Primes of the Y pentomino

5 × 10
9 × 20, 9 × 30, 9 × 45, 9 × 55
10 × 14, 10 × 16, 10 × 23, 10 × 27
11 × 20, 11 × 30, 11 × 35, 11 × 45
12 × 50, 12 × 55, 12 × 60, 12 × 65, 12 × 70, 12 × 75, 12 × 80, 12 × 85, 12 × 90, 12 × 95
13 × 20, 13 × 30, 13 × 35, 13 × 45
14 × 15
15 × 15, 15 × 16, 15 × 17, 15 × 19, 15 × 21, 15 × 22, 15 × 23
17 × 20, 17 × 25
18 × 25, 18 × 35
22 × 25
complete

smallest rectangle: 5 × 10




smallest odd rectangle: 15 × 15

The 5 × 10 rectangle was given by Klarner [4, Figure 2]. It has a tiling in which reflections are not used. Marshall [6, Figures 5, 6, 7, 8] shows how to generalize this to get an infinite family of rectifiable polyominoes. In the meantime, many others [1, 2, 3, 5, 7] have looked for more prime rectangles. The smallest odd rectangle was found by Haselgrove [3].

Also see Torsten Sillke's Y pentomino page.

References

[1] James Bitner, Tiling 5n × 12 Rectangles with Y-pentominoes, Journal of Recreational Mathematics 7 (1974), pp. 276-278.
[2] C.J. Bouwkamp and D.A. Klarner, Packing a Box with Y-pentacubes, Journal of Recreational Mathematics 3 (1970) pp. 10-26.
[3] Jenifer Haselgrove, Packing a Square with Y-pentominoes, Journal of Recreational Mathematics 7 (1974), p. 229.
[4] David A. Klarner, Some Results Concerning Polyominoes, Fibonacci Quarterly 3 (1965), pp. 9-20.
[5] David A. Klarner, Letter to the Editor, Journal of Recreational Mathematics 3 (1970), p. 258.
[6] William Rex Marshall, Packing Rectangles with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 77 (1997), no. 2, pp. 181-192.
[7] Karl Scherer, Some New Results on Y-pentominoes, Journal of Recreational Mathematics 12 (1979-1980), pp. 201-204.

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Updated May 18, 2005.