### 3.7.243. Squared squares

A constraint that can be used for modelling the squared squares problemΒ [CroftFalconerGuy91]Β [LintWilson92] (also called the perfect squared squares problemΒ [Duijvestijn78]): a perfect squared square of order $n$ is a square that can be tiled with $n$ smaller squares such that each of the smaller squares has a different integer size. It is called simple if it does not contain a subset of at least two squares, corresponding to a square or to a rectangle. Duijvestijn has shown in 1962 that no instances exist with less than 21 squaresΒ [Duijvestijn78]. A single solution depicted by FigureΒ 3.7.71 exists with 21 squares, where the squares have sizes 2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42, 50 and must be packed into a square of size 112.

A catalogue of such simple squared squares of orders 21 through 25 is provided inΒ [BouwkampDuijvestijn92]. The following table contains all the problem instances from the previous catalogue. The different fields respectively give the problem number, the number of squares, the size of the master square and a list of the square sizes. The two last columns present a solution as a ground instance of geost, and a graphical view (see the description below). Problems 166 and 167, 168 and 169, 182 and 183 are identical, but have two non-isomorphic solutions. A much bigger table can be found at the following link http://www.squaring.net/.

When the size of the squares is known four constraint programming approach are respectively reported in

 1 21 112 2,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50 sol view 2 22 110 2,3,4,6,7,8,12,13,14,15,16,17,18,21,22,23,24,26,27,28,50,60 sol view 3 22 110 1,2,3,4,6,8,9,12,14,16,17,18,19,21,22,23,24,26,27,28,50,60 sol view 4 22 139 1,2,3,4,7,8,10,17,18,20,21,22,24,27,28,29,30,31,32,38,59,80 sol view 5 22 147 1,3,4,5,8,9,17,20,21,23,25,26,29,31,32,40,43,44,47,48,52,55 sol view 6 22 147 2,4,8,10,11,12,15,19,21,22,23,25,26,32,34,37,41,43,45,47,55,59 sol view 7 22 154 2,5,9,11,16,17,19,21,22,24,26,30,31,33,35,36,41,46,47,50,52,61 sol view 8 22 172 1,2,3,4,9,11,13,16,17,18,19,22,24,33,36,38,39,42,44,53,75,97 sol view 9 22 192 4,8,9,10,12,14,17,19,26,28,31,35,36,37,41,47,49,57,59,62,71,86 sol view 10 23 110 1,2,3,4,5,7,8,10,12,13,14,15,16,19,21,28,29,31,32,37,38,41,44 sol view 11 23 139 1,2,7,8,12,13,14,15,16,18,19,20,21,22,24,26,27,28,32,33,38,59,80 sol view 12 23 140 1,2,3,4,5,8,10,13,16,19,20,23,27,28,29,31,33,38,42,45,48,53,54 sol view 13 23 140 2,3,4,7,8,9,12,15,16,18,22,23,24,26,28,30,33,36,43,44,47,50,60 sol view 14 23 145 1,2,3,4,6,8,9,12,15,20,22,24,25,26,27,29,30,31,32,34,36,61,84 sol view 15 23 180 2,4,8,10,11,12,15,19,21,22,23,25,26,32,33,34,37,41,43,45,47,88,92 sol view 16 23 188 2,4,8,10,11,12,15,19,21,22,23,25,26,32,33,34,37,45,47,49,51,92,96 sol view 17 23 208 1,3,4,9,10,11,12,16,17,18,22,23,24,40,41,60,62,65,67,70,71,73,75 sol view 18 23 215 1,3,4,9,10,11,12,16,17,18,22,23,24,40,41,60,66,68,70,71,74,76,79 sol view 19 23 228 2,7,9,10,15,16,17,18,22,23,25,28,36,39,42,56,57,68,69,72,73,87,99 sol view 20 23 257 2,3,9,11,14,15,17,20,22,24,28,29,32,33,49,55,57,60,63,66,79,123,134 sol view 21 23 332 1,15,17,24,26,30,31,38,47,48,49,50,53,56,58,68,83,89,91,112,120,123,129 sol view 22 24 120 3,4,5,6,8,9,10,12,13,14,15,16,17,19,20,23,25,32,33,34,40,41,46,47 sol view 23 24 186 2,3,4,7,8,9,12,15,16,18,22,23,24,26,28,30,33,36,43,46,47,60,90,96 sol view 24 24 194 2,3,7,9,10,16,17,18,19,20,23,25,28,34,36,37,42,53,54,61,65,68,69,72 sol view 25 24 195 2,4,7,10,11,16,17,18,21,26,27,30,39,41,42,45,47,49,52,53,54,61,63,80 sol view 26 24 196 1,2,5,10,11,15,17,18,20,21,24,26,29,31,32,34,36,40,44,47,48,51,91,105 sol view 27 24 201 1,3,4,6,9,10,11,12,17,18,20,21,22,23,26,38,40,46,50,52,53,58,98,103 sol view 28 24 201 1,4,5,8,9,10,11,15,16,18,19,20,22,24,26,39,42,44,49,52,54,56,93,108 sol view 29 24 203 1,2,5,10,11,15,17,18,20,21,24,26,29,31,32,34,36,40,44,48,54,58,98,105 sol view 30 24 247 3,5,6,9,12,14,19,23,24,25,28,32,34,36,40,45,46,48,56,62,63,66,111,136 sol view 31 24 253 2,4,5,9,13,18,20,23,24,27,28,31,38,40,44,50,61,70,72,77,79,86,88,104 sol view 32 24 255 3,5,10,11,16,17,20,22,23,25,26,27,28,32,41,44,52,53,59,63,65,74,118,137 sol view 33 24 288 2,7,9,10,15,16,17,18,22,23,25,28,36,39,42,56,57,60,68,72,73,87,129,159 sol view 34 24 288 1,5,7,8,9,14,17,20,21,26,30,32,34,36,48,51,54,59,64,69,72,93,123,165 sol view 35 24 290 2,3,8,9,11,12,14,17,21,30,31,33,40,42,45,48,59,61,63,65,82,84,124,166 sol view 36 24 292 1,2,3,8,12,15,16,17,20,22,24,26,29,33,44,54,57,60,63,67,73,102,117,175 sol view 37 24 304 3,5,7,11,12,17,20,22,25,29,35,47,48,55,56,57,69,72,76,92,96,100,116,132 sol view 38 24 304 3,4,7,12,16,20,23,24,27,28,30,32,33,36,37,44,53,57,72,76,85,99,129,175 sol view 39 24 314 2,4,11,12,16,17,18,19,28,29,40,44,47,59,62,64,65,78,79,96,97,105,113,139 sol view 40 24 316 3,9,10,12,13,14,15,23,24,33,36,37,48,52,54,55,57,65,66,78,79,93,144,172 sol view 41 24 326 1,6,10,11,14,15,18,24,29,32,43,44,53,56,63,65,71,80,83,101,104,106,119,142 sol view 42 24 423 2,9,15,17,27,29,31,32,33,36,47,49,50,60,62,77,105,114,123,127,128,132,168,186 sol view 43 24 435 1,2,8,10,13,19,23,33,44,45,56,74,76,78,80,88,93,100,112,131,142,143,150,192 sol view 44 24 435 3,5,9,11,12,21,24,27,30,44,45,50,54,55,63,95,101,112,117,123,134,140,178,200 sol view 45 24 459 8,9,10,11,16,30,36,38,45,55,57,65,68,84,95,98,100,116,117,126,135,144,180,198 sol view 46 24 459 4,6,9,10,17,21,23,25,31,33,36,38,45,50,83,115,117,126,133,135,144,146,180,198 sol view 47 24 479 5,6,17,23,24,26,28,29,35,43,44,52,60,68,77,86,130,140,150,155,160,164,174,175 sol view 48 25 147 3,4,5,6,8,9,10,12,13,14,15,16,17,19,20,23,25,27,32,33,34,40,41,73,74 sol view 49 25 208 1,2,3,4,5,7,8,11,12,17,18,24,26,28,29,30,36,39,44,45,50,59,60,89,119 sol view 50 25 213 3,5,6,7,13,16,17,20,21,23,24,25,26,28,31,35,36,47,49,56,58,74,76,81,90 sol view 51 25 215 1,4,6,7,11,15,24,26,27,33,37,39,40,41,42,43,45,47,51,55,60,62,63,69,83 sol view 52 25 216 1,2,3,4,5,7,8,11,16,17,18,19,25,30,32,33,39,41,45,49,54,59,64,103,113 sol view 53 25 236 1,2,4,9,11,12,13,14,15,16,19,24,38,40,44,46,47,48,59,64,65,70,81,85,107 sol view 54 25 242 1,3,6,7,9,13,14,16,17,19,23,25,26,28,30,31,47,51,54,57,60,64,67,111,131 sol view 55 25 244 1,2,4,5,7,10,15,17,19,20,21,22,26,27,30,37,40,41,45,65,66,68,70,110,134 sol view 56 25 252 4,7,10,11,12,13,23,25,29,31,32,34,36,37,38,40,42,44,62,67,68,71,77,108,113 sol view 57 25 253 2,4,5,6,9,10,12,14,20,24,27,35,36,37,38,42,43,45,50,54,63,66,70,120,133 sol view 58 25 260 1,4,6,7,10,15,24,26,27,28,29,31,33,34,37,38,44,65,70,71,77,78,83,100,112 sol view 59 25 264 3,7,8,12,16,18,19,20,22,24,26,31,34,37,38,40,42,53,54,61,64,69,70,130,134 sol view 60 25 264 3,8,12,13,16,18,20,21,22,24,26,29,34,38,40,42,43,47,54,59,64,70,71,130,134 sol view 61 25 264 1,3,4,6,9,10,11,12,16,17,18,20,21,22,39,42,54,56,61,66,68,69,73,129,135 sol view 62 25 265 1,3,4,6,9,10,11,12,16,17,18,20,21,22,39,42,54,56,62,66,68,69,74,130,135 sol view 63 25 273 1,4,8,10,11,12,17,19,21,22,27,29,30,33,37,43,52,62,65,86,88,89,91,96,120 sol view 64 25 273 1,6,9,14,16,17,18,21,22,23,25,31,32,38,44,46,48,50,54,62,65,68,78,133,140 sol view 65 25 275 2,3,7,13,17,24,25,31,33,34,35,37,41,49,51,53,55,60,68,71,74,81,94,100,107 sol view 66 25 276 1,5,8,9,11,18,19,21,30,36,41,44,45,46,47,51,53,58,63,69,71,84,87,105,120 sol view 67 25 280 5,6,11,17,18,20,21,24,27,28,32,34,41,42,50,53,54,55,68,78,85,88,95,97,117 sol view 68 25 280 2,3,7,8,14,18,30,36,37,39,44,50,52,54,56,60,63,64,65,72,75,78,79,96,106 sol view 69 25 284 1,2,11,12,14,16,18,19,23,26,29,37,38,39,40,42,59,68,69,77,78,97,106,109,110 sol view 70 25 286 1,4,5,7,10,12,15,16,20,23,28,30,32,33,35,37,53,54,64,68,74,79,80,133,153 sol view 71 25 289 2,3,5,8,13,14,17,20,21,32,36,41,50,52,60,61,62,68,74,76,83,87,100,102,104 sol view 72 25 289 2,3,4,5,7,12,16,17,19,21,23,25,29,31,32,44,57,64,65,68,72,76,84,140,149 sol view 73 25 290 1,2,10,11,13,14,15,17,18,28,29,34,36,38,50,56,60,69,77,80,85,91,94,111,119 sol view 74 25 293 5,6,11,17,18,20,21,24,27,28,32,34,41,42,50,54,55,66,68,78,85,88,95,110,130 sol view 75 25 297 2,7,8,9,10,15,16,17,18,23,25,26,28,36,38,43,53,60,61,68,69,77,99,137,160 sol view 76 25 308 1,3,4,7,10,12,13,23,25,34,37,38,39,43,44,45,62,77,79,85,87,108,113,115,116 sol view 77 25 308 1,5,6,7,8,9,13,16,19,28,33,36,38,43,45,48,70,71,73,84,86,102,104,120,133 sol view 78 25 309 7,8,14,16,23,24,25,26,31,33,34,39,48,56,59,60,62,70,76,82,92,100,101,108,117 sol view 79 25 311 2,7,8,9,10,15,16,17,18,23,25,26,28,36,38,43,53,60,61,68,83,91,99,151,160 sol view 80 25 314 1,6,7,11,16,22,26,29,32,36,38,44,51,53,64,69,70,73,74,75,85,87,101,116,128 sol view 81 25 316 1,3,9,12,21,26,30,33,34,35,38,39,40,41,53,56,59,69,79,85,96,103,111,117,120 sol view 82 25 317 1,5,6,7,8,9,16,17,19,32,37,40,42,47,49,52,59,75,81,92,94,110,112,113,126 sol view 83 25 320 2,7,8,9,12,14,15,21,23,35,38,44,46,49,53,54,56,63,96,101,103,105,108,112,116 sol view 84 25 320 3,8,9,11,17,18,22,25,26,27,29,30,31,33,35,49,51,67,72,73,80,85,95,152,168 sol view 85 25 320 1,4,6,7,8,13,14,16,24,28,30,33,34,38,41,42,57,60,69,78,81,90,92,150,170 sol view 86 25 320 3,4,6,8,9,14,15,16,24,28,30,31,34,38,39,42,59,60,71,78,79,90,92,150,170 sol view 87 25 322 3,4,8,9,10,16,18,20,22,23,24,28,31,38,44,47,64,65,68,76,80,81,97,144,178 sol view 88 25 322 3,4,8,10,15,16,18,19,20,22,24,28,35,38,44,53,59,64,68,76,80,85,93,144,178 sol view 89 25 323 2,3,4,7,10,13,15,18,23,32,34,35,36,42,46,50,57,60,66,72,78,87,98,159,164 sol view 90 25 323 3,8,9,11,17,18,22,25,26,27,29,30,31,33,35,49,51,67,72,73,83,88,95,155,168 sol view 91 25 323 2,6,9,11,13,14,18,19,20,23,27,28,29,42,46,48,60,64,72,74,79,82,98,146,177 sol view 92 25 325 3,5,6,11,12,13,18,23,25,28,32,37,40,43,45,46,51,79,92,99,103,108,112,114,134 sol view 93 25 326 1,4,8,10,12,16,21,22,24,27,28,35,36,37,38,46,49,68,70,75,88,90,93,158,168 sol view 94 25 327 2,9,10,12,13,16,19,21,23,26,36,44,46,52,55,61,62,74,84,87,100,103,104,120,140 sol view 95 25 328 2,3,4,7,8,10,14,17,26,27,28,36,38,40,42,45,53,58,73,74,79,94,102,152,176 sol view 96 25 334 1,4,8,10,12,16,21,22,24,27,28,35,36,37,38,46,49,68,75,78,88,93,98,166,168 sol view 97 25 336 2,3,4,7,8,10,14,17,26,27,28,36,38,40,45,50,53,58,73,74,79,94,110,152,184 sol view 98 25 338 1,4,8,10,12,16,19,22,24,25,28,36,37,38,39,46,53,68,70,73,94,96,101,164,174 sol view 99 25 338 4,5,8,10,12,15,16,21,22,24,28,33,36,38,43,46,57,68,70,77,94,96,97,164,174 sol view 100 25 340 1,4,5,6,11,13,16,17,22,24,44,46,50,51,52,53,61,64,66,79,84,85,92,169,171 sol view 101 25 344 2,3,8,11,14,17,19,21,23,25,27,36,39,44,48,53,56,71,77,83,86,89,98,169,175 sol view 102 25 359 7,8,9,10,14,17,18,23,25,27,29,31,40,41,43,46,69,74,82,85,90,98,102,172,187 sol view 103 25 361 2,6,7,8,9,14,20,22,26,27,32,34,36,47,49,56,66,67,74,82,89,98,107,156,205 sol view 104 25 363 1,4,6,12,13,20,21,25,26,27,28,32,37,41,45,53,58,64,69,91,97,102,106,155,208 sol view 105 25 364 2,3,4,6,8,9,13,14,16,19,23,24,28,29,52,57,64,75,82,91,98,100,109,173,191 sol view 106 25 367 1,4,6,12,13,20,21,25,26,27,28,32,37,41,49,53,58,64,69,91,97,102,110,155,212 sol view 107 25 368 1,6,15,16,17,18,22,25,31,33,39,42,45,46,47,48,51,69,72,88,91,96,112,160,208 sol view 108 25 371 1,2,7,8,20,21,22,24,26,28,30,38,43,46,50,51,64,65,70,90,95,102,109,160,211 sol view 109 25 373 3,6,7,8,15,17,22,23,31,32,35,41,43,60,62,68,79,87,104,105,114,120,121,138,148 sol view 110 25 378 2,3,10,17,18,20,21,22,24,27,31,38,41,48,51,56,68,78,80,85,87,96,117,165,213 sol view 111 25 378 1,2,7,13,15,17,18,25,27,29,30,31,42,43,46,56,61,68,73,93,100,105,112,161,217 sol view 112 25 380 4,7,17,18,19,20,21,26,31,33,35,40,45,48,49,60,67,73,79,81,87,107,113,186,194 sol view 113 25 380 4,5,6,9,13,15,16,17,22,24,33,38,44,49,50,56,60,67,82,84,95,108,121,177,203 sol view 114 25 381 12,13,21,23,25,27,35,36,42,45,54,57,59,60,79,82,84,85,92,95,96,100,110,111,186 sol view 115 25 384 1,4,8,9,11,12,19,21,27,32,35,44,45,46,47,51,60,67,84,89,96,108,120,180,204 sol view 116 25 384 1,4,8,9,11,12,15,17,19,25,26,31,32,37,44,57,60,81,84,96,99,108,120,180,204 sol view 117 25 384 3,5,7,11,12,17,20,22,25,29,35,47,48,55,56,57,69,72,76,80,96,100,116,172,212 sol view 118 25 385 1,2,7,13,15,17,18,25,27,29,30,31,43,46,49,56,61,68,73,93,100,105,119,161,224 sol view 119 25 392 4,7,8,15,23,26,29,30,31,32,34,43,48,55,56,68,77,88,98,106,116,135,141,151,153 sol view 120 25 392 10,12,14,16,19,21,25,27,31,35,39,41,51,52,54,55,73,92,98,115,121,123,129,148,171 sol view 121 25 392 1,4,5,8,11,14,16,21,22,24,27,28,30,31,52,64,81,83,96,97,98,99,114,195,197 sol view 122 25 393 4,8,16,20,23,24,25,27,29,37,44,45,50,53,64,66,68,69,73,85,91,101,116,186,207 sol view 123 25 396 1,4,5,14,16,32,35,36,46,47,48,49,68,69,73,93,94,97,99,104,110,111,125,126,160 sol view 124 25 396 1,4,5,8,11,14,16,21,22,24,27,28,30,31,52,64,81,83,98,99,100,101,114,197,199 sol view 125 25 396 3,8,9,11,14,16,17,18,31,32,41,45,48,56,60,66,73,75,81,82,98,99,117,180,216 sol view 126 25 398 2,6,7,11,15,17,23,28,29,39,44,46,53,56,58,65,68,99,100,119,120,134,144,145,154 sol view 127 25 400 3,6,21,23,24,26,29,35,37,40,41,47,53,55,64,76,79,81,99,100,121,122,137,142,179 sol view 128 25 404 3,6,7,14,17,20,21,26,28,31,32,39,46,53,54,68,71,80,88,92,100,111,113,199,205 sol view 129 25 404 4,7,10,11,12,13,16,18,20,23,25,28,29,32,47,62,70,88,93,96,101,114,127,189,215 sol view 130 25 408 2,3,7,13,16,18,20,27,30,33,41,43,46,52,54,57,72,79,84,100,105,108,116,195,213 sol view 131 25 412 3,11,12,15,21,26,32,39,43,47,54,60,68,73,83,85,86,87,89,99,114,129,139,144,169 sol view 132 25 413 5,7,17,20,34,38,39,48,56,57,59,60,64,65,70,72,75,81,105,106,110,125,148,153,155 sol view 133 25 416 2,4,7,11,13,24,25,30,35,37,39,40,44,58,62,65,82,104,112,120,128,135,143,153,169 sol view 134 25 416 1,2,3,8,12,15,16,17,20,22,24,26,29,31,64,75,85,88,91,94,98,104,133,179,237 sol view 135 25 421 1,2,4,5,7,9,12,16,20,22,23,35,38,48,56,83,94,104,116,118,128,140,150,153,177 sol view 136 25 421 5,11,12,17,18,20,23,26,29,36,38,40,44,51,55,59,72,92,97,102,105,107,117,199,222 sol view 137 25 422 2,4,7,13,16,18,20,23,28,29,38,43,46,51,59,68,74,79,86,93,100,111,132,179,243 sol view 138 25 425 3,4,5,9,10,12,13,14,16,19,20,31,46,48,56,79,102,104,116,126,128,140,142,157,181 sol view 139 25 441 5,6,7,16,18,23,24,27,38,39,47,51,52,62,66,72,80,84,92,101,102,118,120,219,222 sol view 140 25 454 1,2,11,17,29,34,35,46,48,51,53,55,63,69,79,87,88,91,109,134,136,143,150,161,184 sol view 141 25 456 5,7,10,11,13,15,18,19,31,49,50,52,59,60,63,72,77,115,128,129,135,142,148,179,193 sol view 142 25 465 6,9,13,14,19,21,24,25,31,32,53,56,64,73,74,82,91,111,125,127,137,139,153,173,201 sol view 143 25 472 7,9,13,15,26,34,35,44,47,51,58,61,65,81,87,103,104,115,118,123,128,133,136,148,221 sol view 144 25 477 3,5,12,16,19,22,25,26,37,41,49,72,76,77,82,86,87,115,117,135,141,149,167,169,193 sol view 145 25 492 2,9,15,17,27,29,31,32,33,36,47,49,50,60,62,69,77,105,114,123,127,128,132,237,255 sol view 146 25 492 3,5,9,11,12,21,24,27,30,44,45,50,54,55,57,63,95,101,112,117,123,134,140,235,257 sol view 147 25 503 4,15,16,19,22,23,25,27,33,34,50,62,67,87,88,93,100,113,135,143,149,157,167,179,211 sol view 148 25 506 1,7,24,26,33,35,40,45,47,51,55,69,87,90,93,96,117,125,134,145,146,147,160,162,199 sol view 149 25 507 2,3,7,11,13,15,28,34,43,50,57,64,80,83,86,89,107,115,116,127,149,163,175,183,217 sol view 150 25 512 1,7,8,9,10,15,22,32,34,46,51,65,69,71,91,105,109,111,136,139,152,157,173,200,203 sol view 151 25 512 1,6,7,8,9,13,17,19,35,45,47,57,62,73,88,93,104,107,128,130,151,163,184,198,221 sol view 152 25 513 6,9,10,17,19,24,28,29,37,39,64,65,68,81,98,99,102,115,145,147,153,159,165,189,201 sol view 153 25 517 5,6,7,16,20,24,28,33,38,43,63,71,80,83,86,92,98,122,132,148,164,166,173,180,205 sol view 154 25 524 9,12,20,21,33,35,37,39,54,55,61,62,87,90,98,101,125,132,135,141,145,159,163,164,220 sol view 155 25 527 11,12,13,14,19,30,41,47,50,52,59,68,71,81,94,97,107,132,147,151,155,169,175,183,197 sol view 156 25 528 2,9,15,17,27,29,31,32,33,36,47,49,50,60,62,69,77,123,127,128,132,141,150,255,273 sol view 157 25 529 9,12,20,21,33,35,37,39,54,55,61,62,87,90,98,101,125,132,140,141,145,159,163,169,225 sol view 158 25 531 6,9,10,17,19,24,29,31,39,40,67,68,71,84,101,102,105,118,151,153,159,165,171,195,207 sol view 159 25 532 16,18,26,27,33,39,41,50,51,55,69,71,84,87,91,94,132,133,141,143,164,168,169,173,195 sol view 160 25 534 11,13,15,17,18,27,38,44,49,52,60,61,68,81,87,94,107,135,149,153,159,171,174,189,210 sol view 161 25 535 2,8,26,27,36,41,45,57,62,77,88,95,97,99,101,102,109,114,117,118,141,147,168,192,226 sol view 162 25 536 1,8,21,30,31,32,33,41,44,46,49,55,57,61,84,91,113,134,137,139,150,155,176,205,247 sol view 163 25 536 3,5,9,11,12,21,24,27,30,44,45,50,54,55,57,63,95,117,123,134,140,145,156,257,279 sol view 164 25 540 1,7,8,9,10,14,19,34,36,51,58,69,81,83,97,109,111,115,136,149,152,167,183,208,221 sol view 165 25 540 6,13,15,25,28,36,43,47,55,57,58,59,60,65,82,89,91,107,124,127,144,163,183,233,250 sol view 166 25 540 8,9,10,11,16,30,36,38,45,55,57,65,68,81,84,95,98,100,116,117,126,135,144,261,279 sol view 167 25 540 8,9,10,11,16,30,36,38,45,55,57,65,68,81,84,95,98,100,116,117,126,135,144,261,279 sol view 168 25 540 4,6,9,10,17,21,23,25,31,33,36,38,45,50,81,83,115,117,126,133,135,144,146,261,279 sol view 169 25 540 4,6,9,10,17,21,23,25,31,33,36,38,45,50,81,83,115,117,126,133,135,144,146,261,279 sol view 170 25 541 3,4,11,13,16,17,21,25,26,44,46,64,75,86,87,97,106,109,133,141,165,185,191,215,217 sol view 171 25 541 3,5,27,32,33,37,47,50,53,56,57,69,71,78,97,98,109,111,126,144,165,169,183,189,232 sol view 172 25 544 1,7,24,26,33,35,40,45,47,51,55,69,87,90,93,96,117,125,134,145,147,184,198,199,200 sol view 173 25 544 6,8,20,21,23,41,42,48,59,61,77,80,81,85,90,92,93,102,115,132,139,168,198,207,244 sol view 174 25 547 3,5,16,22,26,27,35,47,49,59,67,71,72,85,87,102,103,111,137,144,150,197,200,203,207 sol view 175 25 549 4,10,14,24,26,31,34,36,38,40,43,48,59,63,74,89,97,105,117,124,136,152,156,241,308 sol view 176 25 550 1,2,5,13,19,20,25,30,39,43,58,59,73,75,76,90,95,103,116,128,130,132,172,262,288 sol view 177 25 550 1,11,16,23,24,27,29,36,41,43,44,47,59,70,71,80,99,103,111,116,128,156,167,227,323 sol view 178 25 551 3,5,24,25,26,30,35,36,39,40,42,57,68,76,94,109,120,128,152,162,166,175,176,200,223 sol view 179 25 552 5,17,18,22,25,27,32,33,39,59,62,87,91,100,102,111,112,135,137,149,165,168,183,201,204 sol view 180 25 552 1,3,4,7,8,9,10,15,18,19,21,41,52,54,73,93,95,123,125,136,138,153,168,261,291 sol view 181 25 556 6,8,10,13,19,25,32,37,49,54,58,76,84,91,92,100,107,128,145,156,165,185,195,205,206 sol view 182 25 556 3,12,13,15,19,23,27,34,35,39,42,45,48,52,53,87,140,145,158,166,171,184,189,201,227 sol view 183 25 556 3,12,13,15,19,23,27,34,35,39,42,45,48,52,53,87,140,145,158,166,171,184,189,201,227 sol view 184 25 556 1,5,7,8,9,10,12,14,20,27,31,43,47,50,74,93,97,121,125,139,143,153,167,264,292 sol view 185 25 562 2,3,5,8,13,19,20,29,33,47,53,54,64,65,76,93,119,123,142,157,161,180,184,221,259 sol view 186 25 570 3,9,10,33,36,38,40,42,50,51,60,69,72,75,77,90,113,140,141,151,152,189,200,229,230 sol view 187 25 575 4,6,14,16,31,39,63,69,74,81,88,103,107,111,115,120,131,132,133,147,156,159,164,198,218 sol view 188 25 576 1,4,9,11,15,19,22,34,36,53,60,76,82,84,104,126,127,128,153,156,165,174,183,219,237 sol view 189 25 576 8,9,10,11,16,30,36,38,45,55,57,65,68,81,84,95,98,100,116,135,144,153,162,279,297 sol view 190 25 576 4,6,9,10,17,21,23,25,31,33,36,38,45,50,81,83,115,133,135,144,146,153,162,279,297 sol view 191 25 580 2,5,7,10,12,13,19,21,22,29,36,40,61,65,74,101,135,139,161,179,183,192,205,209,236 sol view 192 25 580 5,6,11,13,16,17,21,25,34,44,54,68,80,88,100,112,120,135,142,145,170,173,195,215,265 sol view 193 25 580 11,12,16,17,29,32,39,41,53,55,59,60,68,70,81,84,92,124,125,128,129,156,171,280,300 sol view 194 25 593 13,14,15,35,48,51,55,67,73,79,83,91,94,105,109,116,119,124,133,150,171,173,196,217,226 sol view 195 25 595 4,13,18,19,22,35,40,48,58,61,62,77,78,82,83,86,118,149,163,168,187,192,202,206,240 sol view 196 25 601 7,8,25,34,41,42,46,48,54,55,62,70,71,74,98,103,116,143,168,169,190,192,193,218,240 sol view 197 25 603 7,11,12,14,21,25,32,40,52,56,60,67,68,81,91,92,132,144,149,163,177,191,196,235,263 sol view 198 25 603 13,23,26,27,35,44,45,49,53,54,57,66,75,99,101,110,122,126,144,158,175,180,189,234,270 sol view 199 25 607 6,8,10,13,19,25,32,37,49,54,58,76,84,91,92,100,107,128,156,185,196,205,206,216,246 sol view 200 25 609 9,14,15,17,32,45,47,58,67,74,76,79,80,83,97,111,125,126,150,170,186,188,215,224,235 sol view 201 25 611 1,10,22,26,32,41,45,54,57,61,62,66,85,86,87,95,97,101,119,132,136,167,176,268,343 sol view 202 25 614 15,22,24,31,33,49,53,54,57,60,63,68,74,81,83,104,109,151,155,163,167,217,229,230,234 sol view 203 25 634 15,17,24,26,33,43,44,54,57,60,63,73,79,81,88,109,119,160,161,172,173,227,234,235,239 sol view 204 25 643 2,9,21,29,38,40,41,42,58,62,67,76,82,83,85,96,104,166,172,186,192,201,207,250,270 sol view 205 25 644 7,9,13,18,19,22,31,49,53,61,66,68,71,87,93,94,119,164,178,192,199,206,227,239,253 sol view 206 25 655 10,14,15,21,25,26,31,40,51,53,54,57,65,83,84,86,151,152,173,193,194,215,216,246,288 sol view 207 25 661 5,7,17,18,23,31,36,38,41,64,73,77,83,84,102,106,111,161,175,196,203,210,238,248,262 sol view

For each squared square instance, the graphical view shows:

Within the context of root and trashing, each graphical view from left to right consists of:

1. A two-dimensional view, which displays the compulsory part of the squares. In this context a square has a compulsory part if it has a compulsory part on both dimensions. Completely fixed squares are coloured in light grey or white, while not completely fixed squares are coloured in light red.

2. A cumulative view on the first dimension, which displays the squares that have a compulsory part with respect to the first dimension. Yellow squares correspond to squares that have a compulsory part in the first dimension but no compulsory part in the second dimension. All squares that are below the fat yellow profile correspond to squares for which the origin in the first dimension is fixed.

3. A cumulative view on the second dimension, which displays the squares that have a compulsory part with respect to the second dimension. Orange squares correspond to squares that have a compulsory part in the second dimension but no compulsory part in the first dimension. All squares that are below the fat orange profile correspond to squares for which the origin in the second dimension is fixed.

Within the context of left branch, each graphical view consists of similar graphic views. The only difference is that these views show how the compulsory part of an object increases as we go down a branch of the search tree. In this context the additional part of the compulsory part of an object is coloured in lighter blue as we go down a branch. This allows to see how the compulsory part of an object increases along a branch of the search tree. Finally not fat profile is shown in this context since we would have too many of them (i.e., one for each node of the path).

Let us comment the view associated with the first instance of the squared squares problems:

1. In the context of root, the biggest square (i.e., the light red square labelled with a 1) has a compulsory part thanks to the symmetry constraint. It also has a compulsory part with respect to the first dimension (note that we draw a rectangle since we consider the full height of the corresponding square). Finally it also has a compulsory part with respect to the second dimension.

2. In the context of trashing, the biggest square (i.e., the white square labelled with a 1) is completely fixed at the lower leftmost corner of the Β placement space. The second biggest square (i.e., the light red square labelled with a 2) is not completely fixed but has a compulsory part in both dimensions. Finally a third square has a tiny compulsory part. We retrieve these three squares in the cumulative views associated with the first and second dimension. Finally, within the cumulative view associated with the first dimension we also have a yellow square labelled with 8, which corresponds to a square that only has a compulsory part in the first dimension. On the one hand, the fat yellow profile corresponding to the first dimension consists of the three squares 1, 3, 8, which are completely fixed with respect to the first dimension. On the other hand the fat orange profile corresponding to the second dimension consists of the two squares 1 and 2.

3. In the context of left branch, we can observe how the compulsory part grows as we go down the leftmost branch of the search tree. For instance at the root node we have a small compulsory part for the biggest square (i.e., the square labelled by 1). Then, as we go down the leftmost branch, this compulsory part gradually increases until the square becomes completely fixed in the next two steps. We can also observe that the heuristics tends to gradually create compulsory parts on the cumulative view associated with the first dimension (in light blue on the second picture).