L Shapes from Two Hexominoes

Introduction

Here are the smallest known L-shaped (6-sided) polyominoes that can be formed by specified pairs of hexominoes, using at least one of each.

If a pair of hexominoes can form a rectangle, they can form an L shape by joining two rectangles, one with each orientation. If the rectangle is a square, three squares can be joined to form an L shape.

Many of the solutions shown were formed in this way, often from rectangular tilings discovered by Mike Reid. They are not all known to be the smallest possible. If you find a smaller solution or solve an unsolved case, please write.

Patrick M. Hamlyn improved some of my solutions.

Table of Results

Of 595 possible pairs of hexominoes, 356 are known to be able to tile an L-shaped polyomino.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 • 2 8 8 2 9 2 9 27 12 29 9 3 5 8 3 35 9 29 27 7 27 ? 8 5 27 200 29 3 18 2 479 21 ? ? 2 • 6 2 2 5 2 9 5 7 15 6 3 3 4 3 6 8 16 14 3 10 10 3 12 5 14 10 2 3 2 13 10 13 21 8 6 • 19 5 24 6 15 13 19 22 6 6 7 14 2 3 11 16 19 6 8 23 11 8 19 20 41 8 32 3 20 33 192 66 8 2 19 • 5 43 5 × × 34 × × 6 7 × 4 18 × × × 10 31 × × 14 × × × 2 32 4 × ? × × 2 2 5 5 • 48 2 ? 25 54 ? 2 3 5 ? 3 29 3 6 ? 3 6 5 4 6 5 6 2 3 6 2 5 ? ? ? 9 5 24 43 48 • 2 ? 54 300 ? 35 13 12 188 36 14 22 28 ? 6 204 16 13 ? 196 ? ? 8 ? 5 328 216 ? ? 2 2 6 5 2 2 • 2 7 2 6 3 3 5 10 3 9 3 5 19 2 5 3 3 8 7 7 5 2 5 2 5 20 7 8 9 9 15 × ? ? 2 • × 800 × × 36 28 × 28 ? × × × 28 ? × × ? × × × 6 ? 3 × ? × × 27 5 13 × 25 54 7 × • 37 × × 19 8 × 8 ? × × × 7 ? × × ? × × × 16 89 10 × ? × × 12 7 19 34 54 300 2 800 37 • ? 26 10 8 73 12 304 16 ? ? 11 75 ? 7 ? ? 216 ? 4 272 3 66 ? ? ? 29 15 22 × ? ? 6 × × ? • × 6 32 × 20 6 × × × 7 28 × × ? × × × 8 ? 8 × ? × × 9 6 6 × 2 35 3 × × 26 × • 15 28 × 16 ? × × × 8 ? × × ? × × × 11 13 5 × ? × × 3 3 6 6 3 13 3 36 19 10 6 15 • 6 6 4 8 18 14 10 3 6 24 19 10 10 10 13 4 13 3 6 6 3 4 5 3 7 7 5 12 5 28 8 8 32 28 6 • 10 3 10 3 8 6 3 23 44 4 19 20 30 15 5 4 5 7 8 7 50 8 4 14 × ? 188 10 × × 73 × × 6 10 • 4 ? × × × 7 16 × × 6 × × × 4 ? 8 × ? × × 3 3 2 4 3 36 3 28 8 12 20 16 4 3 4 • 18 29 8 3 4 36 38 8 10 10 8 10 4 47 3 4 8 36 92 35 6 3 18 29 14 9 ? ? 304 6 ? 8 10 ? 18 • 240 200 ? 3 ? ? ? ? 47 336 ? 4 18 10 53 28 ? ? 9 8 11 × 3 22 3 × × 16 × × 18 3 × 29 240 • × × 22 ? × × ? × × × 18 58 7 × ? × × 29 16 16 × 6 28 5 × × ? × × 14 8 × 8 200 × • × 16 ? × × ? × × × 4 ? 10 × ? × × 27 14 19 × ? ? 19 × × ? × × 10 6 × 3 ? × × • 10 ? × × ? × × × 32 ? ? × ? × × 7 3 6 10 3 6 2 28 7 11 7 8 3 3 7 4 3 22 16 10 • 8 25 4 25 17 22 22 4 13 4 16 4 34 40 27 10 8 31 6 204 5 ? ? 75 28 ? 6 23 16 36 ? ? ? ? 8 • ? ? ? ? ? ? 8 224 10 28 ? ? ? ? 10 23 × 5 16 3 × × ? × × 24 44 × 38 ? × × × 25 ? • × ? × × × 3 63 9 × ? × × 8 3 11 × 4 13 3 × × 7 × × 19 4 × 8 ? × × × 4 ? × • 32 × × × 16 2 4 × ? × × 5 12 8 14 6 ? 8 ? ? ? ? ? 10 19 6 10 ? ? ? ? 25 ? ? 32 • ? ? ? 10 ? 6 ? ? ? ? 27 5 19 × 5 196 7 × × ? × × 10 20 × 10 47 × × × 17 ? × × ? • × × 17 ? 10 × ? × × 200 14 20 × 6 ? 7 × × 216 × × 10 30 × 8 336 × × × 22 ? × × ? × • × 26 ? 14 × ? × × 29 10 41 × 2 ? 5 × × ? × × 13 15 × 10 ? × × × 22 ? × × ? × × • 18 ? 6 × ? × × 3 2 8 2 3 8 2 6 16 4 8 11 4 5 4 4 4 18 4 32 4 8 3 16 10 17 26 18 • 5 3 82 42 5 42 18 3 32 32 6 ? 5 ? 89 272 ? 13 13 4 ? 47 18 58 ? ? 13 224 63 2 ? ? ? ? 5 • 9 ? ? ? ? 2 2 3 4 2 5 2 3 10 3 8 5 3 5 8 3 10 7 10 ? 4 10 9 4 6 10 14 6 3 9 • 15 ? 57 22 479 13 20 × 5 328 5 × × 66 × × 6 7 × 4 53 × × × 16 28 × × ? × × × 82 ? 15 • ? × × 21 10 33 ? ? 216 20 ? ? ? ? ? 6 8 ? 8 28 ? ? ? 4 ? ? ? ? ? ? ? 42 ? ? ? • ? ? ? 13 192 × ? ? 7 × × ? × × 3 7 × 36 ? × × × 34 ? × × ? × × × 5 ? 57 × ? • × ? 21 66 × ? ? 8 × × ? × × 4 50 × 92 ? × × × 40 ? × × ? × × × 42 ? 22 × ? × •

800 Tiles

Last revised 2022-12-15.

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Col. George Sicherman [ HOME | MAIL ]