3.7.230. 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 3.7.59 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.

Figure 3.7.59. A simple perfect squared squares of order 21
ctrs/squared_squares

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 inΒ [AggounBeldiceanu93], inΒ [VanHentenryck94], inΒ [SimonisBourreauBeldiceanu99], inΒ [BeldiceanuCarlssonPoderSadekTruchet07] and inΒ [BeldiceanuCarlssonPoder08].

1211122,4,6,7,8,9,11,15,16,17,18,19,24,25,27,29,33,35,37,42,50solview
2221102,3,4,6,7,8,12,13,14,15,16,17,18,21,22,23,24,26,27,28,50,60solview
3221101,2,3,4,6,8,9,12,14,16,17,18,19,21,22,23,24,26,27,28,50,60solview
4221391,2,3,4,7,8,10,17,18,20,21,22,24,27,28,29,30,31,32,38,59,80solview
5221471,3,4,5,8,9,17,20,21,23,25,26,29,31,32,40,43,44,47,48,52,55solview
6221472,4,8,10,11,12,15,19,21,22,23,25,26,32,34,37,41,43,45,47,55,59solview
7221542,5,9,11,16,17,19,21,22,24,26,30,31,33,35,36,41,46,47,50,52,61solview
8221721,2,3,4,9,11,13,16,17,18,19,22,24,33,36,38,39,42,44,53,75,97solview
9221924,8,9,10,12,14,17,19,26,28,31,35,36,37,41,47,49,57,59,62,71,86solview
10231101,2,3,4,5,7,8,10,12,13,14,15,16,19,21,28,29,31,32,37,38,41,44solview
11231391,2,7,8,12,13,14,15,16,18,19,20,21,22,24,26,27,28,32,33,38,59,80solview
12231401,2,3,4,5,8,10,13,16,19,20,23,27,28,29,31,33,38,42,45,48,53,54solview
13231402,3,4,7,8,9,12,15,16,18,22,23,24,26,28,30,33,36,43,44,47,50,60solview
14231451,2,3,4,6,8,9,12,15,20,22,24,25,26,27,29,30,31,32,34,36,61,84solview
15231802,4,8,10,11,12,15,19,21,22,23,25,26,32,33,34,37,41,43,45,47,88,92solview
16231882,4,8,10,11,12,15,19,21,22,23,25,26,32,33,34,37,45,47,49,51,92,96solview
17232081,3,4,9,10,11,12,16,17,18,22,23,24,40,41,60,62,65,67,70,71,73,75solview
18232151,3,4,9,10,11,12,16,17,18,22,23,24,40,41,60,66,68,70,71,74,76,79solview
19232282,7,9,10,15,16,17,18,22,23,25,28,36,39,42,56,57,68,69,72,73,87,99solview
20232572,3,9,11,14,15,17,20,22,24,28,29,32,33,49,55,57,60,63,66,79,123,134solview
21233321,15,17,24,26,30,31,38,47,48,49,50,53,56,58,68,83,89,91,112,120,123,129solview
22241203,4,5,6,8,9,10,12,13,14,15,16,17,19,20,23,25,32,33,34,40,41,46,47solview
23241862,3,4,7,8,9,12,15,16,18,22,23,24,26,28,30,33,36,43,46,47,60,90,96solview
24241942,3,7,9,10,16,17,18,19,20,23,25,28,34,36,37,42,53,54,61,65,68,69,72solview
25241952,4,7,10,11,16,17,18,21,26,27,30,39,41,42,45,47,49,52,53,54,61,63,80solview
26241961,2,5,10,11,15,17,18,20,21,24,26,29,31,32,34,36,40,44,47,48,51,91,105solview
27242011,3,4,6,9,10,11,12,17,18,20,21,22,23,26,38,40,46,50,52,53,58,98,103solview
28242011,4,5,8,9,10,11,15,16,18,19,20,22,24,26,39,42,44,49,52,54,56,93,108solview
29242031,2,5,10,11,15,17,18,20,21,24,26,29,31,32,34,36,40,44,48,54,58,98,105solview
30242473,5,6,9,12,14,19,23,24,25,28,32,34,36,40,45,46,48,56,62,63,66,111,136solview
31242532,4,5,9,13,18,20,23,24,27,28,31,38,40,44,50,61,70,72,77,79,86,88,104solview
32242553,5,10,11,16,17,20,22,23,25,26,27,28,32,41,44,52,53,59,63,65,74,118,137solview
33242882,7,9,10,15,16,17,18,22,23,25,28,36,39,42,56,57,60,68,72,73,87,129,159solview
34242881,5,7,8,9,14,17,20,21,26,30,32,34,36,48,51,54,59,64,69,72,93,123,165solview
35242902,3,8,9,11,12,14,17,21,30,31,33,40,42,45,48,59,61,63,65,82,84,124,166solview
36242921,2,3,8,12,15,16,17,20,22,24,26,29,33,44,54,57,60,63,67,73,102,117,175solview
37243043,5,7,11,12,17,20,22,25,29,35,47,48,55,56,57,69,72,76,92,96,100,116,132solview
38243043,4,7,12,16,20,23,24,27,28,30,32,33,36,37,44,53,57,72,76,85,99,129,175solview
39243142,4,11,12,16,17,18,19,28,29,40,44,47,59,62,64,65,78,79,96,97,105,113,139solview
40243163,9,10,12,13,14,15,23,24,33,36,37,48,52,54,55,57,65,66,78,79,93,144,172solview
41243261,6,10,11,14,15,18,24,29,32,43,44,53,56,63,65,71,80,83,101,104,106,119,142solview
42244232,9,15,17,27,29,31,32,33,36,47,49,50,60,62,77,105,114,123,127,128,132,168,186solview
43244351,2,8,10,13,19,23,33,44,45,56,74,76,78,80,88,93,100,112,131,142,143,150,192solview
44244353,5,9,11,12,21,24,27,30,44,45,50,54,55,63,95,101,112,117,123,134,140,178,200solview
45244598,9,10,11,16,30,36,38,45,55,57,65,68,84,95,98,100,116,117,126,135,144,180,198solview
46244594,6,9,10,17,21,23,25,31,33,36,38,45,50,83,115,117,126,133,135,144,146,180,198solview
47244795,6,17,23,24,26,28,29,35,43,44,52,60,68,77,86,130,140,150,155,160,164,174,175solview
48251473,4,5,6,8,9,10,12,13,14,15,16,17,19,20,23,25,27,32,33,34,40,41,73,74solview
49252081,2,3,4,5,7,8,11,12,17,18,24,26,28,29,30,36,39,44,45,50,59,60,89,119solview
50252133,5,6,7,13,16,17,20,21,23,24,25,26,28,31,35,36,47,49,56,58,74,76,81,90solview
51252151,4,6,7,11,15,24,26,27,33,37,39,40,41,42,43,45,47,51,55,60,62,63,69,83solview
52252161,2,3,4,5,7,8,11,16,17,18,19,25,30,32,33,39,41,45,49,54,59,64,103,113solview
53252361,2,4,9,11,12,13,14,15,16,19,24,38,40,44,46,47,48,59,64,65,70,81,85,107solview
54252421,3,6,7,9,13,14,16,17,19,23,25,26,28,30,31,47,51,54,57,60,64,67,111,131solview
55252441,2,4,5,7,10,15,17,19,20,21,22,26,27,30,37,40,41,45,65,66,68,70,110,134solview
56252524,7,10,11,12,13,23,25,29,31,32,34,36,37,38,40,42,44,62,67,68,71,77,108,113solview
57252532,4,5,6,9,10,12,14,20,24,27,35,36,37,38,42,43,45,50,54,63,66,70,120,133solview
58252601,4,6,7,10,15,24,26,27,28,29,31,33,34,37,38,44,65,70,71,77,78,83,100,112solview
59252643,7,8,12,16,18,19,20,22,24,26,31,34,37,38,40,42,53,54,61,64,69,70,130,134solview
60252643,8,12,13,16,18,20,21,22,24,26,29,34,38,40,42,43,47,54,59,64,70,71,130,134solview
61252641,3,4,6,9,10,11,12,16,17,18,20,21,22,39,42,54,56,61,66,68,69,73,129,135solview
62252651,3,4,6,9,10,11,12,16,17,18,20,21,22,39,42,54,56,62,66,68,69,74,130,135solview
63252731,4,8,10,11,12,17,19,21,22,27,29,30,33,37,43,52,62,65,86,88,89,91,96,120solview
64252731,6,9,14,16,17,18,21,22,23,25,31,32,38,44,46,48,50,54,62,65,68,78,133,140solview
65252752,3,7,13,17,24,25,31,33,34,35,37,41,49,51,53,55,60,68,71,74,81,94,100,107solview
66252761,5,8,9,11,18,19,21,30,36,41,44,45,46,47,51,53,58,63,69,71,84,87,105,120solview
67252805,6,11,17,18,20,21,24,27,28,32,34,41,42,50,53,54,55,68,78,85,88,95,97,117solview
68252802,3,7,8,14,18,30,36,37,39,44,50,52,54,56,60,63,64,65,72,75,78,79,96,106solview
69252841,2,11,12,14,16,18,19,23,26,29,37,38,39,40,42,59,68,69,77,78,97,106,109,110solview
70252861,4,5,7,10,12,15,16,20,23,28,30,32,33,35,37,53,54,64,68,74,79,80,133,153solview
71252892,3,5,8,13,14,17,20,21,32,36,41,50,52,60,61,62,68,74,76,83,87,100,102,104solview
72252892,3,4,5,7,12,16,17,19,21,23,25,29,31,32,44,57,64,65,68,72,76,84,140,149solview
73252901,2,10,11,13,14,15,17,18,28,29,34,36,38,50,56,60,69,77,80,85,91,94,111,119solview
74252935,6,11,17,18,20,21,24,27,28,32,34,41,42,50,54,55,66,68,78,85,88,95,110,130solview
75252972,7,8,9,10,15,16,17,18,23,25,26,28,36,38,43,53,60,61,68,69,77,99,137,160solview
76253081,3,4,7,10,12,13,23,25,34,37,38,39,43,44,45,62,77,79,85,87,108,113,115,116solview
77253081,5,6,7,8,9,13,16,19,28,33,36,38,43,45,48,70,71,73,84,86,102,104,120,133solview
78253097,8,14,16,23,24,25,26,31,33,34,39,48,56,59,60,62,70,76,82,92,100,101,108,117solview
79253112,7,8,9,10,15,16,17,18,23,25,26,28,36,38,43,53,60,61,68,83,91,99,151,160solview
80253141,6,7,11,16,22,26,29,32,36,38,44,51,53,64,69,70,73,74,75,85,87,101,116,128solview
81253161,3,9,12,21,26,30,33,34,35,38,39,40,41,53,56,59,69,79,85,96,103,111,117,120solview
82253171,5,6,7,8,9,16,17,19,32,37,40,42,47,49,52,59,75,81,92,94,110,112,113,126solview
83253202,7,8,9,12,14,15,21,23,35,38,44,46,49,53,54,56,63,96,101,103,105,108,112,116solview
84253203,8,9,11,17,18,22,25,26,27,29,30,31,33,35,49,51,67,72,73,80,85,95,152,168solview
85253201,4,6,7,8,13,14,16,24,28,30,33,34,38,41,42,57,60,69,78,81,90,92,150,170solview
86253203,4,6,8,9,14,15,16,24,28,30,31,34,38,39,42,59,60,71,78,79,90,92,150,170solview
87253223,4,8,9,10,16,18,20,22,23,24,28,31,38,44,47,64,65,68,76,80,81,97,144,178solview
88253223,4,8,10,15,16,18,19,20,22,24,28,35,38,44,53,59,64,68,76,80,85,93,144,178solview
89253232,3,4,7,10,13,15,18,23,32,34,35,36,42,46,50,57,60,66,72,78,87,98,159,164solview
90253233,8,9,11,17,18,22,25,26,27,29,30,31,33,35,49,51,67,72,73,83,88,95,155,168solview
91253232,6,9,11,13,14,18,19,20,23,27,28,29,42,46,48,60,64,72,74,79,82,98,146,177solview
92253253,5,6,11,12,13,18,23,25,28,32,37,40,43,45,46,51,79,92,99,103,108,112,114,134solview
93253261,4,8,10,12,16,21,22,24,27,28,35,36,37,38,46,49,68,70,75,88,90,93,158,168solview
94253272,9,10,12,13,16,19,21,23,26,36,44,46,52,55,61,62,74,84,87,100,103,104,120,140solview
95253282,3,4,7,8,10,14,17,26,27,28,36,38,40,42,45,53,58,73,74,79,94,102,152,176solview
96253341,4,8,10,12,16,21,22,24,27,28,35,36,37,38,46,49,68,75,78,88,93,98,166,168solview
97253362,3,4,7,8,10,14,17,26,27,28,36,38,40,45,50,53,58,73,74,79,94,110,152,184solview
98253381,4,8,10,12,16,19,22,24,25,28,36,37,38,39,46,53,68,70,73,94,96,101,164,174solview
99253384,5,8,10,12,15,16,21,22,24,28,33,36,38,43,46,57,68,70,77,94,96,97,164,174solview
100253401,4,5,6,11,13,16,17,22,24,44,46,50,51,52,53,61,64,66,79,84,85,92,169,171solview
101253442,3,8,11,14,17,19,21,23,25,27,36,39,44,48,53,56,71,77,83,86,89,98,169,175solview
102253597,8,9,10,14,17,18,23,25,27,29,31,40,41,43,46,69,74,82,85,90,98,102,172,187solview
103253612,6,7,8,9,14,20,22,26,27,32,34,36,47,49,56,66,67,74,82,89,98,107,156,205solview
104253631,4,6,12,13,20,21,25,26,27,28,32,37,41,45,53,58,64,69,91,97,102,106,155,208solview
105253642,3,4,6,8,9,13,14,16,19,23,24,28,29,52,57,64,75,82,91,98,100,109,173,191solview
106253671,4,6,12,13,20,21,25,26,27,28,32,37,41,49,53,58,64,69,91,97,102,110,155,212solview
107253681,6,15,16,17,18,22,25,31,33,39,42,45,46,47,48,51,69,72,88,91,96,112,160,208solview
108253711,2,7,8,20,21,22,24,26,28,30,38,43,46,50,51,64,65,70,90,95,102,109,160,211solview
109253733,6,7,8,15,17,22,23,31,32,35,41,43,60,62,68,79,87,104,105,114,120,121,138,148solview
110253782,3,10,17,18,20,21,22,24,27,31,38,41,48,51,56,68,78,80,85,87,96,117,165,213solview
111253781,2,7,13,15,17,18,25,27,29,30,31,42,43,46,56,61,68,73,93,100,105,112,161,217solview
112253804,7,17,18,19,20,21,26,31,33,35,40,45,48,49,60,67,73,79,81,87,107,113,186,194solview
113253804,5,6,9,13,15,16,17,22,24,33,38,44,49,50,56,60,67,82,84,95,108,121,177,203solview
1142538112,13,21,23,25,27,35,36,42,45,54,57,59,60,79,82,84,85,92,95,96,100,110,111,186solview
115253841,4,8,9,11,12,19,21,27,32,35,44,45,46,47,51,60,67,84,89,96,108,120,180,204solview
116253841,4,8,9,11,12,15,17,19,25,26,31,32,37,44,57,60,81,84,96,99,108,120,180,204solview
117253843,5,7,11,12,17,20,22,25,29,35,47,48,55,56,57,69,72,76,80,96,100,116,172,212solview
118253851,2,7,13,15,17,18,25,27,29,30,31,43,46,49,56,61,68,73,93,100,105,119,161,224solview
119253924,7,8,15,23,26,29,30,31,32,34,43,48,55,56,68,77,88,98,106,116,135,141,151,153solview
1202539210,12,14,16,19,21,25,27,31,35,39,41,51,52,54,55,73,92,98,115,121,123,129,148,171solview
121253921,4,5,8,11,14,16,21,22,24,27,28,30,31,52,64,81,83,96,97,98,99,114,195,197solview
122253934,8,16,20,23,24,25,27,29,37,44,45,50,53,64,66,68,69,73,85,91,101,116,186,207solview
123253961,4,5,14,16,32,35,36,46,47,48,49,68,69,73,93,94,97,99,104,110,111,125,126,160solview
124253961,4,5,8,11,14,16,21,22,24,27,28,30,31,52,64,81,83,98,99,100,101,114,197,199solview
125253963,8,9,11,14,16,17,18,31,32,41,45,48,56,60,66,73,75,81,82,98,99,117,180,216solview
126253982,6,7,11,15,17,23,28,29,39,44,46,53,56,58,65,68,99,100,119,120,134,144,145,154solview
127254003,6,21,23,24,26,29,35,37,40,41,47,53,55,64,76,79,81,99,100,121,122,137,142,179solview
128254043,6,7,14,17,20,21,26,28,31,32,39,46,53,54,68,71,80,88,92,100,111,113,199,205solview
129254044,7,10,11,12,13,16,18,20,23,25,28,29,32,47,62,70,88,93,96,101,114,127,189,215solview
130254082,3,7,13,16,18,20,27,30,33,41,43,46,52,54,57,72,79,84,100,105,108,116,195,213solview
131254123,11,12,15,21,26,32,39,43,47,54,60,68,73,83,85,86,87,89,99,114,129,139,144,169solview
132254135,7,17,20,34,38,39,48,56,57,59,60,64,65,70,72,75,81,105,106,110,125,148,153,155solview
133254162,4,7,11,13,24,25,30,35,37,39,40,44,58,62,65,82,104,112,120,128,135,143,153,169solview
134254161,2,3,8,12,15,16,17,20,22,24,26,29,31,64,75,85,88,91,94,98,104,133,179,237solview
135254211,2,4,5,7,9,12,16,20,22,23,35,38,48,56,83,94,104,116,118,128,140,150,153,177solview
136254215,11,12,17,18,20,23,26,29,36,38,40,44,51,55,59,72,92,97,102,105,107,117,199,222solview
137254222,4,7,13,16,18,20,23,28,29,38,43,46,51,59,68,74,79,86,93,100,111,132,179,243solview
138254253,4,5,9,10,12,13,14,16,19,20,31,46,48,56,79,102,104,116,126,128,140,142,157,181solview
139254415,6,7,16,18,23,24,27,38,39,47,51,52,62,66,72,80,84,92,101,102,118,120,219,222solview
140254541,2,11,17,29,34,35,46,48,51,53,55,63,69,79,87,88,91,109,134,136,143,150,161,184solview
141254565,7,10,11,13,15,18,19,31,49,50,52,59,60,63,72,77,115,128,129,135,142,148,179,193solview
142254656,9,13,14,19,21,24,25,31,32,53,56,64,73,74,82,91,111,125,127,137,139,153,173,201solview
143254727,9,13,15,26,34,35,44,47,51,58,61,65,81,87,103,104,115,118,123,128,133,136,148,221solview
144254773,5,12,16,19,22,25,26,37,41,49,72,76,77,82,86,87,115,117,135,141,149,167,169,193solview
145254922,9,15,17,27,29,31,32,33,36,47,49,50,60,62,69,77,105,114,123,127,128,132,237,255solview
146254923,5,9,11,12,21,24,27,30,44,45,50,54,55,57,63,95,101,112,117,123,134,140,235,257solview
147255034,15,16,19,22,23,25,27,33,34,50,62,67,87,88,93,100,113,135,143,149,157,167,179,211solview
148255061,7,24,26,33,35,40,45,47,51,55,69,87,90,93,96,117,125,134,145,146,147,160,162,199solview
149255072,3,7,11,13,15,28,34,43,50,57,64,80,83,86,89,107,115,116,127,149,163,175,183,217solview
150255121,7,8,9,10,15,22,32,34,46,51,65,69,71,91,105,109,111,136,139,152,157,173,200,203solview
151255121,6,7,8,9,13,17,19,35,45,47,57,62,73,88,93,104,107,128,130,151,163,184,198,221solview
152255136,9,10,17,19,24,28,29,37,39,64,65,68,81,98,99,102,115,145,147,153,159,165,189,201solview
153255175,6,7,16,20,24,28,33,38,43,63,71,80,83,86,92,98,122,132,148,164,166,173,180,205solview
154255249,12,20,21,33,35,37,39,54,55,61,62,87,90,98,101,125,132,135,141,145,159,163,164,220solview
1552552711,12,13,14,19,30,41,47,50,52,59,68,71,81,94,97,107,132,147,151,155,169,175,183,197solview
156255282,9,15,17,27,29,31,32,33,36,47,49,50,60,62,69,77,123,127,128,132,141,150,255,273solview
157255299,12,20,21,33,35,37,39,54,55,61,62,87,90,98,101,125,132,140,141,145,159,163,169,225solview
158255316,9,10,17,19,24,29,31,39,40,67,68,71,84,101,102,105,118,151,153,159,165,171,195,207solview
1592553216,18,26,27,33,39,41,50,51,55,69,71,84,87,91,94,132,133,141,143,164,168,169,173,195solview
1602553411,13,15,17,18,27,38,44,49,52,60,61,68,81,87,94,107,135,149,153,159,171,174,189,210solview
161255352,8,26,27,36,41,45,57,62,77,88,95,97,99,101,102,109,114,117,118,141,147,168,192,226solview
162255361,8,21,30,31,32,33,41,44,46,49,55,57,61,84,91,113,134,137,139,150,155,176,205,247solview
163255363,5,9,11,12,21,24,27,30,44,45,50,54,55,57,63,95,117,123,134,140,145,156,257,279solview
164255401,7,8,9,10,14,19,34,36,51,58,69,81,83,97,109,111,115,136,149,152,167,183,208,221solview
165255406,13,15,25,28,36,43,47,55,57,58,59,60,65,82,89,91,107,124,127,144,163,183,233,250solview
166255408,9,10,11,16,30,36,38,45,55,57,65,68,81,84,95,98,100,116,117,126,135,144,261,279solview
167255408,9,10,11,16,30,36,38,45,55,57,65,68,81,84,95,98,100,116,117,126,135,144,261,279solview
168255404,6,9,10,17,21,23,25,31,33,36,38,45,50,81,83,115,117,126,133,135,144,146,261,279solview
169255404,6,9,10,17,21,23,25,31,33,36,38,45,50,81,83,115,117,126,133,135,144,146,261,279solview
170255413,4,11,13,16,17,21,25,26,44,46,64,75,86,87,97,106,109,133,141,165,185,191,215,217solview
171255413,5,27,32,33,37,47,50,53,56,57,69,71,78,97,98,109,111,126,144,165,169,183,189,232solview
172255441,7,24,26,33,35,40,45,47,51,55,69,87,90,93,96,117,125,134,145,147,184,198,199,200solview
173255446,8,20,21,23,41,42,48,59,61,77,80,81,85,90,92,93,102,115,132,139,168,198,207,244solview
174255473,5,16,22,26,27,35,47,49,59,67,71,72,85,87,102,103,111,137,144,150,197,200,203,207solview
175255494,10,14,24,26,31,34,36,38,40,43,48,59,63,74,89,97,105,117,124,136,152,156,241,308solview
176255501,2,5,13,19,20,25,30,39,43,58,59,73,75,76,90,95,103,116,128,130,132,172,262,288solview
177255501,11,16,23,24,27,29,36,41,43,44,47,59,70,71,80,99,103,111,116,128,156,167,227,323solview
178255513,5,24,25,26,30,35,36,39,40,42,57,68,76,94,109,120,128,152,162,166,175,176,200,223solview
179255525,17,18,22,25,27,32,33,39,59,62,87,91,100,102,111,112,135,137,149,165,168,183,201,204solview
180255521,3,4,7,8,9,10,15,18,19,21,41,52,54,73,93,95,123,125,136,138,153,168,261,291solview
181255566,8,10,13,19,25,32,37,49,54,58,76,84,91,92,100,107,128,145,156,165,185,195,205,206solview
182255563,12,13,15,19,23,27,34,35,39,42,45,48,52,53,87,140,145,158,166,171,184,189,201,227solview
183255563,12,13,15,19,23,27,34,35,39,42,45,48,52,53,87,140,145,158,166,171,184,189,201,227solview
184255561,5,7,8,9,10,12,14,20,27,31,43,47,50,74,93,97,121,125,139,143,153,167,264,292solview
185255622,3,5,8,13,19,20,29,33,47,53,54,64,65,76,93,119,123,142,157,161,180,184,221,259solview
186255703,9,10,33,36,38,40,42,50,51,60,69,72,75,77,90,113,140,141,151,152,189,200,229,230solview
187255754,6,14,16,31,39,63,69,74,81,88,103,107,111,115,120,131,132,133,147,156,159,164,198,218solview
188255761,4,9,11,15,19,22,34,36,53,60,76,82,84,104,126,127,128,153,156,165,174,183,219,237solview
189255768,9,10,11,16,30,36,38,45,55,57,65,68,81,84,95,98,100,116,135,144,153,162,279,297solview
190255764,6,9,10,17,21,23,25,31,33,36,38,45,50,81,83,115,133,135,144,146,153,162,279,297solview
191255802,5,7,10,12,13,19,21,22,29,36,40,61,65,74,101,135,139,161,179,183,192,205,209,236solview
192255805,6,11,13,16,17,21,25,34,44,54,68,80,88,100,112,120,135,142,145,170,173,195,215,265solview
1932558011,12,16,17,29,32,39,41,53,55,59,60,68,70,81,84,92,124,125,128,129,156,171,280,300solview
1942559313,14,15,35,48,51,55,67,73,79,83,91,94,105,109,116,119,124,133,150,171,173,196,217,226solview
195255954,13,18,19,22,35,40,48,58,61,62,77,78,82,83,86,118,149,163,168,187,192,202,206,240solview
196256017,8,25,34,41,42,46,48,54,55,62,70,71,74,98,103,116,143,168,169,190,192,193,218,240solview
197256037,11,12,14,21,25,32,40,52,56,60,67,68,81,91,92,132,144,149,163,177,191,196,235,263solview
1982560313,23,26,27,35,44,45,49,53,54,57,66,75,99,101,110,122,126,144,158,175,180,189,234,270solview
199256076,8,10,13,19,25,32,37,49,54,58,76,84,91,92,100,107,128,156,185,196,205,206,216,246solview
200256099,14,15,17,32,45,47,58,67,74,76,79,80,83,97,111,125,126,150,170,186,188,215,224,235solview
201256111,10,22,26,32,41,45,54,57,61,62,66,85,86,87,95,97,101,119,132,136,167,176,268,343solview
2022561415,22,24,31,33,49,53,54,57,60,63,68,74,81,83,104,109,151,155,163,167,217,229,230,234solview
2032563415,17,24,26,33,43,44,54,57,60,63,73,79,81,88,109,119,160,161,172,173,227,234,235,239solview
204256432,9,21,29,38,40,41,42,58,62,67,76,82,83,85,96,104,166,172,186,192,201,207,250,270solview
205256447,9,13,18,19,22,31,49,53,61,66,68,71,87,93,94,119,164,178,192,199,206,227,239,253solview
2062565510,14,15,21,25,26,31,40,51,53,54,57,65,83,84,86,151,152,173,193,194,215,216,246,288solview
207256615,7,17,18,23,31,36,38,41,64,73,77,83,84,102,106,111,161,175,196,203,210,238,248,262solview

For each squared square instance, the graphical view shows:

  • root: the status just after posting the 𝚐𝚎𝚘𝚜𝚝 constraint,

  • leftbranch: the status just after the last node on the leftmost branch of the search tree,

  • trashing: the status at the first node of the search tree where trashing occurs (i.e.Β trashing occurs at a given node if we do not immediately fail at this node but we immediately fail on all children of that node).

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 leftbranch, 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.,Β 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.,Β 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.

    Figure 3.7.60. the root view for instance squared_squares_001
    figpstrick/squared_squares_id001_root
  2. In the context of trashing, the biggest square (i.e.,Β 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.,Β 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.

    Figure 3.7.61. the trashing view for instance squared_squares_001
    figpstrick/squared_squares_id001_thrashing
  3. In the context of leftbranch, 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.,Β 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).

    Figure 3.7.62. the leftbranch view for instance squared_squares_001
    figpstrick/squared_squares_id001_leftbranch