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 [6, Figure 2]. It has a tiling in which reflections are not used. Marshall [8, 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, 4, 5, 7, 10] have looked for more prime rectangles. The complete set of primes was determined by Fogel, Goldenberg and Liu [4]. Also see [9, Example 5.2] for more analysis. 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] Andrejs Cibulis and Ilvars Mizniks, Tiling Rectangles with Pentominoes, Latvijas Universitātes Zinātniskie Raksti 612 (1998) pp. 57-61.
[4] Julian Fogel, Mark Goldenberg and Andy Liu, Packing Rectangles with Y-Pentominoes, Mathematics and Informatics Quarterly 11 (2001), no. 3, pp. 133-137.
[5] Jenifer Haselgrove, Packing a Square with Y-pentominoes, Journal of Recreational Mathematics 7 (1974), p. 229.
[6] David A. Klarner, Some Results Concerning Polyominoes, Fibonacci Quarterly 3 (1965), no. 1, pp. 9-20.
[7] David A. Klarner, Letter to the Editor, Journal of Recreational Mathematics 3 (1970), p. 258.
[8] William Rex Marshall, Packing Rectangles with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 77 (1997), no. 2, pp. 181-192.
[9] Michael Reid, Klarner Systems and Tiling Boxes with Polyominoes, Journal of Combinatorial Theory, Series A 111 (2005), no. 1, pp. 89-105.
[10] Karl Scherer, Some New Results on Y-pentominoes, Journal of Recreational Mathematics 12 (1979-1980), pp. 201-204.

Data for prime rectangles | Rectifiable polyominoes | Polyomino page | Home page | E-mail

Updated August 25, 2011.