Nine L-ements clans The following presents an analysis of closely related solutions for the "Nine L-ements" puzzle. It is based on the view that when you find one solution, then often a number of related solutions can easily be found from it by rearranging only a few pieces. (C) Rick Eason 2003 We first look at solutions which are equal after interchanging two pieces. The following gives group #, pieces interchanged, and solution numbers 1 E I 1 2 2 E I 3 7 27 31 3 G H 3 4 5 6 4 E I 4 8 28 32 5 E I 5 9 29 33 6 E I 6 10 30 34 7 G H 7 8 9 10 8 E I 11 12 13 14 9 G H 11 15 19 23 10 G H 12 16 20 24 11 G H 13 17 21 25 12 G H 14 18 22 26 13 E I 15 16 17 18 14 E I 19 20 21 22 15 E I 23 24 25 26 16 G H 27 28 29 30 17 G H 31 32 33 34 18 F H 35 39 19 G H 36 37 20 D G 40 44 50 54 21 F H 40 41 42 43 22 D G 41 45 51 55 23 D G 42 46 52 56 24 D G 43 47 53 57 25 F H 44 45 46 47 26 F H 50 51 52 53 27 F H 54 55 56 57 28 G H 58 59 60 61 29 A D 63 169 30 A D 64 168 31 A D 65 167 32 A D 66 138 33 E I 66 70 90 94 34 F H 66 67 68 69 35 A D 67 133 36 E I 67 71 91 95 37 A D 68 143 38 E I 68 72 92 96 39 A D 69 148 40 E I 69 73 93 97 41 A D 70 137 42 F H 70 71 72 73 43 A D 71 132 44 A D 72 142 45 A D 73 147 46 A D 74 151 47 E I 74 78 82 86 48 F H 74 75 76 77 49 A D 75 155 50 E I 75 79 83 87 51 A D 76 159 52 E I 76 80 84 88 53 A D 77 163 54 E I 77 81 85 89 55 A D 78 152 56 F H 78 79 80 81 57 A D 79 156 58 A D 80 160 59 A D 81 164 60 A D 82 154 61 F H 82 83 84 85 62 A D 83 158 63 A D 84 162 64 A D 85 166 65 A D 86 153 66 F H 86 87 88 89 67 A D 87 157 68 A D 88 161 69 A D 89 165 70 A D 90 140 71 F H 90 91 92 93 72 A D 91 135 73 A D 92 145 74 A D 93 150 75 A D 94 139 76 F H 94 95 96 97 77 A D 95 134 78 A D 96 144 79 A D 97 149 80 B D 98 99 81 B D 100 101 82 B D 102 103 83 E I 105 106 84 B E 106 107 85 E I 108 110 86 B E 109 110 87 B E 111 112 88 B D 113 114 89 A E 115 138 90 E I 115 127 91 F H 115 116 117 118 92 A E 116 133 93 E I 116 128 94 A E 117 143 95 E I 117 129 96 A E 118 148 97 E I 118 130 98 E I 119 120 99 B E 121 122 100 B E 123 124 101 E I 124 125 102 A E 127 136 137 103 F H 127 128 129 130 104 A E 128 131 132 105 A E 129 141 142 106 A E 130 146 147 107 F H 131 136 141 146 108 E I 132 133 134 135 109 F H 132 137 142 147 110 F H 133 138 143 148 111 F H 134 139 144 149 112 F H 135 140 145 150 113 E I 137 138 139 140 114 E I 142 143 144 145 115 E I 147 148 149 150 116 E I 151 152 153 154 117 F H 151 155 159 163 118 F H 152 156 160 164 119 F H 153 157 161 165 120 F H 154 158 162 166 121 E I 155 156 157 158 122 E I 159 160 161 162 123 E I 163 164 165 166 "Clans" of solutions based on interchanging two pieces: You can arrive at other solutions in each clan by interchanging two pieces, but possibly multiple such interchanges are required. For example, from above using group 2, if pieces E and I are interchanged on solution 3, then you can generate solution 7, and from group 7, if pieces G and H are interchanged on solution 7, then you can generate solution 8. Therefore solution 3 and solution 8 are in the same "clan." The folowing gives Clan number followed by the solutions in that clan. 1 123 124 125 2 121 122 3 119 120 4 113 114 5 111 112 6 108 109 110 7 105 106 107 8 102 103 9 100 101 10 98 99 11 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 12 66 67 68 69 70 71 72 73 90 91 92 93 94 95 96 97 115 116 117 118 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 13 65 167 14 64 168 15 63 169 16 58 59 60 61 17 40 41 42 43 44 45 46 47 50 51 52 53 54 55 56 57 18 36 37 19 35 39 20 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 21 3 4 5 6 7 8 9 10 27 28 29 30 31 32 33 34 22 1 2 Rogue solutions with no close relatives: 38 48 49 62 104 126 We can also allow interchanging two "pairs" of pieces. In particular, pieces E and I form a block which is the same size as a block formed by pieces G and H, and solutions containing both blocks can interchange these pairs to arrive at other solutions. Pairs which can swap places: the following gives a solution number followed by the two group numbers (above) and the pieces in the pair. 3 2(E,I) 3(G,H) 7 2(E,I) 7(G,H) 27 2(E,I) 16(G,H) 31 2(E,I) 17(G,H) 4 3(G,H) 4(E,I) 5 3(G,H) 5(E,I) 6 3(G,H) 6(E,I) 8 4(E,I) 7(G,H) 28 4(E,I) 16(G,H) 32 4(E,I) 17(G,H) 9 5(E,I) 7(G,H) 29 5(E,I) 16(G,H) 33 5(E,I) 17(G,H) 10 6(E,I) 7(G,H) 30 6(E,I) 16(G,H) 34 6(E,I) 17(G,H) 11 8(E,I) 9(G,H) 12 8(E,I) 10(G,H) 13 8(E,I) 11(G,H) 14 8(E,I) 12(G,H) 15 9(G,H) 13(E,I) 19 9(G,H) 14(E,I) 23 9(G,H) 15(E,I) 16 10(G,H) 13(E,I) 20 10(G,H) 14(E,I) 24 10(G,H) 15(E,I) 17 11(G,H) 13(E,I) 21 11(G,H) 14(E,I) 25 11(G,H) 15(E,I) 18 12(G,H) 13(E,I) 22 12(G,H) 14(E,I) 26 12(G,H) 15(E,I) So, by allowing swapping two pairs of pieces, we can merge clans 20 and 21 above, creating a the following set of clans. The following gives clan number followed by the solutions in that clan. 1 123 124 125 2 121 122 3 119 120 4 113 114 5 111 112 6 108 109 110 7 105 106 107 8 102 103 9 100 101 10 98 99 11 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 12 66 67 68 69 70 71 72 73 90 91 92 93 94 95 96 97 115 116 117 118 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 13 65 167 14 64 168 15 63 169 16 58 59 60 61 17 40 41 42 43 44 45 46 47 50 51 52 53 54 55 56 57 18 36 37 19 35 39 20 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 21 1 2 Rogue solutions with no close relatives: 38 48 49 62 104 126 Finally, we can also allow groups of three pieces to be interchanged. The following solutions are equivalent after rearranging exactly three pieces. The following gives group # followed by the pieces interchanged, and the solution numbers 1 F G I 1 42 2 D E F 38 39 3 E F H 48 49 4 E F I 63 64 5 E F H 64 65 6 A B D 66 75 138 155 7 A D E 66 115 138 8 A B D 67 74 133 151 9 A D E 67 116 133 10 A B D 68 77 143 163 11 A D E 68 117 143 12 A B D 69 76 148 159 13 A D E 69 118 148 14 A B D 70 79 137 156 15 A D E 70 127 136 137 16 A B D 71 78 132 152 17 A D E 71 128 131 132 18 A B D 72 81 142 164 19 A D E 72 129 141 142 20 D E H 72 102 21 A B D 73 80 147 160 22 A D E 73 130 146 147 23 A B D 82 95 134 154 24 A B D 83 94 139 158 25 A B D 84 97 149 162 26 A B D 85 96 144 166 27 A B D 86 91 135 153 28 A B D 87 90 140 157 29 A B D 88 93 150 161 30 A B D 89 92 145 165 31 E F H 98 100 32 E F H 99 101 33 E F I 100 102 34 E F I 101 103 35 A B H 104 126 36 A B H 105 130 37 B E I 105 106 107 38 A B H 106 118 39 B E I 108 109 110 40 F H I 109 111 41 F H I 110 112 42 A E I 115 127 136 137 138 139 140 43 A E I 116 128 131 132 133 134 135 44 A E I 117 129 141 142 143 144 145 45 A E I 118 130 146 147 148 149 150 46 F H I 121 123 47 F H I 122 124 48 B E I 123 124 125 49 E F H 167 168 50 E F I 168 169 A final set of clans based on combinations of the above: rearrange two pieces, swap two pairs, and/or rearrange three pieces. The following gives clan number followed by the solutions in that clan. 1 104 126 2 48 49 3 121 122 123 124 125 4 119 120 5 113 114 6 108 109 110 111 112 7 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 105 106 107 115 116 117 118 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 8 63 64 65 167 168 169 9 58 59 60 61 10 1 2 40 41 42 43 44 45 46 47 50 51 52 53 54 55 56 57 11 36 37 12 35 38 39 13 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 Rogue solutions with no close relatives: 62 Number of solutions processed in this file: 169