Primes of the 14-omino

42 × 230, 42 × 272, 42 × 314, 42 × 464, 42 × 506, 42 × 548, 42 × 698, 42 × 740, 42 × 782, 42 × 932, 42 × 974, 42 × 976, 42 × 990, 42 × 1016, 42 × 1018, 42 × 1032, 42 × 1044, 42 × 1060, 42 × 1074, 42 × 1086, 42 × 1128, 42 × 1166, 42 × 1208, 42 × 1210, 42 × 1224, 42 × 1250, 42 × 1252, 42 × 1266, 42 × 1278, 42 × 1294, 42 × 1308, 42 × 1320, 42 × 1362, 42 × 1458, 42 × 1460, 42 × 1474, 42 × 1484, 42 × 1488, 42 × 1500, 42 × 1502, 42 × 1512, 42 × 1516, 42 × 1530, 42 × 1542, 42 × 1544, 42 × 1554, 42 × 1558, 42 × 1572, 42 × 1584, 42 × 1586, 42 × 1596, 42 × 1600, 42 × 1606, 42 × 1612, 42 × 1648, 42 × 1654, 42 × 1692, 42 × 1728, 42 × 1770, 42 × 1812, 42 × 1846, 42 × 1888, 42 × 1892, 42 × 2080, 42 × 3121, 42 × 3135, 42 × 3149, 42 × 3163, 42 × 3177, 42 × 3189, 42 × 3191, 42 × 3199, 42 × 3203, 42 × 3205, 42 × 3217, 42 × 3219, 42 × 3225, 42 × 3231, 42 × 3233, 42 × 3239, 42 × 3241, 42 × 3243, 42 × 3245, 42 × 3247, 42 × 3251, 42 × 3253, 42 × 3255, 42 × 3257, 42 × 3259, 42 × 3261, 42 × 3265, 42 × 3267, 42 × 3271, 42 × 3273, 42 × 3275, 42 × 3279, 42 × 3281, 42 × 3283, 42 × 3285, 42 × 3287, 42 × 3289, 42 × 3293, 42 × 3295, 42 × 3297, 42 × 3299, 42 × 3301, 42 × 3303, 42 × 3307, 42 × 3309, 42 × 3311, 42 × 3313, 42 × 3315, 42 × 3317, 42 × 3319, 42 × 3321, 42 × 3323, 42 × 3325, 42 × 3327, 42 × 3329, 42 × 3331, 42 × 3333, 42 × 3335, 42 × 3337, 42 × 3339, 42 × 3341, 42 × 3343, 42 × 3345, 42 × 3347, 42 × 3349, 42 × 3353, 42 × 3355, 42 × 3357, 42 × 3359, 42 × 3361, 42 × 3363, 42 × 3367, 42 × 3369, 42 × 3371, 42 × 3373, 42 × 3375, 42 × 3377, 42 × 3381, 42 × 3383, 42 × 3385, 42 × 3387, 42 × 3389, 42 × 3391, 42 × 3395, 42 × 3397, 42 × 3399, 42 × 3401, 42 × 3403, 42 × 3405, 42 × 3409, 42 × 3411, 42 × 3413, 42 × 3415, 42 × 3417, 42 × 3423, 42 × 3425, 42 × 3427, 42 × 3431, 42 × 3437, 42 × 3439, 42 × 3441, 42 × 3443, 42 × 3445, 42 × 3451, 42 × 3453, 42 × 3457, 42 × 3459, 42 × 3465, 42 × 3467, 42 × 3479, 42 × 3493, 42 × 3499, 42 × 3507, 42 × 3521, 42 × 3535
52 × 224, 52 × 231, 52 × 238, 52 × 245, 52 × 252, 52 × 259, 52 × 266, 52 × 273, 52 × 280, 52 × 287, 52 × 294, 52 × 301, 52 × 308, 52 × 315, 52 × 322, 52 × 329, 52 × 336, 52 × 343, 52 × 350, 52 × 357, 52 × 364, 52 × 371, 52 × 378, 52 × 385, 52 × 392, 52 × 399, 52 × 406, 52 × 413, 52 × 420, 52 × 427, 52 × 434, 52 × 441
56 × 94, 56 × 101, 56 × 108, 56 × 115, 56 × 122, 56 × 129, 56 × 134, 56 × 136, 56 × 141, 56 × 143, 56 × 148, 56 × 150, 56 × 155, 56 × 157, 56 × 162, 56 × 164, 56 × 169, 56 × 171, 56 × 176, 56 × 178, 56 × 183, 56 × 185, 56 × 190, 56 × 192, 56 × 197, 56 × 199, 56 × 204, 56 × 206, 56 × 211, 56 × 213, 56 × 218, 56 × 220, 56 × 225, 56 × 227, 56 × 232, 56 × 234, 56 × 239, 56 × 241, 56 × 246, 56 × 248, 56 × 253, 56 × 255, 56 × 260, 56 × 262, 56 × 267, 56 × 269, 56 × 274, 56 × 276, 56 × 281, 56 × 283, 56 × 288, 56 × 290, 56 × 295, 56 × 297, ...
63 × 80, ...
70 × 80, ...
77 × 80, ...
80 × 84, 80 × 91, 80 × 98, 80 × 105, 80 × 112, 80 × 119
...

smallest rectangle: 63 × 80




smallest known odd rectangle: 42 × 3121

The first rectangle found for this shape was 42 × 230 , see [1, Figure 5], but the 63 × 80 is the smallest rectangle tiled by this shape. The 42 × 3121 rectangle is the shortest rectangle with width 42 and odd length. It is also the smallest odd rectangle that has been found so far for this polyomino, but it seems likely that there should be a smaller one.

In [2, Theorem 4.3], it is proved that there is a constant C such that if m, n ≥ C and mn is a multiple of 14 , then this polyomino tiles an m × n rectangle. The value C = 3306 is known to be sufficient, but the best value is probably much smaller.

References

[1] Michael Reid, Tiling Rectangles and Half Strips with Congruent Polyominoes, Journal of Combinatorial Theory, Series A 80 (1997), no. 1, pp. 106-123.
[2] Michael Reid, Asymptotically Optimal Box Packing Theorems, The Electronic Journal of Combinatorics 15 (2008), no. 1, R78, 19 pp.

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Updated August 25, 2011.