--+--+--+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ +--+--+ +
| | | |
+--+--+ +--+--+
| |
+ +
| |
+--+--+--+--+--+--+
(5 levels deep)
.............................................................................
* *
*****
+--+--+--+--+ +--+--+--+--+
|x | |x | | |x |x |
+--+ +--+ +--+ +--+--+--+--+--+
| | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 3, 5 levels 2, 4
3x5x14 co-minimal
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +--+ +
| | | | | | |
+ +--+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
5x6x7 co-minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u | | |ud| |u | |u | | |ud| d d | |ud| |u | |u | | | d| d d |
+--+--+ +--+ +--+ + + +--+ +--+ + +--+--+--+--+--+--+ + + +--+ +--+ + +--+--+--+--+--+--+
| |u | |u | |u | | |ud| |ud| |ud| | |ud| |ud| |ud| | |ud| |ud| |ud| | | d| | d| | d|
+ +--+--+--+--+--+ +--+--+ +--+ +--+ + +--+ +--+ +--+ +--+--+ +--+ +--+ + +--+ +--+ +--+
|u | | | d| | |u | | | d| | | | | |
+ +--+ +--+ +--+ +--+--+--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+--+--+ + +--+ +--+ +--+
| |u | |u | |u | | ud| ud| ud| | |ud| |ud| |ud| | ud| ud| ud| | | d| | d| | d|
+ +--+--+--+--+--+ +--+--+--+--+--+--+ + +--+ +--+ +--+ +--+--+--+--+--+--+ + +--+ +--+ +--+
|u | | | d| | |u | | | d| | | | | |
+ +--+ +--+ +--+ +--+--+ +--+ +--+ + +--+--+--+--+--+ +--+--+ +--+ +--+ + +--+ +--+ +--+
| |u | |u | |u | | |ud| |ud| |ud| | |ud| |ud| |ud| | |ud| |ud| |ud| | | d| | d| | d|
+--+--+--+--+--+--+ + + +--+ +--+ + +--+--+ +--+ +--+ + + +--+ +--+ + +--+--+--+--+--+--+
|u | u u | |ud| | d| | d| | |ud| | |ud| |u | |u | | | d| d d |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4 level 5
.............................................................................
*
*
*****
5x6x7 minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u |u |u |u | | d| d| d| d| d| d|
+--+--+--+--+--+--+ + + + + + + +
| | | | | | | | | |
+ +--+--+--+--+ + + + + + + + +
| | | | | | | | | | | |
+ + +--+--+ + + +--+--+--+--+--+--+
| | |u |u | | | | d| d |
+ + +--+--+ + + +--+--+--+--+--+--+
| | |u |u | | | | d| d |
+ + +--+--+ + + +--+--+--+--+--+--+
| | |u |u | | | | d| d |
+--+ +--+--+ +--+ +--+--+--+--+--+--+
|u | | |u | | d | d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 3, 4 level 5
.............................................................................
*
******
+--+--+ +--+--+
|x | | |x |
+--+ +--+--+--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 3, 4, 5, 6 level 2
3x6x14 by Andrew Clarke
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+--+--+--+ +--+ +
| | | | |
+ +--+ +--+--+--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
*
******
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | | | x | |x |
+ +--+ +--+ +--+ +--+--+ +--+--+--+
| | | | | |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+ +--+ +--+ +--+ +--+--+--+--+--+--+
| |x | |x | |x | | x | x | x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 3, 5, 6 level 4
6x6x7
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ +--+ +
| | |
+ + +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(6 levels deep)
also: 6x7x10, 6x8x14
.............................................................................
*
****
**
3x7x8 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u |u |u | |u | |ud|ud|ud| u ud| | d| d| d| | d d|
+ + + +--+ +--+ + + + +--+--+--+--+ +--+--+--+ +--+--+
|u |u |u | | |u | | d| d| d| u ud| | | | d d|
+--+--+--+ + +--+--+ + + + +--+--+--+--+ + +--+--+--+ +--+--+
| | | |u u | | | | | d ud| | | | | d|
+ +--+ + +--+--+ + + + +--+--+--+--+ + +--+--+--+ +--+
| | | |u u | | | | | d ud| | | | | d|
+--+ + +--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+ + +--+
|u | | | | |ud u | | | | | d d| | | |
+--+ +--+--+--+ + +--+--+--+--+ + + + +--+--+ + +--+ +
|u | | | | |ud u | | | | | d d| | | |
+--+--+ +--+--+--+ + +--+--+--+--+ + + + +--+--+ + +--+--+--+
|u u | | | |ud d |u |u |u | | d| | | d| d| d|
+--+--+ +--+--+--+ +--+--+--+--+ + + + +--+ +--+ + + +
|u u | |u |u |u | |ud d |ud|ud|ud| | d| | d| d| d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2 level 3
6x6x7
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| |u | | | |ud| | |u |u |u ud| |u |u |
+--+ +--+ +--+ +--+ +--+ +--+ + +--+--+--+ +--+ +
|u | |u | |u | |ud| |ud| |ud| |ud|u |u ud| |u |ud|
+--+ +--+--+--+ +--+ +--+ +--+--+--+ +--+ +--+--+--+--+ +--+--+
|u | | | |u | |ud| | | |ud| | d|u u | | d|
+ + +--+ + + + + + +--+ + + + +--+--+--+--+ +--+
|u | |u | | |u | | d| |ud| | | d| | |u ud| | |
+--+--+--+ +--+--+--+ + +--+--+ +--+--+ + + +--+--+--+--+--+ +
| u u | |u u | | |u d| | d ud| | |u |ud|u u |ud|u |
+--+--+--+ +--+--+--+ + +--+--+ +--+--+ + + +--+--+--+--+--+ +
| u u | |u u | | |ud d| | d ud| | |u |ud|u u |ud|u |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2 level 3
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| d|ud| d d |ud| d| | |ud u | |u ud| | | d d| | d d |
+ +--+--+--+--+--+ + + +--+--+ +--+--+ + +--+--+--+ +--+--+--+
| d|ud| d d |ud| d| | |ud u | |u ud| | | d d| | d d |
+ +--+--+--+--+--+ + + +--+--+ +--+--+ + +--+--+--+ +--+--+--+
| | d ud| | | |u | |ud| | |u | | d| | d| | | d|
+--+--+--+--+ +--+ + + +--+ + + + + + +--+ + + +
|u | d d | |u | |ud| | | |ud| | d| | | | d|
+--+--+--+--+ +--+--+ +--+ +--+--+--+ +--+ +--+ +--+--+--+ +--+
|ud| d| d ud| | d|ud| |ud| |ud| |ud| | d| | d| | d|
+ +--+--+--+ +--+ + +--+ +--+ +--+ +--+ +--+ +--+
| d| d| d ud| | d| d| | |ud| | | | d| |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 4 level 5 level 6
.............................................................................
*****
**
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
| |u | | |u d|u | | d|u d| | | d| |
+ +--+ + +--+--+--+ +--+--+--+ + +--+ +
| |u | | |u d|u | | d|u d| | | d| |
+ +--+ + +--+--+--+ +--+--+--+ + +--+ +
| |u | | |u d|u | | d|u d| | | d| |
+ +--+ +--+ +--+--+--+--+ +--+--+--+--+ + +--+ +--+
| |u | |u | |u d|u d| | d|u | d|u | | | d| | d|
+ +--+ +--+ +--+--+--+--+ +--+ +--+ + + +--+ +--+
| | | |u |u |u |u | | d| d| d| d| | | |
+--+ +--+ + +--+--+--+--+ + +--+ +--+ +--+ +--+ +
|u | |u | | | d u | d u | |u | d|u | d| | d| | d| |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
levels 1, 2, 3, 4 level 5 level 6 levels 7, 8, 9, 10
6x7x10
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(10 levels deep)
.............................................................................
*
2****
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u | | u | | d | | d|
+--+ +--+--+--+ +--+ +--+ +--+--+--+ +--+
|u | |u | | d| | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
2*
****
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u | |u | | d| | d|
+--+ +--+--+--+ +--+ +--+ +--+--+--+ +--+
|u | | u | | d | | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
**2*
**
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u |u | | | | d| d |
+ +--+--+--+ + + +--+--+--+ +
| |u | u | | d | d| |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
**
2***
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u | | u | | d | | d|
+--+ +--+ +--+ +--+ +--+ +--+
|u | |u | | d| | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
2**
***
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u | |u | | d| | d|
+--+ +--+ +--+ +--+ +--+ +--+
|u | | u | | d | | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
2*
*2*
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u |u | u |u | | d| d | d| d|
+--+ +--+--+--+ +--+ +--+ +--+--+--+ +--+
|u | u |u | u | | d | d| d | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
_________
/ /\
____ _/________/ \
/ /\ / \ \ /
/___/ \/ \ ____\/
\ \ / /\ \ /
\ \/____/ \___\/
\ \ /
\________\/
2x2x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u | u | u | u | | d | d | d | d|
+--+ +--+--+--+ +--+ +--+ +--+--+--+ +--+
|u |u |u |u | | d| d| d| d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
2
*****
5x5x28 by Patrick Hamlyn
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| | |u | | d| | | | | | |
+--+ +--+ + + +--+--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+
| |u |u |u | | |u |ud|ud|ud|u | | d|ud|ud|ud| d| |u |ud|ud|ud|u | | d| d| d| d| d|
+ +--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ +--+--+ +--+--+
|u | | | | d | | | | | | | | |
+--+--+ +--+ + +--+--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+ + +--+--+ +--+
|u |u |u |u |u | | d|ud|ud|ud|ud| |u |ud|ud|ud|ud| | d|ud|ud|ud|ud| | d| d| d| d|
+ +--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| | | | | | |u | | d| | |
+--+--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ + + +--+ +--+
|u |u |u |u |u | | d|ud|ud|ud| d| |u |ud|ud|ud|u | | d|ud|ud|ud| d| | | d| d| d| |
+--+--+ +--+--+ +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ +
| | | | | | | | | u | | | | d|
+--+ +--+--+ + +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ + +--+ +--+--+
|u |u |u |u | |ud|ud|ud|ud|u | |ud|ud|ud|ud| d| |ud|ud|ud|ud|u | | d| d| d| d| d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+ +
| | |u | | d| | | | | | |
+--+ +--+ + + +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+
|u |u |u |u | | |ud|ud|ud|ud|u | |ud|ud|ud|ud| d| |ud|ud|ud|ud|u | | d| d| d| d| d|
+ +--+ +--+--+ +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+--+
| | |u | | d| | | | | | |
+--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+
| u |u |u | | |u |ud|ud|ud|u | | d|ud|ud|ud| d| |u |ud|ud|ud|u | | d| d| d| d| d|
+ +--+--+ +--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ +--+--+ +--+--+
| | | | | | | | | | | | | |
+--+--+ +--+--+ +--+--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+ + +--+--+ +--+
|u |u |u |u |u | | d|ud|ud|ud|ud| |u |ud|ud|ud|ud| | d|ud|ud|ud|ud| | d| d| d| d|
+ +--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ +
| | | | | | |u | | d| | |
+--+--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ + + +--+--+--+
|u |u |u |u |u | | d|ud|ud|ud| d| |u |ud|ud|ud|u | | d|ud|ud|ud| d| | | d| d| d| |
+--+--+ +--+--+ +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+--+ +--+--+--+ + +
| | | | | | | | | u | | | | d|
+--+ +--+--+ + +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+
|u |u |u |u | |ud|ud|ud|ud|u | |ud|ud|ud|ud| d| |ud|ud|ud|ud|u | | d| d| d| d| d|
+--+--+ +--+--+ +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+ +
| | | | | | | | | | | |
+--+ +--+--+ + +--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+
|u |u |u |u | |ud|ud|ud|ud|u | |ud|ud|ud|ud| d| |ud|ud|ud|ud|u | | d| d| d| d| d|
+ +--+ +--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
| | |u | | d| | | | | | |
+--+--+--+--+ + +--+--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+
| u |u |u | | |u |ud|ud|ud|u | | d|ud|ud|ud| d| |u |ud|ud|ud|u | | d| d| d| d| d|
+ +--+--+ +--+ +--+--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ +--+--+ +--+--+
| | | | | | | | | | | | | |
+--+--+ +--+--+ + +--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+ + +--+--+ +--+
|u |u |u |u |u | | d|ud|ud|ud|ud| |u |ud|ud|ud|ud| | d|ud|ud|ud|ud| | d| d| d| d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ +
| | | | | | |u | | d| | |
+--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+--+ + + +--+--+--+
|u |u |u |u |u | | d|ud|ud|ud| d| |u |ud|ud|ud|u | | d|ud|ud|ud| d| | | d| d| d |
+--+--+ +--+--+ +--+--+--+--+--+ + +--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+ +
| | | | | | | | | | | | | |
+--+ +--+--+ + +--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+ + + +--+ +--+--+
|u |u |u |u | |ud|ud|ud|ud|u | |ud|ud|ud|ud| d| |ud|ud|ud|ud|u | | d| d| d| d| d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3 level 4 level 5
same heptacube, allowing reflections
2x3x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u |u |u | | d| d| d|
+ +--+--+--+--+--+ + +--+ +--+--+--+--+--+
| | | | | | | |
+--+--+--+--+--+ +--+ + +--+--+--+--+--+ +
|u |u |u | | d| d| d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
22
***
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u |u u | | d| d d| |u u |u | | d d| d|
+ + + + +--+--+--+ +--+--+--+ + + + +
|u | | | d|u u | | | d d|u | | | d|
+--+--+--+--+ + + + + +--+--+--+ + +--+--+--+--+
| | u | | | | d| |u u | | | d d| |
+ + + +--+--+--+ + +--+--+--+--+ + + +
|u u | u | | d d | d| | | u u | | | d d|
+--+--+--+--+ +--+--+--+--+ + +--+--+--+ +--+--+--+--+
|u u | u | | d ud|u | d| |u | d|ud u | | d | d d|
+ +--+ + + +--+--+ + + +--+--+ + + +--+ +
|u |u | u | | d|ud u | d| |u | d d|u | | d | | d|
+--+--+ +--+ +--+--+ +--+ +--+--+ +--+ +--+--+ +--+
| |u | |u | d| | d|u | | d|
+--+ + +--+ + +--+ + + +
|u u | |ud d| | d|u | | d|
+--+--+ +--+--+ +--+--+ +--+--+
level 1 level 2 level 3 level 4
4x4x14
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
same heptacube, allowing reflections
2x3x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u u |u u |u u | | d d| d d| d d|
+ +--+--+--+ + +--+--+ +--+--+--+
| | | | | | | |
+--+--+--+ +--+--+ + +--+--+--+ +
|u u |u u |u u | | d d| d d| d d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
also: 4x4x7
.............................................................................
2**
2 *
+--+--+ +--+--+
|u u | | d d|
+ +--+--+--+ +--+--+--+--+
| | |u |u | | | d| d|
+ + +--+ + +--+ + + +
|u | u | | d| | | d|
+--+--+--+--+ +--+ + +--+
|u u |u u | | d d| d d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
2x4x7 minimal
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(2 levels deep)
.............................................................................
****
2*
2x4x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| |u |u | u |u | | d| d | d| d| |
+ +--+--+ + + + + +--+--+ +
| | | | | | | |
+ +--+--+--+--+--+ + + +--+--+--+--+--+ +
| | | | | | | |
+ + +--+--+ + + +--+--+ + +
| u |u |u |u | | | | d| d| d| d |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
*2*
***
2x4x7 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u u u |u u u |u | | d d d| d d d| d|
+--+ +--+--+--+--+ + +--+--+--+--+ +--+ +
|u | |u | |u | | d| d | | d|
+ +--+--+ + + + + +--+--+ +
|u | | u |u | | d| | d| | d|
+ +--+ +--+--+--+--+ + +--+--+--+--+ +--+
|u |u u u |u u u | | d| d d d| d d d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
+--+--+--+ +--+--+--+ +--+--+--+
|u |u u | |ud|ud ud| | d| d d|
+ +--+--+ + +--+ + + +--+--+
|u | | |ud | | | d| |
+--+ +--+ +--+--+--+--+ +--+ +--+
|u |u |u | |ud| d|u |ud| | d| d| d|
+ + + + + +--+--+ + + + + +
|u | |u | |ud | ud| | d| | d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3
3x4x7
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(3 levels deep)
also: 3x6x14
.............................................................................
2**
***
+--+--+--+ +--+--+--+ +--+--+--+
| | | |u | | d|
+ + + +--+ + +
| u | | | d| | |
+--+--+--+--+ + +--+--+ +--+--+--+--+
| |u | u | | d|u |ud| | d| d|
+ +--+--+ + +--+--+--+ + + + +
|u u u |u | | d d d|ud| | | d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3
3x4x7 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(3 levels deep)
same heptacube, allowing reflections
2x3x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u |u |u u u | | d| d| d d d|
+ +--+--+--+ + +--+ +--+--+--+
| | | | | | | |
+--+--+--+ +--+ + +--+--+--+ +
|u u u |u |u | | d d d| d| d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
22*
**
2x6x7 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u |u |u u | u | | d| d| d d | d|
+ + +--+--+ + + + + +--+ +
|u |u | | u | | d| d| | | d|
+ +--+--+ +--+ + +--+--+ +--+--+ +--+
| |u | u u |u | | | d| | d d| d |
+--+ +--+--+--+ +--+ + +--+--+--+ +
| |u |u u |u | | | d| d d| | d |
+ +--+ +--+--+ + +--+ +--+--+ +--+--+
|u | |u | u | | d| | | d | d|
+ +--+--+ + + + +--+ + + +
|u |u u |u | u | | d| d d| d | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
2**
*2
allowing reflections
2x6x7
+--+--+--+--+--+--+ +--+--+--+--+--+--+
|u u | |u u | | d d| | d d|
+ +--+--+ + +--+ +--+ + +--+--+ +
| | u u | | | | | | d d| | |
+--+--+--+--+--+ + + +--+--+--+ +--+
|u |u | u u | | d| d| | d d|
+ +--+ +--+--+--+ + +--+ + +--+--+
|u | u |u |u | | d | d| d| d|
+--+--+--+ +--+ + +--+--+ + +--+ +
|u u |u | u | | d d| | d | d|
+ +--+--+--+--+--+ +--+ +--+--+--+ +
| | |u u | | | | | d d| | |
+--+ + +--+--+ + + +--+--+ + +--+
|u u | | u u | | d d | | d d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2
.............................................................................
****
3
allowing reflections
+--+--+ +--+--+ +--+--+ +--+--+
|u |u | |ud|ud| |ud|ud| | d| d|
+--+ +--+--+ +--+--+--+--+ +--+--+--+--+ + +--+--+--+
|u | | | d | | u | | | d|
+ +--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+ +
| | | | | | | | | |
+ + +--+--+ + +--+--+--+ +--+--+--+ + +--+--+ + +
| |u |u |u | |u |ud|ud|ud| |ud|ud|ud| d| | d| d| d| |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
4x4x7
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(4 levels deep)
.............................................................................
2**
2*
allowing reflections
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
|u |u u | |ud| d d| | d |u | | d|
+--+ + + +--+--+ + + + +--+ +
| | | |u | | | d |u | | | d|
+ + +--+--+ +--+ +--+ +--+--+--+--+ + +--+--+--+
|u | | u | | d|u u |ud| |u |ud d| d| | d| d| |
+ +--+ + + +--+--+ + + +--+--+--+ + +--+ +
|u u | u | | d d|u | d| |u u |ud ud| | d d| d d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
4x4x7
+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+
(4 levels deep)
also: 3x4x14, 3x4x21, 3x6x14, 3x6x21, 4x5x7, 5x6x7
.............................................................................
***2
**
allowing reflections
4x4x7
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u u | | d d| | | |u u | | d d|
+ +--+--+--+ +--+--+ + + +--+--+ +--+--+--+ +
| | | |u | | d| |u | | d| | | |
+--+ + +--+ +--+--+--+--+ +--+--+--+--+ +--+ + +--+
|u | | |u | | d | d| |u | u | | d| | | d|
+--+ + +--+ + + + + + + +--+ + +--+
| | | | | | | | | | | |
+ + + + +--+--+ + + +--+--+ + + + +
| u |u | | | d| d| | | |u |u | | | d| d |
+--+--+--+--+ + +--+--+ + + +--+--+ + +--+--+--+--+
| |u |u | | |ud|ud| | | |ud|ud| | | d| d| |
+ +--+ +--+ +--+--+ +--+ +--+ +--+--+ +--+ +--+ +
|u u |u |u | |ud d| d|ud| |ud|u |u ud| | d| d| d d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*2
*2*
4x4x7 (minimal) by Patrick Hamlyn
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u u |u | | d d| d| |u |u u | | d| d d|
+--+ +--+ + +--+ + + + + +--+ + +--+ +--+
| |u | |u | |u | d| ud| |ud |u | d| | d| | d| |
+ +--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+ +
|u u |u u | | d d| d d| |u u |u u | | d d| d d|
+ +--+ +--+ + +--+ +--+ +--+ +--+ + +--+ +--+ +
|u |u |u |u | | d| d| d| d| |u |u |u |u | | d| d| d| d|
+--+ +--+ + +--+ +--+ + + +--+ +--+ + +--+ +--+
|u u |u u | | d d| d d| |u u |u u | | d d| d d|
+--+ +--+ + +--+--+--+--+ +--+--+--+--+ + +--+ +--+
|u | |u | | |ud | d|u | | d|u | ud| | | d| | d|
+ +--+ +--+ + + +--+ +--+ + + +--+ +--+ +
|u | u u | | d | d d| |u u | u | | d d | d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
same heptacube, allowing reflections
2x6x7 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u u | |u u |u | | d d | d d| d|
+--+ + + +--+ + +--+ +--+--+ +--+ +
|u |u |u u |u |u u | | d| d| d d| d| d d|
+ +--+--+ +--+--+--+ + +--+--+--+--+--+ +
|u u |u | |u u | | | d d| d| d d| |
+ +--+ +--+ +--+ + +--+--+ +--+ +--+--+
| |u u | |u |u u | | | d d | d| d d|
+--+--+--+ +--+--+ + + +--+--+--+--+--+ +
|u u |u |u u |u |u | | d d| d| d d| d| d|
+ +--+ + + +--+ + +--+ +--+--+ +--+
|u |u u | |u u | | d| d d | d d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2
.............................................................................
Octacubes
.............................................................................
*
*******
1x2x8 minimal
+--+--+--+--+--+--+--+--+
| | |
+ +--+--+--+--+--+--+ +
| | |
+--+--+--+--+--+--+--+--+
also: 1x9x16, 1x12x44, 1x20x20
also: 3x7x8 by Andrew Clarke
.............................................................................
**
******
1x2x8 minimal
+--+--+--+--+--+--+--+--+
| | |
+ +--+--+--+--+ +
| | |
+--+--+--+--+--+--+--+--+
.............................................................................
***
*****
1x2x8 minimal
+--+--+--+--+--+--+--+--+
| | |
+ +--+--+ +
| | |
+--+--+--+--+--+--+--+--+
.............................................................................
***
*
****
1x4x4 minimal
+--+--+--+--+
| |
+--+--+--+ +
| | |
+ +--+--+ +
| | |
+ +--+--+--+
| |
+--+--+--+--+
.............................................................................
* *
* *
****
1x4x4 minimal
+--+--+--+--+
| |
+--+ +--+ +
| | | | |
+ + + + +
| | | | |
+ +--+ +--+
| |
+--+--+--+--+
.............................................................................
*
* **
****
1x4x4 minimal
+--+--+--+--+
| |
+ +--+ +
| | | |
+--+ + +--+
| | | |
+ +--+ +
| |
+--+--+--+--+
.............................................................................
**
**
****
1x4x4 minimal
+--+--+--+--+
| |
+--+--+ +
| | |
+ + +
| | |
+ +--+--+
| |
+--+--+--+--+
also: 1x5x16, 1x6x8, 1x7x16 by David Klarner
.............................................................................
*
***
****
1x4x4 minimal
+--+--+--+--+
| |
+--+ +
| | |
+ +--+--+ +
| | |
+ +--+
| |
+--+--+--+--+
.............................................................................
*
***
****
1x12x16 (co-minimal) by David Klarner
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | | | |
+ + +--+ +--+--+ + +
| | | | | | | |
+ +--+--+ + +--+--+ + + +--+--+ +
| | | | | | | | | |
+ +--+--+ +--+ +--+ +--+ +--+--+ +
| | | | | | | |
+--+ +--+--+--+--+ + +--+--+ +--+--+
| | | | | | | | |
+ +--+--+--+--+ + +--+ + + +
| | | | | | |
+ +--+--+ +--+--+--+--+ +--+--+ +
| | | | | | |
+ + + +--+ + +--+--+--+--+ +
| | | | | | | | |
+--+--+ +--+--+ + +--+--+--+--+ +--+
| | | | | | | |
+ +--+--+ +--+ +--+ +--+ +--+--+ +
| | | | | | | | | |
+ +--+--+ + + +--+--+ + +--+--+ +
| | | | | | | |
+ + +--+--+ +--+ + +
| | | | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
4x6x8 co-minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u | u u u | | d| d|u |ud d d| |u u ud| d|u |u | | d d d | d| d|
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
|u |u | u u u | | d| d|u |ud d d| |u u ud| d|u |u | | d d d | d| d|
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
|u |u | | | | |ud|ud| |u |u | | d| d| |ud|ud| | | | | d| d|
+ + + + + + +--+--+ +--+--+ +--+--+ +--+--+ + + + + + +
| | | |u |u | |u |u | |ud|ud| |ud|ud| | d| d| | d| d| | | |
+--+--+ + + + +--+--+ + + + + + + +--+--+ + + + +--+--+
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ +--+--+ + + + + +--+--+ + + + + + + +--+--+ + + + + +--+--+ +
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ +--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+ +
| | u u u | | |u |ud d d| |u u ud| d| | | d d d | |
+ +--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+ +
| | u u u | | |u |ud d d| |u u ud| d| | | d d d | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*
****
***
4x7x8 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| | |u | |u u u | u u ud| |ud d d | d d d| | d| | |
+--+ + +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ + +--+
|u | | | |ud u u |u u u | | d d d| d d ud| | | | d|
+--+ + +--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+ + +--+
|u | | | | |ud u u |u u u | | d d d| d d ud| | | | | d|
+--+ +--+--+ + +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ + +--+--+ +--+
|u | | | | |ud u u |u u u | | d d d| d d ud| | | | | d|
+--+--+ +--+--+--+ + +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ + +--+--+--+ +--+--+
|u | | | | |ud u u |u u u | | d d d| d d ud| | | | | d|
+--+ + +--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+ + +--+
| | | |u | |u u u | u u ud| |ud d d | d d d| | d| | | |
+ +--+--+--+ +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ +--+--+--+ +
| | |u | |u u u | u u ud| |ud d d | d d d| | d| | |
+ +--+ +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ +--+ +
| | |u | |u u u | u u ud| |ud d d | d d d| | d| | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2 level 3 level 4
also: 1x36x40, 8x8x9, many others
.............................................................................
*
***
****
5x8x12
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| |u u |u |u | |u |u |u | ud ud|ud|ud| |ud|ud|ud u | d ud|ud|ud|
+--+ +--+--+ + + +--+--+--+--+--+--+ + + + + +--+--+--+--+ + +
|u | |u u |u |u | | d| |u | ud ud|ud|ud| |u |u |ud u | d ud| d| d|
+ +--+ +--+--+--+--+--+ + + +--+--+--+--+ + + +--+--+--+--+--+--+--+--+
|u |u | | |u u | | |ud|ud|u | ud ud| | | | d| d|ud u | d ud|u |u |
+ + +--+ +--+--+ + + + +--+--+--+--+--+--+ +--+ +--+--+--+--+ +--+
|u |u | | | | | d|ud| |u |u | |u |ud| |ud| d|
+ +--+ + +--+ + +--+--+--+ +--+ + + +--+--+ +--+ +
| | | | | |u | | | |u | |ud| | | |ud|
+--+ + +--+--+ + +--+ +--+ +--+ + + + + +--+ +--+ + +
| | | | | | | | | |u | |u | | | | d|
+--+ +--+--+ +--+ + + +--+--+ + + +--+ +--+--+ +--+
| | | | | |u | | | | | | d| | | |u |
+ +--+--+ + +--+ + + +--+ +--+ +--+ + + +--+ +--+ + +
| | | | | |u | | | |u | |ud| | | |ud|
+ +--+ + +--+ + + +--+ +--+--+--+ + +--+ +--+--+ +
| | | |u |u | |u |u | |ud| d| | d|ud| |ud|u |
+ +--+--+ +--+ + + +--+--+--+--+--+--+ + + +--+ +--+--+--+--+ +--+
| |u u | | |u |u | | | |ud ud |u |ud|ud| |u |u |ud d|u ud| d| d|
+--+--+--+--+--+ +--+ + + + +--+--+--+--+ + + +--+--+--+--+--+--+--+--+
|u |u |u u | |u | |ud|ud|ud ud |u | | d| | d| d|ud d|u ud|u |u |
+ + +--+--+ +--+ + + +--+--+--+--+--+--+ + + +--+--+--+--+ + +
|u |u |u u | | |ud|ud|ud ud |u |u |u | |ud|ud|ud d|u ud|ud|ud|
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 level 2 level 3
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|ud|ud|ud ud | d| d| d| | d| d| d d| |
+ + +--+--+--+--+--+--+ + + +--+--+ +--+
|ud|ud|ud ud | d| |u | | d| d| d d| | d|
+ + +--+--+--+--+ + + +--+--+--+--+--+ +--+ +
| | |ud ud | d|ud|ud| | | d d| | | d| d|
+--+--+--+--+--+--+ + + + +--+--+ +--+ + +
| d| d| |ud|u | | | | | d| d|
+ +--+ +--+--+--+ + +--+ + +--+ +
| d| | | | d| | | | | |
+ + +--+ +--+ +--+ + +--+--+ + +--+
| d| | | | | | | | | |
+ + +--+--+ + + +--+ +--+--+ +--+
| | | | | d| | | | | |
+--+ +--+ +--+ + + +--+ + +--+--+ +
| d| | | | d| | | | | |
+--+--+--+ +--+ + + +--+ + +--+ +
|u |ud| | d| d| | d| d| | | |
+ + +--+--+--+--+--+--+ + + +--+ +--+--+ +
|ud|ud| d| ud ud| | | | d| d| | | d d| |
+ + +--+--+--+--+ + + + +--+ +--+--+--+--+--+
|u | | d| ud ud|ud|ud| | d| | d d| d| d|
+--+--+--+--+--+--+ + + +--+ +--+--+ + +
| d| d| d| ud ud|ud|ud| | | d d| d| d|
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 4 level 5
also: 1x41x48, others
.............................................................................
* *
****
**
4x4x4 minimal
+--+--+--+--+ +--+--+--+--+
| |x | | | x | |
+--+ +--+--+ + +--+--+ +
| | | | | |
+ + + + +--+ +--+
|x | |x | |x | |x |
+--+ +--+--+ + +--+--+ +
| |x | | | x | |
+--+--+--+--+ +--+--+--+--+
levels 1, 4 levels 2, 3
.............................................................................
*
****
***
4x4x5 minimal
+--+--+--+--+--+ +--+--+--+--+--+
| |u |u | | d| d| |
+ +--+--+--+--+ +--+--+--+--+ +
| | | | | |
+--+ + + + + +--+
|u | |u | | d| | d|
+ +--+--+--+--+ +--+--+--+--+ +
| |u |u | | d| d| |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2, 3 level 4
.............................................................................
* *
******
+--+--+--+--+ +--+--+--+--+
|u |u |u |u | | d| d| d| d|
+--+--+--+--+--+--+ +--+ + + + +--+
|u u | | d| | | | | d|
+--+--+--+--+--+--+ +--+ + + + +--+
|u u | | d| | | | | d|
+--+--+--+--+--+--+ +--+ + + + +--+
|u u | | d| | | | | d|
+--+--+--+--+--+--+ +--+ + + + +--+
|u u | | d| | | | | d|
+--+--+--+--+--+--+ +--+ + + + +--+
|u |u |u |u | | d| d| d| d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
2x6x8 (minimal) by Andrew Clarke
+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+ + + +
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+
(2 levels deep)
.............................................................................
* *
******
+--+--+ +--+--+ +--+--+ +--+--+
| |x | | |x | | x | | x |
+ +--+--+ +--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 4, 5 levels 3, 6
4x6x6 minimal
+--+--+--+--+--+--+
| |
+--+--+ +--+--+ +
| | | | |
+ +--+ +--+
| |
+ +
| |
+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
**
***
***
4x5x8 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|u u | | | | d d | |u |u | | | | d| d|
+--+--+ + + +--+--+--+ + + + + + + + +
|u u | | | | d d | |u |u | | | | d| d|
+--+--+ +--+--+ + +--+--+--+--+ + + + + +--+--+ +--+--+
| | | | | | | | | | | | | | | | |
+ +--+--+ +--+--+ + + + +--+--+--+--+ +--+--+ +--+--+ +
| | |u u | |u |u | | d d| | d| d| | |
+ + +--+--+ + + + +--+--+--+ + + + + +
| | |u u | |u |u | | d d| | d| d| | |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 levels 2, 3 level 4
.............................................................................
*
* **
****
4x6x8 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u u u | | d| d|ud|u | d d| |u u | d|ud|u |u | | d d d| d| d|
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
|u |u |u u u | | d| d|ud|u | d d| |u u | d|ud|u |u | | d d d| d| d|
+ + +--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+ + +
| | | |u |u | |u |u | |ud|ud| |ud|ud| | d| d| | d| d| | | |
+ + + + + + +--+--+ +--+--+ +--+--+ +--+--+ + + + + + +
|u |u | | | | |ud|ud| |u |u | | d| d| |ud|ud| | | | | d| d|
+--+--+--+ + + + +--+--+--+ +--+--+ +--+--+ +--+--+--+ + + + +--+--+--+
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ +--+--+ + + + + +--+--+ + + + + + + +--+--+ + + + + +--+--+ +
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ +--+--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+--+ +
| |u u u | | |ud|u | d d| |u u | d|ud| | | d d d| |
+ +--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+ +
| |u u u | | |ud|u | d d| |u u | d|ud| | | d d d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
**
* *
****
4x6x8 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u u | | d| d|ud|u | d| |u | d|ud|u |u | | d d| d| d|
+ + +--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+ + +
| | |u u | | | |ud|u | d| |u | d|ud| | | | d d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | | |u |u | |u |u | | d| d| |ud|ud| | d| d| | d| d| | | |
+ + +--+ + + + +--+--+--+ +--+--+ +--+--+ +--+--+--+ + + + +--+ + +
|u |u | | | | | |ud|ud| | |u |u | | d| d| | |ud|ud| | | | | | d| d|
+--+--+ + + + + +--+--+ + + + + + + + + +--+--+ + + + + +--+--+
| | | | | | | |u |u | | | | | | | | | | |
+ +--+--+ + + + + +--+--+ +--+--+ +--+--+ +--+--+ + + + + +--+--+ +
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | |u u | | | |ud|u | d| |u | d|ud| | | | d d| | |
+ +--+--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+--+ +
| |u u | | |ud|u | d| |u | d|ud| | | d d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
***
*
****
4x6x8 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| | |u | | | |ud|u | | d|ud| | | | d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | |u | | | |ud|u | | d|ud| | | | d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | | | |u |u | |u |u | | |ud|ud| |ud|ud| | | d| d| | d| d| | | | |
+ + + + + + + +--+--+ + +--+--+ +--+--+ + +--+--+ + + + + + + +
|u |u | | | | | |ud|ud| | |u |u | | d| d| | |ud|ud| | | | | | d| d|
+--+--+ + + + + +--+--+ + + + + + + + + +--+--+ + + + + +--+--+
| | | | | | | | | | | | | | | | | | | |
+ +--+--+ + + + + +--+--+ + + + + + + +--+--+ + + + + +--+--+ +
| | | | | | | | | | | | | | | | | | | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | |u | | | |ud|u | | d|ud| | | | d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | |u | | | |ud|u | | d|ud| | | | d| | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*
***
*
***
4x6x8 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u |u | | d| d|ud|u d| |u d|ud|u |u | | d| d| d| d|
+--+--+--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+--+--+
| | |u |u | | | |ud|u d| |u d|ud| | | | d| d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | | |u |u | |u |u | |ud|ud| |ud|ud| | d| d| | d| d| | | |
+ + + +--+ + + +--+--+ +--+--+--+ +--+--+--+ +--+--+ + + +--+ + + +
|u |u | | | | | |ud|ud| | |u |u | | d| d| | |ud|ud| | | | | | d| d|
+--+--+ + + + + +--+--+ + + + + + + + + +--+--+ + + + + +--+--+
| | | | | | | | | | | | | | | | | | | |
+ +--+--+ +--+--+ + +--+--+ + + + + + + +--+--+ + +--+--+ +--+--+ +
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
| | |u |u | | | |ud|u d| |u d|ud| | | | d| d| | |
+--+ +--+--+--+--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+--+--+--+ +--+
| |u |u | | |ud|u d| |u d|ud| | | d| d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
**
***
*
**
4x6x8 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u |u u | | d| d|ud|u d d| |u u d|ud|u |u | | d d| d| d| d|
+ + +--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+ + +
|u |u |u |u u | | d| d|ud|u d d| |u u d|ud|u |u | | d d| d| d| d|
+--+--+--+--+--+--+ + + +--+--+--+--+ +--+--+--+--+ + + +--+--+--+--+--+--+
| | | |u |u | |u |u | |ud|ud| |ud|ud| | d| d| | d| d| | | |
+ + + + + + +--+--+ +--+--+ +--+--+ +--+--+ + + + + + +
|u |u | | | | |ud|ud| |u |u | | d| d| |ud|ud| | | | | d| d|
+--+--+ +--+--+--+ +--+--+ +--+ + + + + +--+ +--+--+ +--+--+--+ +--+--+
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+ +--+--+ + + + + +--+--+ + + + + + + +--+--+ + + + + +--+--+ +
| | |u |u | | | | d| d| |u |u | | | | d| d| | |
+--+ +--+--+--+--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+--+--+--+ +--+
| |u |u u | | |ud|u d d| |u u d|ud| | | d d| d| |
+ +--+--+--+--+ + +--+--+--+--+ +--+--+--+--+ + +--+--+--+--+ +
| |u |u u | | |ud|u d d| |u u d|ud| | | d d| d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
* **
*****
+--+--+--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+ +--+--+ +--+--+--+--+
|x | |x | |x |x |
+--+ +--+ +--+--+ +
|x | | |x | |
+--+ + +--+--+--+
|x | | |x | |
+--+ +--+ +--+--+ +
|x | |x | |x |x |
+--+--+--+ +--+--+--+
level 1 level 2
+--+--+ +--+--+
| |x | | x |
+--+ +--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+ +--+ +--+--+--+--+
|x | | |x | |
+--+ +--+ +--+--+ +
|x | |x | |x |x |
+--+ +--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+--+--+--+ +--+--+--+--+
level 1 level 2
5x5x8 (minimal) by Andrew Clarke
+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + +--+ +
| | | |
+ + + +
| | | |
+ + +--+ +
| | | |
+--+--+--+--+--+--+--+--+
(5 levels deep)
.............................................................................
*
*
******
4x7x8 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| | |u | |u |ud| |ud| d| | d| | |
+ +--+--+--+--+ +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ +--+--+--+--+ +
| | | | |u | |u |ud| |ud| d| | d| | | | |
+ + +--+--+ + +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ + +--+--+ + +
| | | | | | |u | |u |ud| |ud| d| | d| | | | | | |
+ + + + + + +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ + + + + + +
| | | | | | |u | |u |ud| |ud| d| | d| | | | | | |
+ + + + + + +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ + + + + + +
| | | | | | |u | |u |ud| |ud| d| | d| | | | | | |
+ + + + + + +--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+ + + + + + +
| | | | | | |u | |u |ud| |ud| d| | d| | | | | | |
+--+ + + + +--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+ + + + +--+
|u | | | | | |ud| u | | d |ud| | | | | | d|
+--+--+ + +--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+ + +--+--+
| | |u | |u |ud| |ud| d| | d| | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2 level 3 level 4
also: 6x6x8
.............................................................................
* *
****
* *
4x7x8 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| |x |x |x | | x | x | x | |
+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +
|x | |x |x |x | |x | x | x | |x |
+--+ +--+--+--+--+ + + +--+--+--+--+ +--+
| |x | |x |x | | | |x | x | |x | |
+ +--+ +--+--+ +--+ +--+ +--+--+ +--+ +
| | | | | | | | | |
+--+ +--+ + +--+ + + +--+ + +--+ +--+
|x | |x | |x | |x | |x | |x | |x | |x |
+ +--+ +--+--+ +--+ +--+ +--+--+ +--+ +
| |x | |x |x | | | |x | x | |x | |
+--+ +--+--+--+--+ + + +--+--+--+--+ +--+
|x | |x |x |x | |x | x | x | |x |
+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +
| |x |x |x | | x | x | x | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 3 levels 2, 4
.............................................................................
* * *
*****
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| x | x | x | | |x | |x | |x |
+--+--+--+--+--+--+ + +--+ +--+ +--+
| | | | | | | |
+ + + + + + +--+--+--+--+--+
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 3, 5 levels 2, 4
3x5x16 co-minimal
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +--+ + +
| | | | | | | | |
+ + +--+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
5x6x8 co-minimal
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| | | | | x | | | |x |
+ + +--+--+ + +--+--+ +--+ + +--+ +--+ +--+
|x | | x | |x | x | |x | | |x | |x | |x |
+--+ +--+--+ +--+--+--+ +--+ +--+--+--+--+--+--+
| | | | | x | | | |x |
+ + +--+--+ + +--+--+ +--+ + +--+ +--+ +--+
|x | | x | |x | x | |x | | |x | |x | |x |
+--+ +--+--+ +--+--+--+ +--+ +--+--+--+--+--+--+
| | | | | x | | | |x |
+ +--+--+--+--+ +--+--+ +--+--+ +--+ +--+ +--+
|x |x |x |x | x | |x |x | |x | |x | |x |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
levels 1, 3, 5 levels 2, 4
.............................................................................
** *
*****
+--+--+--+--+--+--+ +--+--+--+--+--+--+
|x | |x | |x | |x |x | x |
+--+ +--+ +--+ +--+--+--+--+--+--+
| | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 3 levels 2, 4, 5
3x5x16
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+--+ +--+ + +
| | | | | | |
+ + +--+ +--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
.............................................................................
* **
*****
+--+--+--+--+--+--+ +--+--+--+--+--+--+
| x |x | x | | |x |x | |x |
+--+--+--+--+--+--+ + +--+--+ +--+
| | | | | | | |
+ + + + + + +--+--+--+--+--+
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 4, 5 levels 2, 3
+--+--+ +--+--+
|x |x | |x |x |
+--+--+ + +--+--+--+ +--+--+--+--+--+--+--+
|x | | | | x | |x | |x |
+--+--+--+--+ +--+--+ +--+ +--+--+ +--+
|x | x |x | x | |x | |x |x | |x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 4, 5 levels 2, 3
3x5x16
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+--+ + +
| | | | | | |
+ + +--+--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
4x5x12
+--+--+--+--+--+--+--+--+--+--+--+--+
| | |
+ +--+--+ + +
| | | | |
+--+--+ +--+--+--+--+ +--+--+
| | | | |
+ + +--+--+ +
| | |
+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
also: 5x8x8, 5x8x10
.............................................................................
* *
******
+--+--+--+--+ +--+--+--+--+
| |x | |x | | x | x |
+ +--+ +--+--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 3, 5 levels 4, 6
3x6x16
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+--+--+ +--+ + +
| | | | | | |
+ + +--+ +--+--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
**
******
+--+--+--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+--+ +--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2, 5, 6 levels 3, 4
6x6x8 minimal
+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+ +--+--+--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+
(6 levels deep)
also: 6x8x8
.............................................................................
* *
******
+--+--+--+--+ +--+--+--+--+
|x | |x | | |x |x |
+--+ +--+ +--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 3, 5, 6 levels 2, 4
+--+--+--+--+ +--+--+--+--+
| |x | |x | | x | x |
+--+ +--+ +--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 3, 5, 6 levels 2, 4
3x6x16
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+--+ +--+ +--+ +
| | | | | | |
+ +--+ +--+ +--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(6 levels deep)
6x6x8
+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +--+--+ +
| | | | | | |
+ +--+--+ +--+ +--+ +
| | | |
+ +--+--+--+--+--+--+ +
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
* *
******
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
|u | |u | |u | |ud |ud| ud| |ud| |ud| |ud| |ud| |ud| |ud| |ud |ud| ud| | d| | d| | d|
+--+ +--+ +--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+ +--+ +--+
|u | | | d| |u | | | d| | |u | |u | | d| | d| |
+ +--+ +--+ +--+--+--+--+ + +--+ +--+ + +--+ +--+ +--+--+--+--+ + +--+ +--+
| |u | |u | | ud| ud| | |ud| |ud| | |ud| |ud| | ud| ud| | | d| | d|
+ +--+ +--+ +--+--+--+--+ + +--+ +--+ + +--+ +--+ +--+--+--+--+ + +--+ +--+
| |u | |u | | ud| ud| | |ud| |ud| | |ud| |ud| | ud| ud| | | d| | d|
+ +--+ +--+ +--+--+--+--+ + +--+ +--+ + +--+ +--+ +--+--+--+--+ + +--+ +--+
|u | | | d| |u | | | d| | |u | |u | | d| | d| |
+--+ +--+ +--+ +--+--+ +--+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+--+ +--+--+ +--+ +--+ +--+
|u | |u | |u | |ud |ud| ud| |ud| |ud| |ud| |ud| |ud| |ud| |ud |ud| ud| | d| | d| | d|
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
level 1 level 2 level 3 level 4 level 5 level 6
+--+--+--+--+ +--+--+--+--+
| |x |x | | | x |x |
+--+ +--+--+ +--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 3, 4, 6 levels 2, 5
6x6x8 minimal
+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + +--+ +
| | | |
+ + + +
| | | |
+ + +--+ +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
***
*****
+--+--+ +--+--+
|x |x | |x |x |
+--+--+--+--+ +--+--+ + +
|x | | |x | | |
+--+ +--+ +--+--+--+--+--+
| | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 5 levels 2, 3, 4
5x8x8 minimal
+--+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +--+--+
| | | |
+ +--+--+--+--+ +
| | | |
+--+--+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+--+
(5 levels deep)
.............................................................................
*
*******
+--+--+ +--+--+
|x | | |x |
+--+ +--+--+--+--+--+ +--+--+--+--+--+--+--+
| | | | | | | | | |
+--+--+--+--+--+--+--+ + + + + + + + +
|x |x |x |x |x |x |x | |x |x |x |x |x |x |x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 3, 4, 5, 6, 7 level 2
3x7x16 by Andrew Clarke
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+--+--+--+--+ +--+ +
| | | | |
+ +--+ +--+--+--+--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(7 levels deep)
.............................................................................
**
******
+--+ +--+
|x | |x |
+--+--+--+ +--+--+ +
|x | | |x | |
+--+ +--+--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 4, 5, 6 levels 2, 3
6x8x8
+--+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+--+--+--+ + +
| | | |
+ + +--+--+ +
| | | |
+ +--+--+ +
| | | |
+ +--+--+ + +
| | | |
+ + +--+--+--+
| | |
+ + +
| | |
+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
*
*******
7x7x8
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| |x | |x | |x | |x | | x | x | x | x |
+ +--+ +--+ +--+ +--+ +--+--+--+--+--+--+--+--+
| |x | |x | |x | |x | | x | |x |x |x |
+ +--+ +--+ +--+ +--+ +--+--+ +--+--+--+--+ +
| | | | | | | |
+ +--+ +--+ +--+ +--+ +--+--+--+--+--+--+--+--+
| |x | |x | |x | |x | | x | x | x | x |
+ +--+ +--+ +--+ +--+ +--+--+--+--+--+--+--+--+
| |x | |x | |x | |x | | x | x | x | x |
+ +--+ +--+ +--+ +--+ +--+--+--+--+--+--+--+--+
| |x | |x | |x | |x | | x | x | x | x |
+ +--+ +--+ +--+ +--+ +--+--+--+--+--+--+--+--+
| |x | |x | |x | |x | | x | x | x | x |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
levels 1, 2, 4, 5, 6, 7 level 3
.............................................................................
*
*
*
*****
6x6x8 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u | |u | |ud| |ud| |ud|
+ +--+--+--+--+--+ + + +--+--+--+--+ +--+ +--+--+--+--+
| | | | | | | |u | | |
+ + +--+--+--+--+ + + +--+--+--+ + +--+ +--+--+--+ +
| | |u | | | |ud|u | | | | | |ud|ud|u | |
+ + +--+--+--+--+ + + +--+--+ + + + + +--+--+ + +
| | |u | | | |ud|u | | | | | |ud|ud| | |
+--+ +--+--+--+--+ +--+ +--+--+ + + + + +--+--+ + +
| | |u | | | |ud|u | | | | | |ud|ud| | |
+ +--+--+--+--+--+ + +--+--+--+ + + + +--+--+--+ + +
| |u |u | | |ud|ud|u | | | | |ud|ud|ud| | |
+ +--+--+--+--+--+ + +--+--+--+ +--+ + +--+--+--+ +--+
| | u | | | |ud| | | |ud|
+ +--+--+--+--+--+ + +--+--+--+--+--+ +--+--+--+--+--+--+
| |u | | |ud| |u |ud|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|ud| d| |ud| | | d| |
+--+--+--+--+--+--+ +--+--+--+--+--+ + +--+--+--+--+--+ +
|ud| | | |ud| | | | d | |
+--+ +--+--+--+ + +--+ +--+--+--+ + +--+--+--+--+--+ +
| | |ud|ud|ud| | | | | d|ud|ud| | | d| d| |
+ + +--+--+--+ + + + +--+--+--+ + +--+--+--+--+--+ +
| | |ud|ud| | | | | | d|ud| | | | d| | |
+ + +--+--+ + + + + +--+--+ +--+ +--+--+--+--+ +--+
| | |ud|ud| | | | | | d|ud| | | | d| | |
+ + +--+--+ + + + + +--+--+ + + +--+--+--+--+ + +
| | d|ud|ud| | | | | | d|ud| | | | d| | |
+ +--+--+--+ +--+ + +--+--+--+ + + +--+--+--+--+ + +
| | | d| | | | | | | |
+--+--+--+--+ +--+ +--+--+--+--+ + + +--+--+--+--+--+ +
|ud| |ud| |ud| | d| | d| |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 4 level 5 level 6
+--+--+ +--+--+ +--+--+
|u |u | |ud|ud| | d| d|
+--+--+ +--+--+ + + +
| |u | | | d| | | |
+ +--+ + + + + + +
| | | | | | | | |
+ + + + + + + + +
| | | | | | | | |
+--+--+--+--+--+ + +--+ +--+--+--+--+--+ + +--+ +--+--+--+--+--+--+--+--+
|u | | |u | |ud| | |ud| | d | d|
+--+--+--+--+--+--+ +--+ +--+--+--+--+--+--+ +--+ +--+--+--+--+--+--+--+--+
|u |u | |u | |ud| d | |ud| | d | d|
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
levels 1, 2, 3 level 4 level 5
5x8x10
+--+--+--+--+--+--+--+--+--+--+
| | | | |
+ + + +--+--+--+--+ +
| | | | |
+ + + + +
| | | | |
+ + + +--+--+--+--+ +
| | | | | |
+ + +--+--+--+--+ + +
| | | | | |
+ +--+--+--+--+ + + +
| | | | |
+ + + + +
| | | | |
+ +--+--+--+--+ + + +
| | | | |
+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
also: 5x6x12 by Patrick Hamlyn
also: 5x8x12, 5x8x14
.............................................................................
*
*******
7x7x16
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| | | | | |u | | | | | d | | | | |u | | | |
+--+--+--+ +--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
|u |u |u | |u |u |u | | d| d| d|u | d| d| d| |u |u |u |ud|u |u |u | | d| d| d|ud| d| d| d|
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 level 2 level 3 level 4
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| d | | | | |u | | | | | d | | | | |u | | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
|u |u | ud|u |u |u | | d| d|u | d| d| d| d| |u |u | d|u |u |u |u | | d| d|u | d| d| d| d|
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 5 level 6 level 7 level 8
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| d | | | | |u | | | | | d | | | | | | | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + + + + + +
|u |u | d|u |u |u |u | | d| d|u | d| d| d| d| |u |u | d|u |u |u |u | | d| d|u |ud| d| d| d|
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 9 level 10 level 11 level 12
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
|u |u | d|ud|u |u |u | | d| d|u |ud| d| d| d| |u |u | d| d|u |u |u | | d| d| | d| d| d|
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ + + + +--+ + + + +--+--+--+--+--+--+--+ + + + +--+ + + +
| u | | | | | d| | | | | u | | | | | d| | | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 13 level 14 level 15 level 16
.............................................................................
* *
******
+--+--+--+--+--+ +--+--+--+--+--+
| |x |x |x | | |x |x |x |
+ +--+--+--+ +--+ +--+--+ +--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + +--+ + + +
|x |x | |x |x |x | |x |x | |x |x |x |
+--+--+ +--+--+--+ +--+--+ +--+--+--+
levels 1, 3, 4, 5 levels 2, 6
+--+--+ +--+--+ +--+--+ +--+--+
| |x | | |x | | x | | x |
+ +--+--+--+ +--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 3, 4, 5 levels 2, 6
+--+--+ +--+--+
|x |x | |x |x |
+--+--+--+--+--+--+--+ + + +--+--+--+--+--+
| |x | | | | | | | x |
+--+ +--+--+--+ +--+ +--+--+ + + +--+--+
|x | |x |x |x | |x | |x |x |x |x | x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 3, 4, 5 levels 2, 6
+--+--+ +--+--+
|x |x | |x |x |
+--+--+--+--+--+--+--+ +--+ + +--+--+--+--+
| |x | | | | | | | x |
+--+ +--+--+--+ +--+ + +--+--+ + +--+--+
|x | |x |x |x | |x | |x | x |x |x | x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 3, 4, 5 levels 2, 6
6x8x8 by Robert Reid
+--+--+--+--+--+--+--+--+
| | |
+ +--+--+--+ +--+--+ +
| | | | | |
+--+ +--+ + +
| | | |
+ +--+ + +
| | | |
+ +--+--+--+--+--+--+ +
| | | |
+ + +--+ +
| | | |
+ + +--+ +--+
| | | | | |
+ +--+--+ +--+--+--+ +
| | |
+--+--+--+--+--+--+--+--+
(6 levels deep)
6x10x16
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+--+ +--+--+--+--+ +--+--+ +
| | | | | | | | |
+--+--+ +--+--+ + +--+--+ + +
| | | | |
+ + + + +
| | | | |
+ +--+--+--+--+--+--+--+--+--+--+--+--+ +
| | | | |
+ + + + +
| | | | |
+ + +--+--+ + +--+--+ +--+--+
| | | | | | | | |
+ +--+--+ +--+--+--+--+ +--+--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
22*
***
+--+--+ +--+--+ +--+--+ +--+--+
|u u | | d ud| |u | d| | d |
+--+ + +--+ + + + + + +
|u |u | | d|ud| |u | d| | d |
+ +--+--+--+ + +--+--+--+ +--+--+--+--+ + +--+--+
|u | | | d |u u | | |ud ud| | | d d|
+ + + + +--+--+ + +--+ + +--+--+--+ +
|u | u u | | d | d d| |u u u |u | | d d d| d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
4x4x6 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+--+ +
| | |
+ + +
| | |
+--+--+--+--+--+--+
(4 levels deep)
same octacube, allowing reflections
2x3x4 (minimal) by Andrew Clarke
+--+--+--+--+ +--+--+--+--+
|u |u u u | | d| d d d|
+ +--+ + + +--+--+--+
|u | | | d| |
+ +--+--+ + + +
|u |u u | | d| d d|
+--+--+--+--+ +--+--+--+--+
level 1 level 2
.............................................................................
2 2
****
2x4x4 minimal
+--+--+--+--+ +--+--+--+--+
| u |u | | | d| d| |
+--+--+--+--+ + + + + +
|u u | | d| | | d|
+--+--+--+--+ +--+ + +--+
|u u | | d| | | d|
+--+--+--+--+ + + + + +
| u |u | | | d| d| |
+--+--+--+--+ +--+--+--+--+
level 1 level 2
4x4x4 which can be physically assembled in 3d
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u | u | | d|u | d| |u | d|u |u | | d| | d| d|
+--+ + +--+ +--+--+--+--+ +--+ + +--+ + +--+--+ +
| | | | | |u u | | d| | | d| | |
+ + + + + +--+--+--+--+ + + + + + +--+--+--+--+
|u | | |u | | d d| | | | | | | |
+--+ + +--+ +--+--+--+--+ +--+ + +--+ + +--+--+ +
|u | u | | d|u | d| |u | d|u |u | | d| | d| d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
also: 4x4x5, 4x4x6, 4x4x7 which are physically constructible
.............................................................................
**2
**
**
allowing reflections
3x4x8
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|u |u u |u |u |u u | | d|ud ud|ud| d|ud ud|u | | d d| d | d d| d|
+ +--+--+ + +--+--+ + +--+--+--+ +--+--+--+ + + +--+--+--+--+--+--+--+
| |u | | | | | d|u | | | | | d| |
+ + + +--+ + + + + + +--+ + + +
| |u | | | | | d|u | | | | | d| |
+--+--+--+--+--+--+--+ + + +--+--+--+ +--+--+--+ + +--+--+ + +--+--+ +
|u |u u | u |u u | | d|ud ud|u |ud|ud ud|u | | d d| d| d| d d| d|
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 level 2 level 3
4x4x6
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
| u |u | |u | d|ud| |ud|u | d| | d| d |
+ +--+ + +--+--+--+ + + +--+--+--+ + +--+ +
| |u |u | | u |ud| d| |u |ud| d | | d| d| |
+ + + + +--+--+ + + + + +--+--+ + + + +
| |u |u | | | d| d| |u |u | | | d| d| |
+--+--+ +--+ + + +--+ +--+ + + +--+ +--+--+
| |u |u | | | d| d| |u |u | | | d| d| |
+ +--+ + + +--+ + + +--+ + + +--+ +
| |u |u | | d| d| |u |u | | d| d| |
+ +--+ + +--+--+--+ + + +--+--+--+ + +--+ +
| u |u | |u | d|ud| |ud|u | d| | d| d |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
also: 3x4x12
.............................................................................
Nonacubes
.............................................................................
****
*
****
+--+--+--+--+--+--+--+--+--+--+--+
| | x | | |x |
+ +--+--+--+--+ + +--+--+--+--+
| | | |
+ +--+--+--+--+ + +--+--+--+--+
| | x | | |x |
+ +--+--+--+--+ +--+--+--+--+--+
|x | |x |
+--+ +--+
levels 1, 3 level 2
3x4x6 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+
| | |
+ +--+--+--+--+ +
| | |
+ +--+--+--+--+ +
| | |
+ +--+--+--+--+ +
| | |
+--+--+--+--+--+--+
(3 levels deep)
also: 4x5x9
.............................................................................
*
*
*******
3x7x9 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
|u | | | | d |
+--+ +--+--+--+--+ + +--+--+--+--+--+--+--+
|u | | | | | | d |
+--+ + +--+--+ + + +--+--+--+--+--+--+--+
|u | | | | | | | | d |
+--+ + + + + + + +--+--+--+--+--+--+--+
|u | | | | | | | | d |
+--+ + + + + + + +--+--+--+--+--+--+--+
|u | | | | | | | | d |
+--+ + + + + + + +--+--+--+--+--+--+--+
|u | | | | | | | | d |
+--+ + + + + + + +--+--+--+--+--+--+--+
|u | | | | | | | | d |
+--+--+ + + + +--+ +--+--+--+--+--+--+--+
| | | | |u | | d|
+--+--+--+ + +--+--+ +--+--+--+--+--+--+--+
|u | | | | d |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 2 level 3
.............................................................................
***
******
+--+--+--+ +--+--+--+
|x |x |x | |x |x |x |
+--+--+--+ + + + +
| | | | | |
+--+ +--+--+ +--+--+--+--+--+--+
| | | | | | | | |
+--+--+--+--+--+--+ + + + + + + +
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 5, 6 levels 2, 3, 4
6x6x9
+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+--+ +
| | | |
+ +--+--+--+ +
| | | |
+ + + +
| | | |
+ +--+ +--+--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+
(6 levels deep)
.............................................................................
** **
*****
+--+ +--+
|u | | d|
+ + + +
|u | | d|
+--+--+--+--+--+ +--+--+ +--+--+
|u u u u | | d d| | d d|
+--+--+--+--+--+ +--+--+ +--+--+
|u | | d|
+ + + +
|u | | d|
+--+ +--+
level 1 level 2
+--+--+--+--+ +--+--+--+--+
|x | x | |x | |x |
+--+--+--+--+ +--+ +--+
|x | x | |x | |x |
+--+--+--+--+ +--+ +--+--+
|x | |x | |
+--+--+--+--+ +--+ +--+--+
|x | x | |x | |x |
+--+--+--+--+ +--+ +--+
|x | x | |x | |x |
+--+--+--+--+ +--+--+--+--+
levels 1, 2, 4, 5 level 3
5x9x10
+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+--+ +--+--+ +
| | | |
+ +--+--+ +--+--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+
(10 levels deep)
.............................................................................
*** *
*****
+--+--+--+ +--+--+--+
|x |x |x | |x |x |x |
+ + + +--+--+ +--+--+--+--+--+
| | | |x | | |x | |
+--+--+--+--+--+ + +--+ +
| | | | | | | |
+ + + + + + +--+--+--+--+--+
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2, 3, 5 level 4
+--+--+--+ +--+--+--+
|x |x |x | |x |x |x |
+ + + +--+--+--+ +--+--+--+--+--+--+
| | | |x | x | | |x | |x |
+--+--+--+ +--+--+ + +--+ +--+
| | | | | x | | |x |
+ + + +--+--+--+ +--+--+--+--+--+--+
|x |x |x | |x |x |x |
+--+--+--+ +--+--+--+
levels 1, 2, 3, 5 level 4
5x9x10
+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+ +--+--+--+
| | | |
+ +--+--+--+--+ +
| | |
+ + +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
.............................................................................
* *
*******
+--+--+ +--+--+
|x |x | |x |x |
+ + +--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| | | | | | | x | | |x |
+--+--+ + + + +--+--+ +--+ +--+--+--+--+ +--+
|x |x |x |x |x | x | |x | |x |x |x |x | |x |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
levels 1, 6 levels 2, 3, 4, 5, 7
+--+--+ +--+--+ +--+--+ +--+--+
| x | |x | | |x | |x | |
+--+--+--+--+--+--+--+--+ +--+--+ +--+--+--+--+ +
|x | | | | | | | |x | |
+--+--+ + + + + + + +--+--+--+--+--+--+--+--+
|x |x |x |x |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 6 levels 2, 3, 4, 5, 7
7x8x9
+--+--+--+--+--+--+--+--+--+
| | | | |
+ + + + +
| | | | |
+ +--+ +--+ + +
| | | | |
+ + + + +
| | | | |
+ + +--+ + +
| | | | |
+ + + + +
| | | | |
+ + +--+ +--+ +
| | | | |
+ + + + +
| | | | |
+--+--+--+--+--+--+--+--+--+
(7 levels deep)
.............................................................................
**
*******
+--+--+--+--+ +--+--+--+--+
|x | |x | |x | x |
+--+ +--+--+--+--+ +--+--+--+--+--+--+--+
| | | | | | | | | |
+--+--+--+--+--+--+--+ + + + + + + + +
|x |x |x |x |x |x |x | |x |x |x |x |x |x |x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 4, 5, 6, 7 levels 2, 3
+--+--+ +--+--+
|x |x | |x |x |
+--+--+--+--+--+--+ +--+--+--+--+ + +
|x |x |x |x | | |x |x |x |x | | |
+--+--+--+--+ +--+ + + + + +--+--+--+
| | | | | | | | | |
+--+--+--+--+--+--+--+ +--+--+--+--+ + + +
|x |x |x | |x |x |x |
+--+--+--+ +--+--+--+
levels 1, 4, 5, 6, 7 levels 2, 3
7x12x12
+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ + +--+--+--+ +
| | | | |
+ + + +--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + + +--+--+--+--+
| | | | | |
+ + +--+--+--+--+ + +
| | | | | |
+--+--+--+--+ + + +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+ + + +
| | | | |
+ +--+--+--+ + +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+
(7 levels deep)
also: 7x12x15, 7x12x18, 7x12x21
.............................................................................
* *
*******
+--+--+ +--+--+ +--+--+ +--+--+
| |x | | |x | | x | | x |
+ +--+--+--+ +--+--+ +--+--+--+--+--+--+--+
| | | | | | | | | |
+--+--+--+--+--+--+--+ + + + + + + + +
|x |x |x |x |x |x |x | |x |x |x |x |x |x |x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 2, 4, 5, 6 levels 3, 7
+--+ +--+
|x | |x |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+ +
|x |x | |x |x |x | | |x |x | x |x |x | |
+--+--+ +--+--+--+ + + + +--+--+ + +--+
| | | | | | | | | |
+--+--+--+--+--+--+--+ +--+--+ + +--+--+ +
|x |x | |x | |x |x | |x |
+--+--+ +--+ +--+--+ +--+
levels 1, 2, 4, 5, 6 levels 3, 7
7x12x12
+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+ +--+ +--+--+ +
| | | | | | |
+ + + +--+--+ +--+--+
| | | |
+ +--+ + +
| | | |
+ + + +--+--+--+--+--+--+
| | | | | |
+ +--+ +--+--+ +--+ +
| | | | | |
+--+--+--+--+--+--+ + + +
| | | |
+ + +--+ +
| | | |
+--+--+ +--+--+ + + +
| | | | | | |
+ +--+--+ +--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+
(7 levels deep)
.............................................................................
22*
22
4x6x6 minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u u |u u |u u | | d d|ud d| d d| |u u | d| |u u | | d d | d d|
+ + + + + + + + + +--+ + + + +--+--+ +
|u u |u u |u u | | d d| d d| d d| |u u |u u |u u | | d d| d d| d d|
+ +--+--+--+--+ + +--+--+--+--+--+--+ +--+--+ +--+--+ +--+--+ +--+--+
| |u u |u u | | |u | d d| d d | |ud u |u u |u u | | d d| d d| d d|
+--+ + +--+ +--+ + +--+ + +--+--+ + + +--+--+ +
| |u u |u u | | | d d| d d|u | |u u |u u |u ud| | d d| d d| d d|
+ +--+--+--+--+ + +--+--+--+--+--+--+ +--+--+ +--+--+ +--+--+ +--+--+
|u u |u u |u u | | d d| d d| d d| |u u |u u |u u | | d d| d d| d d|
+ + + + + + + + + + +--+ + + +--+--+ +
|u u |u u |u u | | d d| d ud| d d| |u u | | d|u u | | d d | d d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*2**
****
allowing reflections
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
|u |u u | | d|ud ud| |u |ud ud| | d| d d|
+ +--+--+ + + +--+ + +--+--+ + +--+--+
|u | | d| | |u | | | d |
+ +--+--+ + +--+--+ + + +--+ + +--+--+
|u |u u | | d|ud ud| |u |ud ud| | d| d d|
+ +--+--+ + +--+--+ + +--+--+ + +--+--+
|u | | d| |u | | d|
+--+ +--+ +--+ +--+
level 1 level 2 level 3 level 4
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
|u |u u | | d|ud ud| |u |ud ud| | d| d d|
+ +--+--+ + + +--+ + +--+--+ + +--+--+
|u |u | | d| |ud| |u | ud| | d | d|
+ +--+ + + +--+ + + +--+ + + +--+ +
|u | |u | | d| |ud| |u | |ud| | d| | d|
+ + +--+ + + +--+ + + +--+ + + +--+
|u | | d| |u | | d|
+--+ +--+ +--+ +--+
level 1 level 2 level 3 level 4
4x9x12
+--+--+--+--+--+--+--+--+--+--+--+--+
| | | | |
+--+ + + +--+--+ +
| | | | | |
+ + +--+ + +--+ + +--+ +
| | | | | | | | | |
+ +--+ +--+ + +--+--+--+--+ +
| | | | | | |
+ + +--+ + +--+--+--+
| | | | |
+--+--+--+ + +--+ + +
| | | | | | |
+ +--+--+--+--+ + +--+ +--+ +
| | | | | | | | | |
+ +--+ + +--+ + +--+ + +
| | | | | |
+ +--+--+ + + +--+
| | | | |
+--+--+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
also: 4x12x12, 4x12x15
.............................................................................
Dekacubes
.............................................................................
***
***
***
*
+--+--+--+ +--+--+--+
|u |u |u | | d| d| d|
+--+--+--+ + + + +
| | | | | |
+ + + + + +
| | | | | |
+--+ +--+--+ +--+--+--+--+--+--+
|u | |u | | d | d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 levels 2, 3, 4
+--+--+--+ +--+--+--+ +--+--+--+
| | |u |u |u | | d| d| d|
+ +--+ + + + +--+ + + + +--+
| |u | |u |u |u |ud| | d| d| d| d|
+ +--+ + + + + + + + + + +
| | |u |u |u |u | | d| d| d| d|
+--+--+--+--+ +--+--+--+ + + + + + +
|u u u | | d d d|u | | | | | d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 levels 2, 3 level 4
4x4x10 (confirmed minimal by Patrick Hamlyn)
+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ + + +
| | | |
+ +--+ +--+--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
also: 4x7x10
.............................................................................
*
*****
****
4x5x9 minimal
+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+
| |u | u u |u | u u | | |ud u |ud d|ud u |ud d| | d d| d| d d| d| |
+ +--+--+--+--+--+--+--+--+ + +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +
| | | | | | | | |
+ +--+ + + +--+ + + +--+ +
| | | | | | | | |
+--+--+--+--+--+--+--+--+ + +--+--+--+--+--+--+--+--+ + + +--+--+--+--+--+--+--+--+
|u u |u |u u |u | | | d ud|u ud| d ud|u ud| | | | d| d d | d| d d |
+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+
level 1 levels 2, 3, 4 level 5
also 1x30x46 by William Rex Marshall, many others
.............................................................................
**
**
******
+--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+
|u |u | |ud|ud| | d| d| |u |u | |ud|ud| | d| d|
+ + + +--+--+ +--+--+ +--+--+ +--+--+ + + +
| | | | | | |u |u | |ud|ud| | d| d| | | |
+ + +--+ + + +--+ +--+--+--+ +--+--+--+ + + +--+ + + +--+
| | |u | | | |ud| | d| | u | | | |ud| | | | d|
+--+--+--+ + + +--+ +--+--+--+ +--+--+--+ + + +--+ +--+--+--+
| u | |u |u |ud| |ud|ud| d| | d| d|u | | | |ud| | d|
+--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+ +--+--+--+
levels 1, 2 levels 3, 4 levels 5, 6 levels 7, 8 levels 9, 10 levels 11, 12
4x5x12 co-minimal
+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+
(12 levels deep)
4x6x10 co-minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| | |u |u |u |u | |u u | d| d| d| d| | d d |
+ + + + + + + +--+--+ + + + + +--+--+--+--+--+--+
| | |u |u |u |u | |u u | d| d| d| d| | d d |
+ + + + + + + +--+--+--+--+--+--+ +--+--+--+--+--+--+
| | | | | | | | | |u u | | d d|
+ + + + + + + + + +--+--+ +--+--+--+--+--+--+
| | | | | | | | | |u u | | d d|
+ + + + + + + +--+--+ + +--+--+ +--+--+--+--+--+--+
|u |u | | | | | | d| d| | |u |u | | | | | | d| d|
+ + + + + + + + + + + + + + + + + + + + +
|u |u | | | | | | d| d| | |u |u | | | | | | d| d|
+--+--+--+--+--+--+ +--+--+ + +--+--+ + + + + + + +
|u u | | d d| | | | | | | | | |
+--+--+--+--+--+--+ +--+--+ + + + + + + + + +
|u u | | d d| | | | | | | | | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ + + + + + + +
| u u | |u |u |u |u | d d| | d| d| d| d| | |
+--+--+--+--+--+--+ + + + + +--+--+ + + + + + + +
| u u | |u |u |u |u | d d| | d| d| d| d| | |
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 levels 2, 3 level 4
5x6x8 co-minimal
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
|u | | | |ud|u | | | d|ud| | | | d| |
+--+--+--+ + + +--+--+--+--+ + +--+--+ + + +--+ +
|u | | | |ud|u | | | d|ud| | | | d| |
+--+--+--+ + + +--+--+--+--+ + +--+--+--+--+ + + +--+--+--+ +
| u |u | | |u |u |ud|ud| | |ud|ud| d| d| | | d| d | |
+--+--+--+--+ + +--+--+--+--+ + +--+--+--+--+ + +--+--+--+--+ +
|u |u | | |ud|ud|u |u | | | d| d|ud|ud| | | d| d| |
+ +--+--+--+ + +--+--+--+--+ + +--+--+--+--+ + +--+--+--+ + +
| |u | | |u |ud| | |ud| d | | | d | | |
+ +--+ + +--+--+ + +--+--+--+--+ + +--+--+--+ + +
| |u | | |u |ud| | |ud| d | | | d | | |
+--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2 levels 3, 4 levels 5, 6 levels 7, 8
.............................................................................
**
****
**
**
+--+--+ +--+--+
|x |x | |x |x |
+ + + + + +
|x |x | |x |x |
+--+--+ +--+--+--+ +--+--+ +--+ + +
|x |x | |x | | |x |x | |x | | |
+ + + + + + + + + + + + +
|x |x | |x | | |x |x | |x | | |
+--+--+--+--+ + + + +--+ +--+--+
| | | | | | | | | |
+ +--+--+ + + + + + + + +
| | | | | | | | | |
+ +--+--+--+--+ +--+--+ +--+ + +
| |x | |x |x | | | |x | |x |x |
+ + + + + + + + + + + + +
| |x | |x |x | | | |x | |x |x |
+--+--+--+ +--+--+ + + +--+ +--+--+
|x |x | |x |x |
+ + + + + +
|x |x | |x |x |
+--+--+ +--+--+
levels 1, 2, 4 level 3
4x6x10 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+
| | |
+ + +
| | |
+--+--+ +--+ +
| | | |
+ + + +
| | | |
+ +--+ +
| |
+ +
| |
+ +--+ +
| | | |
+ + + +
| | | |
+ +--+ +--+--+
| | |
+ + +
| | |
+--+--+--+--+--+--+
(4 levels deep)
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|x |x |x |x | |x x | |x |x |x |x | x x |
+ + + + + +--+--+ + + + + +--+--+--+--+
|x |x |x |x | |x x | |x |x |x |x | x x |
+--+--+--+--+ +--+--+ + + + + +--+--+--+--+
| | |x x | | | | | | x x |
+ +--+--+ +--+--+ + + + + +--+--+--+--+
| | |x x | | | | | | x x |
+ +--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
| | | | |
+ + + + +
| | | | |
+--+--+ + + +
|x x | |x |x |
+--+--+ + + +
|x x | |x |x |
+--+--+ +--+--+
levels 1, 2, 4 level 3
4x8x10
+--+--+--+--+--+--+--+--+--+--+
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+ +--+--+--+--+--+--+ +
| | |
+ + +
| | |
+ + +
| | |
+ + +
| | |
+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
also: 1x30x32 by William Rex Marshall
also: 6x6x10, many others
.............................................................................
* * *
*******
+--+--+--+--+ +--+--+--+--+
|x |x |x |x | |x |x |x |x |
+ + + + +--+--+--+--+ +--+--+--+--+--+--+--+--+
| | | | | | | x | | |x |
+--+--+--+--+ + +--+--+ + +--+--+ +--+--+ +--+
| x |x |x |x | x | | |x |x | |x |x | |x |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
levels 1, 4, 7 levels 2, 3, 5, 6
5x7x8 minimal
+--+--+--+--+--+--+--+--+
| |
+ +
| |
+--+--+--+--+ +
| | |
+ +--+--+--+--+
| |
+ +
| |
+--+--+--+--+--+--+--+--+
8 levels deep
also: 4x7x15
.............................................................................
2**2
****
4x5x6 minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u u |u u | |ud ud|ud ud|ud ud| |ud ud|ud ud|ud ud| | d d| d d| d d|
+--+ + +--+ +--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+ + +--+ +--+
|u | | | | |u | |ud|u u u u |ud| |ud| d d d d|ud| | d| | | | | d|
+ +--+--+ +--+ + + + +--+--+ + + + +--+--+--+--+ + + +--+--+ +--+ +
|u |u u | u | |ud| |ud ud| |ud| |ud| |ud ud| |ud| | d | d d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+ +--+--+ +--+ +--+--+--+--+--+--+
|u | | | u | |ud|u u u u |ud| |ud| d d d d|ud| | d | | | d|
+ +--+ + +--+ + + +--+--+--+--+ + + +--+--+--+--+ + + +--+ + +--+ +
|u |u u |u u |u | |ud|ud ud|ud ud|ud| |ud|ud ud|ud ud|ud| | d| d d| d d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 level 2 level 3 level 4
.............................................................................
*22*
****
+--+ +--+ +--+ +--+ +--+ +--+ +--+ +--+
|u | |u | |ud| |ud| |ud| |ud| | d| | d|
+ +--+--+ + + +--+--+ + + +--+--+ + + +--+--+ +
|u | |u | |ud u |u ud| |ud d| d ud| | d| | d|
+--+ +--+ +--+--+--+--+ +--+--+--+--+ +--+ +--+
|u u u u | | d d d d| |u u u u | | d d d d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u u |u u | |ud ud|ud ud| |ud ud|ud ud| | d d| d d|
+--+--+--+--+--+--+ +--+ +--+--+ +--+ +--+ +--+--+ +--+ +--+--+--+--+--+--+
|u | u u |u | |ud|u | d d|u |ud| |ud| d|u u | d|ud| | d| d d | d|
+ + + + + +--+--+--+--+ + + +--+--+--+--+ + + + + +
|u | |u | |ud u | |u ud| |ud d| | d ud| | d| | d|
+--+--+--+--+--+--+ +--+--+ +--+--+ +--+--+ +--+--+ +--+--+--+--+--+--+
|u u u u | | d d d d| |u u u u | | d d d d|
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3 level 4
4x4x10 minimal
+--+--+--+--+--+--+--+--+--+--+
| | | |
+ +--+ +--+ +
| | | |
+ + + +
| | | |
+ +--+ +--+ +
| | | |
+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
also: 4x6x10
.............................................................................
2**
2**
2
allowing reflections
+--+--+--+ +--+--+--+ +--+--+--+
|u u u | | u | | | d|
+--+--+ + +--+--+ + + + +
|u |u |u | |u | |u | | d | d|
+ + +--+ + + +--+ + +--+
|u |u | |u | | | d |
+--+ + +--+ + +--+--+
|u u | |u u | | d d|
+--+--+ +--+--+ +--+--+
level 1 level 2 level 3
3x4x5 minimal
+--+--+--+--+--+
| | |
+ + +
| | |
+ +--+ +
| | |
+ + +
| | |
+--+--+--+--+--+
(3 levels deep)
.............................................................................
11-cubes
.............................................................................
***
****
****
4x6x11 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u u | | |ud ud u | | | d d d| |
+--+--+ + +--+--+--+ + +--+--+--+ +
| | | |u |u | | | d| d| |
+ +--+ + + + + + + + + +
| | | |u |u | | | d| d| |
+ +--+--+--+ + + +--+--+--+--+ + + +--+--+--+--+
| | | |u |u | | | d| d| |
+ + + + + + + +--+--+ +
| | | | | | | | | |
+--+--+--+ + +--+--+--+ + + +--+ +
| | | | | | | | |
+ +--+ + + +--+--+--+ + +--+--+--+
| | | | | | | | | |
+ +--+--+ + + + + + + +
| |u |u | | | d| d| | | |
+--+--+--+--+ + + +--+--+--+--+ + + +--+--+--+ +
| |u |u | | | d| d| | | |
+ + + + + + + + + +--+ +
| |u |u | | | d| d| | | |
+ +--+--+--+ + +--+--+--+ + +--+--+
| |u u u | | | d ud ud| | | d d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 levels 2, 3 level 4
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u |u u u | |ud|ud|ud|ud ud ud| | | | | d ud ud| | | d d|
+ + + +--+--+--+ + + + +--+--+--+ + + + +--+--+--+ + +--+--+
|u |u |u |u u u | |ud|ud|ud|ud ud ud| | | | | d ud ud| | | d d|
+ + + +--+--+--+ + + + +--+--+--+ + + + +--+--+--+ + +--+--+--+
|u |u |u | |ud|ud|ud| | | | | | |
+ + + + + + + + +--+--+--+ +--+--+--+
|u |u |u | | d| d| d| |u |u |u | | d| d| d|
+--+--+--+ +--+--+--+ + + + + + + + +
| | |u |u |u | |ud|ud|ud| | d| d| d|
+ + + + + + + + + + + + + +
| | |u |u |u | |ud|ud|ud| | d| d| d|
+ + + + + + + + + + + + + +
| | |u |u |u | |ud|ud|ud| | d| d| d|
+ +--+ +--+--+--+ +--+--+--+ +--+--+--+
| | |u |u | |ud|ud| | d| d|
+--+--+ + + + + + + + + +
|u |u | |ud|ud| |ud|ud| | d| d|
+ + + + + + + + + + + +
|u |u | |ud|ud| |ud|ud| | d| d|
+--+--+ +--+--+ +--+--+ +--+--+
level 1 level 2 level 3 level 4
4x10x11
+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+--+--+ + +
| | | | |
+ + + +--+ +
| | | | |
+ + +--+--+ +
| | | | |
+ + +--+ + +
| | | | |
+ +--+--+ + +
| | | | |
+ +--+ + + +
| | | | |
+ + +--+--+--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+
(4 levels deep)
also: 1x22x25 (by William Rex Marshall and Ingo Wrede, independently)
also: 4x7x22, many others
.............................................................................
***
***
***
*
*
5x6x11 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
| |u u u | |u | d d d| | d| |
+ +--+--+--+ + +--+--+--+--+--+ + +--+--+ +
| | | |u |u |u | | | d| d| d| |
+ + + + + + + + + + + + +
| | | |u |u |u | | | d| d| d| |
+ +--+--+ + + + + + + +--+ + +--+--+--+
| |u |u | | | |ud|ud| | | | d| d| |
+ +--+--+--+--+ + + +--+--+--+--+ + + +--+--+ +
| |u |u |u |u | | | |ud|ud|ud|ud| | | | d| d| |
+--+ + + + + + +--+ + + + + + + +--+--+ +
| |u |u |u |u | | | |ud|ud|ud|ud| | | | |
+ + + + + +--+ + + + + + +--+ + +--+--+ +
| |u |u |u |u | | | |ud|ud|ud|ud| | | | d| d| |
+ +--+--+--+--+ + + +--+--+--+--+ + + +--+--+ +
| |u |u | | | |ud|ud| | | | d| d| |
+ +--+--+ + + + + + + +--+--+--+ + +--+
| | | | |u |u |u | | | d| d| d|
+ + + + + + + + + + + + +
| | | | |u |u |u | | | d| d| d|
+--+--+--+ + +--+--+--+--+--+ + + +--+--+ +
|u u u | | | d d d |u | | | d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2 level 3 levels 4, 5
also: 1x54x55, many others
.............................................................................
*
*****
*****
3x10x11 co-minimal
+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+
| |u |u |u |u | |u | | | d| d| d| d| | d|
+ +--+--+--+--+ +--+ + + + + + + +--+--+--+--+--+
| | |u | | | | | | | | d|
+ + +--+--+--+--+--+ + + + + + + +--+--+--+--+--+
| | | | |u | | | | | | | | d|
+--+--+--+--+--+--+ + +--+ + + + + + + +--+--+--+--+--+
| | | | |u | | | | | | | | d|
+--+ + +--+ + +--+ + + + + + + +--+--+--+--+--+
|u | | |u | | |u | | d| | | | | d| d|
+--+ + +--+ + +--+ +--+--+--+--+--+--+--+--+--+--+--+
|u | | |u | | |u | | d | d| | | | | d|
+--+ + +--+ + +--+ +--+--+--+--+--+ + + + + + +
|u | | | | | | d | | | | | | |
+--+ + +--+--+--+--+--+--+ +--+--+--+--+--+ + + + + + +
|u | | | | | | d | | | | | | |
+--+--+--+--+--+ + + +--+--+--+--+--+ + + + + + +
|u | | | | d | | | | | | |
+--+ +--+--+--+--+ + +--+--+--+--+--+ + + + + + +
|u | |u |u |u |u | | | d | | d| d| d| d| |
+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+
level 1 levels 2, 3
5x6x11 co-minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u |u |u | | | d| d| d| d| d|u | | | d|
+ + + + + + + +--+--+--+--+--+--+ + +--+--+--+--+--+
| | | | | | | | |u | | | d|
+ + + + + + + + +--+ + +--+--+--+--+--+
| | | | | | | | |u | | | d|
+ + + + + + + + +--+--+--+--+--+ + +--+--+--+--+--+
| | | | | | | | | | | | | |
+ + + + + + + +--+ + +--+ + + +
| | | | | |u | |u | | | d| | d| |
+--+--+--+--+--+--+ +--+ + +--+ +--+--+--+--+--+ +
| | u | |u | | | d| | d | |
+ +--+--+--+--+--+ +--+ + +--+ +--+--+--+--+--+--+
| |u | |u | | | d| | d| | | | | |
+ + + +--+ + +--+ + + + + + + +
| | | | | | | | | | | | | |
+--+--+--+--+--+ + +--+--+--+--+--+ + + + + + + + +
|u | | | d| | | | | | | | |
+--+--+--+--+--+ + +--+ + + + + + + + +
|u | | | d| | | | | | | | |
+--+--+--+--+--+ + +--+--+--+--+--+--+ + + + + + + +
|u | | | d|u |u |u |u |u | | | d| d| d| d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
levels 1, 2 level 3 levels 4, 5
.............................................................................
12-cubes
.............................................................................
**
****
******
3x8x10 minimal
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
|u |u | | | d| d| | | | | |
+ + +--+--+ + + + +--+--+ + + +--+--+ + +
|u |u | | | | d| d| | | | | | | |
+ + + + +--+--+ + + + + +--+--+ +--+--+ + + + +
|u |u | | |u |u | |ud|ud| | |ud|ud| | d| d| | | d| d|
+ + + +--+--+ + + + + + +--+--+ + + + + +--+--+ + + +
|u |u | | |u |u | |ud|ud| | |ud|ud| | d| d| | | d| d|
+ + + + +--+--+ +--+--+ + + + + +--+--+ + + + +
| | | | | | | |u |u | | |u |u | | d| d| | | d| d|
+ + + +--+--+ + + + + + + +--+--+ + + + + + + +--+--+ + + +
| | | | | | | |u |u | | |u |u | | d| d| | | d| d|
+--+--+ + + + + + + + + +--+--+ + + + + +--+--+
|u |u | | |u |u | |ud|ud| | |ud|ud| | d| d| | | d| d|
+ + +--+--+ + + + + + +--+--+ + + + + + + +--+--+ + +
|u |u | | |u |u | |ud|ud| | |ud|ud| | d| d| | | d| d|
+--+--+ + + + + +--+--+ + + + + + + + + +--+--+
| | |u |u | | | | d| d| | | | | |
+ +--+--+ + + + +--+--+ + + + + +--+--+ +
| |u |u | | | d| d| | | | |
+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+
level 1 level 2 level 3
also: 1x45x48, many others
.............................................................................
2222
**
**
3x4x4 minimal
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
|u |u u |u | | d|ud ud| d| | d d |
+ +--+--+ + + +--+--+ + + +
|u |u | | d| d| | |
+ + + + + + +--+--+--+--+
|u |u | | d| d| | |
+ +--+--+ + + +--+--+ + + +
|u |u u |u | | d|ud ud| d| | d d |
+--+--+--+--+ +--+--+--+--+ +--+--+--+--+
level 1 level 2 level 3
.............................................................................
13-cubes
.............................................................................
****
****
*****
6x6x13 minimal
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u |u |u |u |u |u | | d| d| d| d| d| d| | | |
+--+--+--+--+--+--+ + + + + + + + + + +
| | | | | | | | | | | | |
+ + + + + + + + + + + + +
| | | | | | | | | | | | |
+ + + +--+--+--+--+--+--+ + + +
| | | | | | | | | | | | |
+ + + + + + + + + + + +--+--+--+--+ +
| | | | | | | | | | | | | |
+ +--+--+--+--+ + + + + + + + + +--+ +--+
| |u |u |u |u | | |u | d| d| d| d|u | | d| | d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+ +--+
| |u | |u | d| | d| |
+--+ +--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
|u | |u | | d|u |u |u |u | d| | | d| d| d| d| |
+--+ +--+ + + + + + + + + +--+--+--+--+ +
| | | | | | | | | | | | | |
+ +--+--+--+--+ + + + + + + + + + + +
| | | | | | | | | | | | |
+ + + +--+--+--+--+--+--+ + + +
| | | | | | | | | | | | |
+ + + + + + + + + + + + +
| | | | | | | | | | | | |
+ + + + + + + + + + +--+--+--+--+--+--+
| | | |u |u |u |u |u |u | | d| d| d| d| d| d|
+--+--+--+--+--+--+ +--+--+--+--+--+--+ +--+--+--+--+--+--+
level 1 levels 2, 3, 4, 5 level 6
.............................................................................
14-cubes
.............................................................................
**
**
*****
*****
+--+--+--+--+ +--+--+--+--+
|x x | | |x x |
+--+--+ + +--+--+--+--+
|x x | | |x x |
+--+--+ +--+ +--+--+--+--+--+
| | | | | | | |
+ + + + + + + +
| | | | | | | |
+--+--+--+--+--+ + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+ + + + + + + + + + + +
|x |x |x |x |x | |x |x |x |x |x |
+--+--+--+--+--+ +--+--+--+--+--+
levels 1, 2, 5 levels 3, 4
5x6x14 minimal
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+ +--+--+ +
| | | | |
+ + + + +
| | | | |
+ +--+--+ +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(5 levels deep)
also: 1x63x80, many others
.............................................................................
******
****
****
4x7x12 (minimal) by Patrick Hamlyn
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u u u u |u | |u | | d d d d| d| | d| | |
+--+--+--+--+--+--+ + + + +--+--+--+--+ + + + + +
| u u u u |u | |u | | d d d d| d| | d| | |
+--+--+--+--+--+--+ + + + +--+--+--+--+ + + + + +
| | |u |u |u | |u | | d| d| d| | d| | |
+ + + + + + + + + + + + + + + +
| | |u |u |u | |u | | d| d| d| | d| | |
+ + + + + + +--+ + + +--+ + +--+--+ + +--+--+
|u | |u |u | | | d| | d| d|u | | | | | | | | d|
+ + + + + + + + + + + + + + + + + + + +
|u | |u |u | | | d| | d| d|u | | | | | | | | d|
+ + +--+--+ + +--+ + +--+--+--+--+--+ + +--+ + +--+--+ + +
|u | | | | | | | | d|u |u | |u | | | d| d| | d|
+ + + + + + + + + + + + + + + + + + + +
|u | | | | | | | | d|u |u | |u | | | d| d| | d|
+--+--+ + +--+--+ + +--+ + + +--+ + + + + + +
| | |u | |u |u |u | | d| | d| d| d| | |
+ + + + + + + + + + + + + + + +
| | |u | |u |u |u | | d| | d| d| d| | |
+ + + + + +--+--+--+--+ + + + +--+--+--+--+--+--+
| | |u | |u |u u u u | | d| | d| d d d d |
+ + + + + +--+--+--+--+ + + + +--+--+--+--+--+--+
| | |u | |u |u u u u | | d| | d| d d d d |
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
level 1 levels 2, 3 level 4
also: 1x66x84
.............................................................................
15-cubes
.............................................................................
*
*******
*******
3x7x15 minimal
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
| u | | | | d|
+--+--+--+--+--+--+--+ +--+ + +--+--+
|u | | d| | | |
+--+--+--+--+--+--+--+ +--+ + + +
|u | | d| | | |
+--+--+--+--+--+--+--+ +--+ + + +
|u | | d| | | |
+--+--+--+--+--+--+--+ +--+ + + +
|u | | d| | | |
+--+--+--+--+--+--+--+ +--+ + + +
|u | | d| | | |
+--+--+--+--+--+--+--+ +--+ + + +
|u | | d| | | |
+--+--+--+--+--+--+--+ +--+--+--+--+--+ +
|u | | d| | |
+--+--+--+--+--+--+--+ +--+--+ +--+--+--+
| u | | | | | d|
+--+--+--+--+--+--+--+ + + + +--+
| u | | | | | d|
+--+--+--+--+--+--+--+ + + + +--+
| u | | | | | d|
+--+--+--+--+--+--+--+ + + + +--+
| u | | | | | d|
+--+--+--+--+--+--+--+ + + + +--+
| u | | | | | d|
+--+--+--+--+--+--+--+ + + + +--+
| u | | | | | d|
+--+--+--+--+--+--+--+ + +--+--+ +--+
| u | | | | d|
+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+
levels 1, 2 level 3
also: 1x30x30 by William Rex Marshall
.............................................................................
16-cubes
.............................................................................
* * * * *
***********
+--+--+ +--+--+--+--+--+--+ +--+--+ +--+--+--+--+--+--+
|x |x | |x |x |x |x |x |x | |x |x | |x |x |x |x |x |x |
+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+ + +--+--+ + + + + + +
|x | | |x | | | | | | | | | | |
+--+ +--+ +--+--+ +--+ +--+ +--+ +--+--+--+--+ + +--+--+--+--+--+--+
|x | |x | |x |x | |x | |x | |x | |x |x |x |x | x | x | x |
+--+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+--+
levels 1, 3, 5, 7, 8, 10 levels 2, 4, 6, 9, 11
+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+
| |x | |x |x | |x | |x | | | x | x |x |x |x |
+ +--+ +--+--+ +--+ +--+ +--+ +--+--+--+--+--+--+--+--+--+--+--+
| | | | | | | | | | | | | |
+--+--+--+--+--+--+--+--+--+--+--+ + + + + + + + + + + + +
|x |x |x |x |x |x |x |x |x |x |x | |x |x |x |x |x |x |x |x |x |x |x |
+--+--+--+--+--+--+--+--+--+--+--+ +--+--+--+--+--+--+--+--+--+--+--+
levels 1, 3, 5, 7, 8, 10 levels 2, 4, 6, 9, 11
11x12x16
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
| | | |
+ + + +
| | | |
+ +--+ + +
| | | |
+ +--+--+--+--+--+--+--+--+--+--+ +
| | | |
+ +--+ + +
| | | |
+ + + +
| | | |
+ +--+--+--+--+--+--+--+--+--+--+--+--+ +
| | | |
+ + + +
| | | |
+ + +--+ +
| | | |
+ +--+--+--+--+--+--+--+--+--+--+ +
| | | |
+ + +--+ +
| | | |
+ + + +
| | | |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
(11 levels deep)
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Updated September 15, 2007.