N-Pentomino: Torsten Sillke, Jan. 1997 Stripes with one hole: ---------------------- 2*N with one hole solution (unique): A A A A A . - A A A A A . 6*N with one hole solution (unique): A A A A A . n A A A A A . n B B B B B . n n B B B B B . - n C C C C C . C C C C C . 8*N with one hole: A A A A A . a A A A A A . a B B B B B . a a B B B B B . b a c c c C C C C C . b c c x x x C C C C C . b b x x x x D D D D D . - b x x x D D D D D . 9*N with one hole: dies out after 57 pieces. 10*N with one hole: ok 11*N with one hole: ok ? - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Rectangles with two holes: -------------------------- 2 * (5n + 1): A A A A A B B B B B - - A A A A A B B B B B 3 * 4: A A A - B B A A - B B B These are all possibilities. Assume there is a ractange a*b with a, b >= 4 having only two holes. Then at least two corners 2*2 blocks must be filled. Starting from a filled corner block both adjacent corner blocks must contain a hole. (Avoiding holes in the adjacent corner blocks create more holes on the border.) Filling the opposite corner block creates new holes. Hole X of Type 1: . . 1 1 2 2 3 3 3 . . . . 8 8 1 1 1 2 2 2 3 3 . . . . 8 8 8 X ^ ^ Filled Corner Hole in the corner block Hole X of Type 2: 9 . 9 9 . . 1 1 2 2 3 3 3 . . . . 8 8 9 1 1 1 2 2 2 3 3 . . . . 8 8 8 X 9 ^ ^ Filled Corner Hole in the corner block - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Rectangles with three holes: ---------------------------- 2 * (5n - 1): A A A B B B B B - - - A A B B B B B A A A A A B B - C C C C C - - A A A A A B B B C C C C C I did not find other rectangle with n*n and m<10 and n<48. Question: Are there other ones? ^^^^^^^^^ There are more cases to analyze than in the two hole case. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Rectanges without the 4 corners: -------------------------------- 6 * 9 (without 4 corners): unique . A A A C B B B . A A A A C B B B B A A A C C C B B B B B B C C C A A A B B B B C A A A A . B B B C A A A . 8 * 8 (without 4 corners): unique 11 * 24 (without 4 corners): 2 solutions . 96 96 96 90 90 90 82 82 82 79 79 2 2 2 2 2 2 2 2 2 2 2 . 96 96 92 90 90 87 82 82 79 79 79 2 2 2 2 2 2 2 2 2 2 2 2 2 97 95 92 92 87 87 86 86 86 80 76 2 2 2 2 2 2 2 2 2 2 2 2 2 97 95 95 92 87 86 86 83 80 80 76 2 2 2 2 2 2 2 2 2 2 2 2 2 97 97 95 92 87 88 83 83 80 76 76 2 2 2 2 2 2 2 2 2 2 2 2 2 98 97 95 93 88 88 83 84 80 76 75 74 2 2 2 2 2 2 2 2 2 2 2 2 98 98 93 93 88 89 83 84 81 77 75 74 74 2 2 2 2 2 2 2 2 2 2 2 99 98 93 94 88 89 84 84 81 77 75 75 74 2 2 2 2 2 2 2 2 2 2 2 99 98 93 94 89 89 84 81 81 77 77 75 74 2 2 2 2 2 2 2 2 2 2 2 99 99 94 94 89 91 91 81 85 85 77 78 78 2 2 2 2 2 2 2 2 2 2 2 . 99 94 91 91 91 85 85 85 78 78 78 2 2 2 2 2 2 2 2 2 2 2 . 0 96 96 96 90 90 90 82 82 82 79 79 2 2 2 2 2 2 2 2 2 2 2 . 96 96 92 90 90 87 82 82 79 79 79 2 2|67| 2 2 2 2 2 2 2 2 2 2 97 95 92 92 87 87 86 86 86|80 74 74 74 67 67| 2 2 2 2 2 2 2 2 2 97 95 95 92 87 86 86 83|80 80 76 73 74 74 67| 2 2 2 2 2 2 2 2 2 97 97 95 92 87 88 83 83|80 76 76 73 68 68 67| 2 2 2 2 2 2 2 2 2 98 97 95 93 88 88 83 84|80 76 77 73 73 68 68 68| 2 2 2 2 2 2 2 2 98 98 93 93 88 89 83 84 81|76 77 75 73 69 69 69| 2 2 2 2 2 2 2 2 99 98 93 94 88 89 84 84 81|77 77 75 69 69 70 70| 2 2 2 2 2 2 2 2 99 98 93 94 89 89 84 81 81|77 75 75 70 70 70| 2 2 2 2 2 2 2 2 2 99 99 94 94 89 91 91 81 85 85|75|78 78 2 2 2 2 2 2 2 2 2 2 2 0 99 94 91 91 91 85 85 85 78 78 78 2 2 2 2 2 2 2 2 2 2 2 . 11 * 34 (without 4 corners): unique . 96 96 96 90 90 90 82 82 82 77 77 72 72 72 67 67 2 2 2 2 2 2 + 96 96 92 90 90 87 82 82 77 77 77 72 72 67 67 67 2 2 2 2 2 2 2 + 97 95 92 92 87 87 86 86 86 78 78 78 69 69 64 64 64 2 2 2 2 2 2 + 97 95 95 92 87 86 86 83 78 78 79 79 70 69 69 69 64 64 2 2 2 2 2 + 97 97 95 92 87 88 83 83 79 79 79 73 70 70 65 A A A 2 2 2 2 2 + 98 97 95 93 88 88 83 84 80 80 80 73 73 70 65 A A A A 2 2 2 2 + 98 98 93 93 88 89 83 84 81 75 80 80 73 70 65 65 A A A 2 2 2 2 + 99 98 93 94 88 89 84 84 81 75 74 74 73 68 68 65 2 2 2 2 2 2 2 + 99 98 93 94 89 89 84 81 81 75 75 74 74 74 68 68 68 2 2 2 2 2 2 + 99 99 94 94 89 91 91 81 85 85 75 76 76 71 71 71 66 66 2 2 2 2 2 + . 99 94 91 91 91 85 85 85 76 76 76 71 71 66 66 66 2 2 2 2 2 2 + 12 * 12 (without 4 corners): unique (symmetry is rot4) . 2 2 2 2 2 2 2 2 2 2 . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 98 2 2 2 2 2 2 2 2 2 2 2 98 2 2 2 90 90 2 2 2 2 2 2 98 98 90 90 90 86 2 2 2 2 2 2 99 98 95 95 95 86 86 2 2 2 2 2 99 95 95 91 91 91 86 2 2 2 2 2 99 99 91 91 92 92 86 2 2 2 2 2 . 99 92 92 92 2 2 2 2 2 2 . 12 * (12+5n) (without 4 corners): . 96 89 89 89 A A A A A 2 2 2 2 2 2 . 96 96 90 90 89 89 A A A A A 2 2 2 2 2 2 96 93 93 90 90 90 B B B B B 2 2 2 2 2 2 96 97 93 93 93 B B B B B 2 2 2 2 2 2 2 97 97 91 91 91 A A A A A 2 2 2 2 2 2 2 97 94 94 94 91 91 A A A A A 2 2 2 2 2 2 97 98 95 94 94 87 B B B B B 2 2 2 2 2 2 98 98 95 92 87 87 86 B B B B B 2 2 2 2 2 98 95 95 92 87 86 86 A A A A A 2 2 2 2 2 98 95 92 92 87 86 A A A A A 2 2 2 2 2 2 99 99 92 88 88 86 B B B B B 2 2 2 2 2 2 . 99 99 99 88 88 88 B B B B B 2 2 2 2 . 14 * (16+5n) (without 4 corners): . 95 87 87 87|80 80 2 2 2 2 2 2 2 2 . 95 95 88 88 87 87|80 80 80 2 2 2 2 2 2 2 95 92 92 88 88 88|81 81 81 2 2 2 2 2 2 2 95 96 92 92 92|81 81 79 2 2 2 2 2 2 2 2 96 96 93 93 93|84 79 79 2 2 2 2 2 2 2 2 96 93 93 89|84 84 79 2 2 2 2 2 2 2 2 2 96 97 89 89|84 85 79 2 78 2 2 2 2 2 2 2 97 97 89|90 84 85 82 2 78 2 2 2 2 2 2 2 97 98 89|90 85 85 82 78 78 2 2 2 2 2 2 2 97 98|90 90 85 82 82 78 2 2 2 2 2 2 2 2 98 98|90 94 94 82 86 86 2 2 2 2 2 2 2 2 98|94 94 94 86 86 86 2 2 2 2 2 2 2 2 2 |99 99 91 91 91 83 83 2 2 2 2 2 2 2 2 2 .|99 99 99 91 91 83 83 83 2 2 2 2 2 2 . 16 * (14+5n) (without 4 corners): . 94 87 87 87| 2 2 2 2 2 2 2 2 . 94 94 88 83 87 87 78| 2 2 2 2 2 2 2 94 95 88 83 83 78 78| 2 2 2 2 2 2 2 94 95 88 88 83 78| 2 2 2 2 2 2 2 2 95 95 89 88 83 78| 2 2 2 2 2 2 2 2 95 96 89 84 84| 2 2 2 2 2 2 2 2 2 96 96 89 89 84 84 84| 2 2 2 2 2 2 2 96 97 90 89 81 81| 2 2 2 2 2 2 2 2 96 97 90 85 85 81 81 81| 2 2 2 2 2 2 97 97 90 90 85 85 85| 2 2 2 2 2 2 2 97 98 93 90 86 86 86 79 79| 2 2 2 2 2 98 98 93 86 86 79 79 79| 2 2 2 2 2 2 98 93 93 91 91 91 82 82| 2 2 2 2 2 2 98 93 91 91 82 82 82| 2 2 2 2 2 2 2 99 99 92 92 92 80 80| 2 2 2 2 2 2 2 . 99 99 99 92 92 80 80 80| 2 2 2 2 . 17 * 17 (without 4 corners): 3 solutions . 61 61 61 51 51 51 X X X X X 22 22 22 10 . 61 61 52 51 51 47 X X X X X 22 22 23 23 10 10 62 58 52 52 47 47 Y Y Y Y Y 23 23 23 19 19 10 62 58 58 52 47 43 43 Y Y Y Y Y 19 19 19 11 10 62 62 58 52 47 48 43 43 43 29 29 29 20 20 20 11 11 63 62 58 53 48 48 X X X X X 29 29 16 20 20 11 63 63 53 53 48 X X X X X 30 24 21 16 16 12 11 64 63 53 54 48 Y Y Y Y Y 30 24 21 21 16 12 12 64 63 53 54 Y Y Y Y Y 30 30 24 24 21 16 13 12 64 64 54 54 X X X X X 30 28 25 24 21 17 13 12 65 64 54 59 59 X X X X X 28 25 25 17 17 13 13 65 59 59 59 55 55 42 37 37 37 28 28 25 17 15 14 13 65 65 55 55 55 45 42 42 34 37 37 28 25 17 15 14 14 66 65 60 60 60 45 45 42 34 31 31 26 26 26 15 15 14 66 60 60 56 56 56 45 42 34 34 31 31 31 26 26 15 14 66 66 56 56 57 57 45 46 46 34 35 35 X X X X X . 66 57 57 57 46 46 46 35 35 35 X X X X X . 18 * (18+5n) (without 4 corners): 0 94 83 83 83|75 75 75 2 2 2 2 2 2 2 2 2 . 94 94 84 84 83 83|76 75 75 2 2 2 2 2 2 2 2 2 94 89 89 84 84 84|76 71 2 2 2 2 2 2 2 2 2 2 94 95 89 89 89|76 76 71 2 2 2 2 2 2 2 2 2 2 95 95 90 90 90|76 71 71 2 2 2 2 2 2 2 2 2 2 95 90 90 81 81 81|71 72 2 2 2 2 2 2 2 2 2 2 95 96 85 85 85 81 81|72 2 2 2 2 2 2 2 2 2 2 96 96 86 86 85 85|72 72 2 2 2 2 2 2 2 2 2 2 96 91 91 86 86 86|72 77 77 2 2 2 2 2 2 2 2 2 96 97 91 91 91|77 77 77 68 2 2 2 2 2 2 2 2 2 97 97 87 87 87|78 78 78 68 68 2 2 2 2 2 2 2 2 97 92 92 92 87 87|79 78 78 68 2 2 2 2 2 2 2 2 97 98 93 92 92|79 79 73 70 68 2 2 2 2 2 2 2 2 98 98 93|88 80 79 73 73 70 69 2 2 2 2 2 2 2 2 98 93 93|88 80 79 73 70 70 69 2 2 2 2 2 2 2 2 98 93|88 88 80 80 73 70 69 69 2 2 2 2 2 2 2 2 99 99|88 82 82 80 74 74 69 2 2 2 2 2 2 2 2 2 0 99 99 99|82 82 82 74 74 74 2 2 2 2 2 2 2 . 22 * 22 (without 4 corners): . 92 92 92 81 81 68 68 68 60 60 2 2 2 2 2 2 2 2 2 2 . 92 92 81 81 81 68 68 60 60 60 2 2 2 2 2 2 2 2 2 2 2 2 93 88 88 88 72 72 72 61 61 2 2 2 2 2 2 2 2 2 2 2 2 2 93 93 82 88 88 73 72 72 61 61 61 2 2 2 2 2 2 2 2 2 2 2 94 93 82 82 73 73 69 69 69 62 62 2 2 2 2 2 2 2 2 2 2 2 94 93 83 82 73 69 69 62 62 62 2 2 2 2 2 2 2 2 2 2 2 2 94 94 83 82 73 74 74 74 63 53 2 2 2 2 2 2 2 2 2 2 2 2 95 94 83 83 74 74 75 75 63 53 2 2 2 2 2 2 2 2 2 2 2 2 95 89 89 83 75 75 75 63 63 53 53 2 2 2 2 2 2 2 2 2 2 2 95 95 89 89 89 78 78 63 65 65 53 2 2 2 2 2 2 2 2 2 2 2 96 95 84 78 78 78 65 65 65 56 56 56 2 2 2 2 2 2 2 2 2 2 96 96 84 79 79 70 70 70 56 56 2 2 2 2 2 2 2 2 2 2 2 2 97 96 84 84 79 79 79 70 70 64 64 2 2 2 2 2 2 2 2 2 2 2 97 96 85 84 76 76 76 64 64 64 52 2 2 2 2 2 2 2 2 2 2 2 97 97 85 80 80 80 76 76 57 54 52 52 2 2 2 2 2 2 2 2 2 2 98 97 85 85 77 80 80 66 57 54 54 52 2 2 2 2 2 2 2 2 2 2 98 90 90 85 77 77 66 66 57 57 54 52 2 2 2 2 2 2 2 2 2 2 98 98 90 90 90 77 66 67 58 57 54 2 2 2 2 2 2 2 2 2 2 2 99 98 91 91 91 77 66 67 58 55 55 2 2 2 2 2 2 2 2 2 2 2 99 91 91 86 86 86 67 67 58 58 55 55 55 2 2 2 2 2 2 2 2 2 99 99 86 86 87 87 67 71 71 58 59 59 2 2 2 2 2 2 2 2 2 2 . 99 87 87 87 71 71 71 59 59 59 2 2 2 2 2 2 2 2 2 2 . impossible: 6*{14,19,24,29,... } 7*{ 7,12,17,22,... and N} 8*{13,18,23,28,... } 9*{11,16,21,26,... and N} 11*{14,19,29,39,44} 13*{13,18,23,28,33} Exercise: construct all possible corner-less rectangles. You have to find augmentation pieces which work in two dimension to derive a finite problem. - - - - - - - - - - - - - - - - - - - - - - - - N5 rectangles with minimal number of V5 pieces. 41 * N5 + 4 * V5 -> 15*15 v v v 92 84 84 79 79 79 71 71 71 v v v v X 92 92 90 84 84 84 79 79 69 71 71 X v v X 92 90 90 85 85 85 80 80 69 69 65 X v X X 92 90 85 85 80 80 80 73 70 69 65 X X X 98 93 90 91 91 91 81 73 73 70 69 65 65 X 98 98 93 91 91 86 81 81 73 74 70 70 66 65 63 98 99 93 93 86 86 81 82 73 74 72 70 66 63 63 98 99 97 93 86 87 81 82 74 74 72 66 66 63 64 99 99 97 94 86 87 82 82 74 72 72 66 67 63 64 99 97 97 94 87 87 82 83 83 72 75 75 67 64 64 X 97 94 94 87 83 83 83 75 75 75 67 67 64 X X X 94 95 95 95 88 88 76 76 76 67 68 X X v X 95 95 88 88 88 77 77 77 76 76 68 X v v X 96 96 89 89 89 78 78 77 77 68 68 X v v v v 96 96 96 89 89 78 78 78 68 v v v 42 * N5 + 3 * V5 -> 15*15 50 50 42 42 42 35 35 27 27 27 19 19 v v v 51 50 50 50 42 42 35 35 35 27 27 19 19 19 v 51 51 43 41 41 41 30 30 30 21 21 17 17 17 v 52 51 43 43 38 41 41 31 30 30 21 21 21 17 17 52 51 44 43 38 38 31 31 28 28 28 22 22 16 13 52 52 44 43 39 38 31 28 28 22 22 22 16 16 13 53 52 44 44 39 38 31 32 32 32 25 25 16 13 13 53 47 47 44 39 39 32 32 25 25 25 18 16 13 14 53 53 47 47 47 39 33 33 26 26 26 18 18 14 14 54 53 48 48 48 36 36 33 33 33 26 26 18 14 15 54 48 48 45 45 45 36 36 36 23 23 23 18 14 15 54 54 45 45 46 46 37 37 37 29 29 23 23 15 15 v 54 46 46 46 37 37 29 29 29 24 24 24 15 v v 49 49 49 40 40 34 34 34 24 24 20 20 20 v v v v 49 49 40 40 40 34 34 20 20 v v v All solutions for 17*17 without all corners: . 61 61 61 51 51 51 X X X X X 22 22 22 10 . 61 61 52 51 51 47 X X X X X 22 22 23 23 10 10 62 58 52 52 47 47 X X X X X 23 23 23 19 19 10 62 58 58 52 47 43 43 X X X X X 19 19 19 11 10 62 62 58 52 47 48 43 43 43 29 29 29 20 20 20 11 11 63 62 58 53 48 48 X X X X X 29 29 16 20 20 11 63 63 53 53 48 X X X X X 30 24 21 16 16 12 11 64 63 53 54 48 X X X X X 30 24 21 21 16 12 12 64 63 53 54 X X X X X 30 30 24 24 21 16 13 12 64 64 54 54 X X X X X 30 28 25 24 21 17 13 12 65 64 54 59 59 X X X X X 28 25 25 17 17 13 13 65 59 59 59 55 55 42 37 37 37 28 28 25 17 15 14 13 65 65 55 55 55 45 42 42 34 37 37 28 25 17 15 14 14 66 65 60 60 60 45 45 42 34 31 31 26 26 26 15 15 14 66 60 60 56 56 56 45 42 34 34 31 31 31 26 26 15 14 66 66 56 56 57 57 45 46 46 34 35 35 X X X X X . 66 57 57 57 46 46 46 35 35 35 X X X X X . . 61 61 61 51 51 51 X X X X X 22 22 22 10 . 61 61 52 51 51 48 X X X X X 22 22 23 23 10 10 62 58 52 52 48 48 X X X X X 23 23 23 19 19 10 62 58 58 52 48 44 44 X X X X X 19 19 19 11 10 62 62 58 52 48 45 44 44 44 28 28 28 20 20 20 11 11 63 62 58 53 X 45 45 X 34 34 34 28 28 16 20 20 11 63 63 53 53 X X 45 X X X 34 34 21 16 16 12 11 64 63 53 54 X X 45 X X X X X 21 21 16 12 12 64 63 53 54 X X 42 X X X X X X 21 16 13 12 64 64 54 54 X X 42 X X X X X X 21 17 13 12 65 64 54 59 59 X 42 42 X X X X X 17 17 13 13 65 59 59 59 55 55 43 42 35 35 X X X 17 15 14 13 65 65 55 55 55 46 43 43 36 35 35 35 X 17 15 14 14 66 65 60 60 60 46 46 43 36 31 31 26 26 26 15 15 14 66 60 60 56 56 56 46 43 36 36 31 31 31 26 26 15 14 66 66 56 56 57 57 46 47 47 36 37 37 X X X X X . 66 57 57 57 47 47 47 37 37 37 X X X X X . . 61 52 52 52 X X X X X 28 28 28 16 16 16 . 61 61 53 53 52 52 X X X X X 22 28 28 17 16 16 61 58 58 53 53 53 X X X X X 22 22 17 17 15 10 61 62 58 58 58 X X X X X 33 33 22 17 15 15 10 62 62 59 59 59 47 47 47 33 33 33 29 22 17 15 10 10 62 59 59 54 47 47 44 44 44 X 29 29 20 20 15 10 11 62 X 54 54 51 44 44 X X X 29 23 23 20 20 20 11 X X 54 51 51 X X X X X 29 24 23 23 23 11 11 X X 54 51 X X X X X X 31 24 21 21 21 11 12 X X 55 51 X X X X X X 31 24 24 X 21 21 12 X X 55 55 X X X X X 31 31 25 24 X X 12 12 X 65 60 55 X X X 45 45 31 32 25 25 X X 12 X 65 65 60 55 X 45 45 45 39 32 32 30 25 X X X X 65 60 60 X X X X X 39 32 30 30 25 X X X X 65 60 X X X X X 39 39 32 30 26 26 26 X X X X X X X X 46 46 39 36 36 30 27 27 26 26 X X . X X X X X 46 46 46 36 36 36 27 27 27 X .