Imperial Earth, Pentominoes, and SAT solvers

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pcallahan
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Joined: April 26th, 2013, 1:04 pm

Imperial Earth, Pentominoes, and SAT solvers

Post by pcallahan » March 6th, 2022, 2:42 pm

Not about CGoL but hopefully of some interest. I imagine at least some others here have written pentomino solvers.

I was recently thinking about Imperial Earth, a science fiction novel I read as a teen, and something the author Arthur C. Clarke said in it about pentomino solving that I doubt was true in 1975 let alone plausible 500 years later, assuming humanity is alive and well and colonizing the solar system:
"Look at that," she said. A pattern of bright lines had appeared on the screen, showing the set of all twelve pentominoes fitted into the six-by-ten frame. It held for a few seconds, then was replaced by another obviously different, though Duncan could not possibly remember the arrangement briefly presented to him. Then came another . . and another, until Grandma canceled the program.

"Even at this fast rate [but see my update below]," she said, "it takes five hours to run through them all. And take my word for it--though no human being has ever checked each one, or ever could-they're all different." For a long time, Duncan stared at the collection of twelve deceptively simple figures. As he slowly assimilated what Grandma had told him, he had the first genuine mathematical revelation of his life. What had at first seemed merely a childish game had opened endless vistas and horizons
Five hours to enumerate polyomino solutions? In 2276 CE? Seriously? Well, maybe Duncan's grandma is not as good as programmer as she thinks. Maybe the Titan colonies are forced to use really slow computers. But I think Clarke was showing his limitations as a futurist. (Totally agree about the endless vistas and horizons though!)

(On slow computers: Dean Hickerson found the first c/3 spaceship on a very limited 6502 device (as I recall) that David Buckingham once described hilariously as a "Fisher-Price computer".)

For all my speculation, I have never tried to solve pentominoes. I just never had a set or the inclination to write a search (and for years I may have suffered from the misapprehension--Clarke's fault!--that it was very hard). Normally, I would satisfy my curiosity by writing a backtracking enumeration. But since we have these fancy new toys called SAT solvers, I thought I'd give that a whirl instead. Note: I'm using Golomb's labeling, not Conway's, so there is an f pentomino, not r pentomino, etc.

I tried different ways to express a 6x10 pentomino packing as a SAT problem, eventually settling on one that was fairly concise and effective. For each letter pentomino, I enumerate all ways of placing it (with translation, rotation, and flip) and introduce a variable for each one. Each placement covers a subset of cell coordinates, and I use that in its name, e.g. W_72_73_81_82_91. ​I introduce another variable for each combination of letter and cell position, e.g. z_62 means "z covers row 6 (out of 0-9), column 2 (out of 0-5)." For reasons I'll explain later, I use 10 rows and 6 columns not the other way around.

The constraints are:
  • For each letter, at least one of the placement variables must be true (one clause per letter).
  • For each placement and cell it covers, either the placement variable is false or the cell variable is true (5 clauses per placement).
  • For each cell position, two letters cannot both cover it. (12 choose 2=66 clauses per cell position.)
Originally I had constraints to limit placement variables to 1 per letter, but that is handled by packing. I.e., if you mistakenly place the same pentomino more than once, you cover cells needed by some other pentomino. You can actually eliminate all the cell variables by making constraints directly between pentominoes. However, that increases the number of constraints by a lot. It tried it and it sometimes helped the SAT solver, but not enough to justify the expansion.

I have intermediate formats I use, but for standardization, here's the SAT problem as a DIMACS file with comments to identify variables. It's entirely unintentional that the number of variables is 2776, 500 years after Clarke's novel set at the US quincentennial, and 1000 years after 1776.

Code: Select all

p cnf 2776 14252
c variable F_00_01_11_12_21: 1
c variable F_00_10_11_12_21: 2
c variable F_01_02_10_11_21: 3
c variable F_01_02_12_13_22: 4
c variable F_01_10_11_12_20: 5
c variable F_01_10_11_12_22: 6
c variable F_01_10_11_21_22: 7
c variable F_01_11_12_13_22: 8
c variable F_01_11_12_20_21: 9
c variable F_02_03_11_12_22: 10
c variable F_02_03_13_14_23: 11
c variable F_02_10_11_12_21: 12
c variable F_02_11_12_13_21: 13
c variable F_02_11_12_13_23: 14
c variable F_02_11_12_22_23: 15
c variable F_02_12_13_14_23: 16
c variable F_02_12_13_21_22: 17
c variable F_03_04_12_13_23: 18
c variable F_03_04_14_15_24: 19
c variable F_03_11_12_13_22: 20
c variable F_03_12_13_14_22: 21
c variable F_03_12_13_14_24: 22
c variable F_03_12_13_23_24: 23
c variable F_03_13_14_15_24: 24
c variable F_03_13_14_22_23: 25
c variable F_04_05_13_14_24: 26
c variable F_04_12_13_14_23: 27
c variable F_04_13_14_15_23: 28
c variable F_04_13_14_15_25: 29
c variable F_04_13_14_24_25: 30
c variable F_04_14_15_23_24: 31
c variable F_05_13_14_15_24: 32
c variable F_10_11_21_22_31: 33
c variable F_10_20_21_22_31: 34
c variable F_11_12_20_21_31: 35
c variable F_11_12_22_23_32: 36
c variable F_11_20_21_22_30: 37
c variable F_11_20_21_22_32: 38
c variable F_11_20_21_31_32: 39
c variable F_11_21_22_23_32: 40
c variable F_11_21_22_30_31: 41
c variable F_12_13_21_22_32: 42
c variable F_12_13_23_24_33: 43
c variable F_12_20_21_22_31: 44
c variable F_12_21_22_23_31: 45
c variable F_12_21_22_23_33: 46
c variable F_12_21_22_32_33: 47
c variable F_12_22_23_24_33: 48
c variable F_12_22_23_31_32: 49
c variable F_13_14_22_23_33: 50
c variable F_13_14_24_25_34: 51
c variable F_13_21_22_23_32: 52
c variable F_13_22_23_24_32: 53
c variable F_13_22_23_24_34: 54
c variable F_13_22_23_33_34: 55
c variable F_13_23_24_25_34: 56
c variable F_13_23_24_32_33: 57
c variable F_14_15_23_24_34: 58
c variable F_14_22_23_24_33: 59
c variable F_14_23_24_25_33: 60
c variable F_14_23_24_25_35: 61
c variable F_14_23_24_34_35: 62
c variable F_14_24_25_33_34: 63
c variable F_15_23_24_25_34: 64
c variable F_20_21_31_32_41: 65
c variable F_20_30_31_32_41: 66
c variable F_21_22_30_31_41: 67
c variable F_21_22_32_33_42: 68
c variable F_21_30_31_32_40: 69
c variable F_21_30_31_32_42: 70
c variable F_21_30_31_41_42: 71
c variable F_21_31_32_33_42: 72
c variable F_21_31_32_40_41: 73
c variable F_22_23_31_32_42: 74
c variable F_22_23_33_34_43: 75
c variable F_22_30_31_32_41: 76
c variable F_22_31_32_33_41: 77
c variable F_22_31_32_33_43: 78
c variable F_22_31_32_42_43: 79
c variable F_22_32_33_34_43: 80
c variable F_22_32_33_41_42: 81
c variable F_23_24_32_33_43: 82
c variable F_23_24_34_35_44: 83
c variable F_23_31_32_33_42: 84
c variable F_23_32_33_34_42: 85
c variable F_23_32_33_34_44: 86
c variable F_23_32_33_43_44: 87
c variable F_23_33_34_35_44: 88
c variable F_23_33_34_42_43: 89
c variable F_24_25_33_34_44: 90
c variable F_24_32_33_34_43: 91
c variable F_24_33_34_35_43: 92
c variable F_24_33_34_35_45: 93
c variable F_24_33_34_44_45: 94
c variable F_24_34_35_43_44: 95
c variable F_25_33_34_35_44: 96
c variable F_30_31_41_42_51: 97
c variable F_30_40_41_42_51: 98
c variable F_31_32_40_41_51: 99
c variable F_31_32_42_43_52: 100
c variable F_31_40_41_42_50: 101
c variable F_31_40_41_42_52: 102
c variable F_31_40_41_51_52: 103
c variable F_31_41_42_43_52: 104
c variable F_31_41_42_50_51: 105
c variable F_32_33_41_42_52: 106
c variable F_32_33_43_44_53: 107
c variable F_32_40_41_42_51: 108
c variable F_32_41_42_43_51: 109
c variable F_32_41_42_43_53: 110
c variable F_32_41_42_52_53: 111
c variable F_32_42_43_44_53: 112
c variable F_32_42_43_51_52: 113
c variable F_33_34_42_43_53: 114
c variable F_33_34_44_45_54: 115
c variable F_33_41_42_43_52: 116
c variable F_33_42_43_44_52: 117
c variable F_33_42_43_44_54: 118
c variable F_33_42_43_53_54: 119
c variable F_33_43_44_45_54: 120
c variable F_33_43_44_52_53: 121
c variable F_34_35_43_44_54: 122
c variable F_34_42_43_44_53: 123
c variable F_34_43_44_45_53: 124
c variable F_34_43_44_45_55: 125
c variable F_34_43_44_54_55: 126
c variable F_34_44_45_53_54: 127
c variable F_35_43_44_45_54: 128
c variable F_40_41_51_52_61: 129
c variable F_40_50_51_52_61: 130
c variable F_41_42_50_51_61: 131
c variable F_41_42_52_53_62: 132
c variable F_41_50_51_52_60: 133
c variable F_41_50_51_52_62: 134
c variable F_41_50_51_61_62: 135
c variable F_41_51_52_53_62: 136
c variable F_41_51_52_60_61: 137
c variable F_42_43_51_52_62: 138
c variable F_42_43_53_54_63: 139
c variable F_42_50_51_52_61: 140
c variable F_42_51_52_53_61: 141
c variable F_42_51_52_53_63: 142
c variable F_42_51_52_62_63: 143
c variable F_42_52_53_54_63: 144
c variable F_42_52_53_61_62: 145
c variable F_43_44_52_53_63: 146
c variable F_43_44_54_55_64: 147
c variable F_43_51_52_53_62: 148
c variable F_43_52_53_54_62: 149
c variable F_43_52_53_54_64: 150
c variable F_43_52_53_63_64: 151
c variable F_43_53_54_55_64: 152
c variable F_43_53_54_62_63: 153
c variable F_44_45_53_54_64: 154
c variable F_44_52_53_54_63: 155
c variable F_44_53_54_55_63: 156
c variable F_44_53_54_55_65: 157
c variable F_44_53_54_64_65: 158
c variable F_44_54_55_63_64: 159
c variable F_45_53_54_55_64: 160
c variable F_50_51_61_62_71: 161
c variable F_50_60_61_62_71: 162
c variable F_51_52_60_61_71: 163
c variable F_51_52_62_63_72: 164
c variable F_51_60_61_62_70: 165
c variable F_51_60_61_62_72: 166
c variable F_51_60_61_71_72: 167
c variable F_51_61_62_63_72: 168
c variable F_51_61_62_70_71: 169
c variable F_52_53_61_62_72: 170
c variable F_52_53_63_64_73: 171
c variable F_52_60_61_62_71: 172
c variable F_52_61_62_63_71: 173
c variable F_52_61_62_63_73: 174
c variable F_52_61_62_72_73: 175
c variable F_52_62_63_64_73: 176
c variable F_52_62_63_71_72: 177
c variable F_53_54_62_63_73: 178
c variable F_53_54_64_65_74: 179
c variable F_53_61_62_63_72: 180
c variable F_53_62_63_64_72: 181
c variable F_53_62_63_64_74: 182
c variable F_53_62_63_73_74: 183
c variable F_53_63_64_65_74: 184
c variable F_53_63_64_72_73: 185
c variable F_54_55_63_64_74: 186
c variable F_54_62_63_64_73: 187
c variable F_54_63_64_65_73: 188
c variable F_54_63_64_65_75: 189
c variable F_54_63_64_74_75: 190
c variable F_54_64_65_73_74: 191
c variable F_55_63_64_65_74: 192
c variable F_60_61_71_72_81: 193
c variable F_60_70_71_72_81: 194
c variable F_61_62_70_71_81: 195
c variable F_61_62_72_73_82: 196
c variable F_61_70_71_72_80: 197
c variable F_61_70_71_72_82: 198
c variable F_61_70_71_81_82: 199
c variable F_61_71_72_73_82: 200
c variable F_61_71_72_80_81: 201
c variable F_62_63_71_72_82: 202
c variable F_62_63_73_74_83: 203
c variable F_62_70_71_72_81: 204
c variable F_62_71_72_73_81: 205
c variable F_62_71_72_73_83: 206
c variable F_62_71_72_82_83: 207
c variable F_62_72_73_74_83: 208
c variable F_62_72_73_81_82: 209
c variable F_63_64_72_73_83: 210
c variable F_63_64_74_75_84: 211
c variable F_63_71_72_73_82: 212
c variable F_63_72_73_74_82: 213
c variable F_63_72_73_74_84: 214
c variable F_63_72_73_83_84: 215
c variable F_63_73_74_75_84: 216
c variable F_63_73_74_82_83: 217
c variable F_64_65_73_74_84: 218
c variable F_64_72_73_74_83: 219
c variable F_64_73_74_75_83: 220
c variable F_64_73_74_75_85: 221
c variable F_64_73_74_84_85: 222
c variable F_64_74_75_83_84: 223
c variable F_65_73_74_75_84: 224
c variable F_70_71_81_82_91: 225
c variable F_70_80_81_82_91: 226
c variable F_71_72_80_81_91: 227
c variable F_71_72_82_83_92: 228
c variable F_71_80_81_82_90: 229
c variable F_71_80_81_82_92: 230
c variable F_71_80_81_91_92: 231
c variable F_71_81_82_83_92: 232
c variable F_71_81_82_90_91: 233
c variable F_72_73_81_82_92: 234
c variable F_72_73_83_84_93: 235
c variable F_72_80_81_82_91: 236
c variable F_72_81_82_83_91: 237
c variable F_72_81_82_83_93: 238
c variable F_72_81_82_92_93: 239
c variable F_72_82_83_84_93: 240
c variable F_72_82_83_91_92: 241
c variable F_73_74_82_83_93: 242
c variable F_73_74_84_85_94: 243
c variable F_73_81_82_83_92: 244
c variable F_73_82_83_84_92: 245
c variable F_73_82_83_84_94: 246
c variable F_73_82_83_93_94: 247
c variable F_73_83_84_85_94: 248
c variable F_73_83_84_92_93: 249
c variable F_74_75_83_84_94: 250
c variable F_74_82_83_84_93: 251
c variable F_74_83_84_85_93: 252
c variable F_74_83_84_85_95: 253
c variable F_74_83_84_94_95: 254
c variable F_74_84_85_93_94: 255
c variable F_75_83_84_85_94: 256
c variable I_00_01_02_03_04: 257
c variable I_00_10_20_30_40: 258
c variable I_01_02_03_04_05: 259
c variable I_01_11_21_31_41: 260
c variable I_02_12_22_32_42: 261
c variable I_03_13_23_33_43: 262
c variable I_04_14_24_34_44: 263
c variable I_05_15_25_35_45: 264
c variable I_10_11_12_13_14: 265
c variable I_10_20_30_40_50: 266
c variable I_11_12_13_14_15: 267
c variable I_11_21_31_41_51: 268
c variable I_12_22_32_42_52: 269
c variable I_13_23_33_43_53: 270
c variable I_14_24_34_44_54: 271
c variable I_15_25_35_45_55: 272
c variable I_20_21_22_23_24: 273
c variable I_20_30_40_50_60: 274
c variable I_21_22_23_24_25: 275
c variable I_21_31_41_51_61: 276
c variable I_22_32_42_52_62: 277
c variable I_23_33_43_53_63: 278
c variable I_24_34_44_54_64: 279
c variable I_25_35_45_55_65: 280
c variable I_30_31_32_33_34: 281
c variable I_30_40_50_60_70: 282
c variable I_31_32_33_34_35: 283
c variable I_31_41_51_61_71: 284
c variable I_32_42_52_62_72: 285
c variable I_33_43_53_63_73: 286
c variable I_34_44_54_64_74: 287
c variable I_35_45_55_65_75: 288
c variable I_40_41_42_43_44: 289
c variable I_40_50_60_70_80: 290
c variable I_41_42_43_44_45: 291
c variable I_41_51_61_71_81: 292
c variable I_42_52_62_72_82: 293
c variable I_43_53_63_73_83: 294
c variable I_44_54_64_74_84: 295
c variable I_45_55_65_75_85: 296
c variable I_50_51_52_53_54: 297
c variable I_50_60_70_80_90: 298
c variable I_51_52_53_54_55: 299
c variable I_51_61_71_81_91: 300
c variable I_52_62_72_82_92: 301
c variable I_53_63_73_83_93: 302
c variable I_54_64_74_84_94: 303
c variable I_55_65_75_85_95: 304
c variable I_60_61_62_63_64: 305
c variable I_61_62_63_64_65: 306
c variable I_70_71_72_73_74: 307
c variable I_71_72_73_74_75: 308
c variable I_80_81_82_83_84: 309
c variable I_81_82_83_84_85: 310
c variable I_90_91_92_93_94: 311
c variable I_91_92_93_94_95: 312
c variable L_00_01_02_03_10: 313
c variable L_00_01_02_03_13: 314
c variable L_00_01_10_20_30: 315
c variable L_00_01_11_21_31: 316
c variable L_00_10_11_12_13: 317
c variable L_00_10_20_30_31: 318
c variable L_01_02_03_04_11: 319
c variable L_01_02_03_04_14: 320
c variable L_01_02_11_21_31: 321
c variable L_01_02_12_22_32: 322
c variable L_01_11_12_13_14: 323
c variable L_01_11_21_30_31: 324
c variable L_01_11_21_31_32: 325
c variable L_02_03_04_05_12: 326
c variable L_02_03_04_05_15: 327
c variable L_02_03_12_22_32: 328
c variable L_02_03_13_23_33: 329
c variable L_02_12_13_14_15: 330
c variable L_02_12_22_31_32: 331
c variable L_02_12_22_32_33: 332
c variable L_03_04_13_23_33: 333
c variable L_03_04_14_24_34: 334
c variable L_03_10_11_12_13: 335
c variable L_03_13_23_32_33: 336
c variable L_03_13_23_33_34: 337
c variable L_04_05_14_24_34: 338
c variable L_04_05_15_25_35: 339
c variable L_04_11_12_13_14: 340
c variable L_04_14_24_33_34: 341
c variable L_04_14_24_34_35: 342
c variable L_05_12_13_14_15: 343
c variable L_05_15_25_34_35: 344
c variable L_10_11_12_13_20: 345
c variable L_10_11_12_13_23: 346
c variable L_10_11_20_30_40: 347
c variable L_10_11_21_31_41: 348
c variable L_10_20_21_22_23: 349
c variable L_10_20_30_40_41: 350
c variable L_11_12_13_14_21: 351
c variable L_11_12_13_14_24: 352
c variable L_11_12_21_31_41: 353
c variable L_11_12_22_32_42: 354
c variable L_11_21_22_23_24: 355
c variable L_11_21_31_40_41: 356
c variable L_11_21_31_41_42: 357
c variable L_12_13_14_15_22: 358
c variable L_12_13_14_15_25: 359
c variable L_12_13_22_32_42: 360
c variable L_12_13_23_33_43: 361
c variable L_12_22_23_24_25: 362
c variable L_12_22_32_41_42: 363
c variable L_12_22_32_42_43: 364
c variable L_13_14_23_33_43: 365
c variable L_13_14_24_34_44: 366
c variable L_13_20_21_22_23: 367
c variable L_13_23_33_42_43: 368
c variable L_13_23_33_43_44: 369
c variable L_14_15_24_34_44: 370
c variable L_14_15_25_35_45: 371
c variable L_14_21_22_23_24: 372
c variable L_14_24_34_43_44: 373
c variable L_14_24_34_44_45: 374
c variable L_15_22_23_24_25: 375
c variable L_15_25_35_44_45: 376
c variable L_20_21_22_23_30: 377
c variable L_20_21_22_23_33: 378
c variable L_20_21_30_40_50: 379
c variable L_20_21_31_41_51: 380
c variable L_20_30_31_32_33: 381
c variable L_20_30_40_50_51: 382
c variable L_21_22_23_24_31: 383
c variable L_21_22_23_24_34: 384
c variable L_21_22_31_41_51: 385
c variable L_21_22_32_42_52: 386
c variable L_21_31_32_33_34: 387
c variable L_21_31_41_50_51: 388
c variable L_21_31_41_51_52: 389
c variable L_22_23_24_25_32: 390
c variable L_22_23_24_25_35: 391
c variable L_22_23_32_42_52: 392
c variable L_22_23_33_43_53: 393
c variable L_22_32_33_34_35: 394
c variable L_22_32_42_51_52: 395
c variable L_22_32_42_52_53: 396
c variable L_23_24_33_43_53: 397
c variable L_23_24_34_44_54: 398
c variable L_23_30_31_32_33: 399
c variable L_23_33_43_52_53: 400
c variable L_23_33_43_53_54: 401
c variable L_24_25_34_44_54: 402
c variable L_24_25_35_45_55: 403
c variable L_24_31_32_33_34: 404
c variable L_24_34_44_53_54: 405
c variable L_24_34_44_54_55: 406
c variable L_25_32_33_34_35: 407
c variable L_25_35_45_54_55: 408
c variable L_30_31_32_33_40: 409
c variable L_30_31_32_33_43: 410
c variable L_30_31_40_50_60: 411
c variable L_30_31_41_51_61: 412
c variable L_30_40_41_42_43: 413
c variable L_30_40_50_60_61: 414
c variable L_31_32_33_34_41: 415
c variable L_31_32_33_34_44: 416
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When I run this in cadical without any special options, it takes over 360 seconds on my MacBook just to find one solution, namely:

Code: Select all

F_32_41_42_43_53
I_05_15_25_35_45
L_81_91_92_93_94
N_23_33_34_44_54
P_03_04_13_14_24
T_55_63_64_65_75
U_00_01_02_10_12
V_30_40_50_51_52
W_73_74_84_85_95
X_11_20_21_22_31
Y_60_70_71_80_90
Z_61_62_72_82_83
Some variations are faster and I did get it to solve in under 5 seconds in one case using the command line option -L1. To get more results, I add clauses to exclude each previous solution. I was able to get 168 solutions in an overnight run this way, so I'm still a long way from beating Clarke's prediction using a SAT solver. I tried kissat and lingeling too. In fact, lingeling sometimes outperformed cadical. I'd be very interested if there are any options that would accelerate the solution. I think it's still magical that it works at all considering that the SAT solver has no explicit geometry and packing is implicit.

But I wasn't about to concede a very silly prediction. "Obviously" we can find the 2339 solutions to pentominoes a lot faster than the excerpt suggests. So I went back to my reliable go-to of a backtracking search. To reduce the search space, I assumed pentominoes were placed as follows. Cells are ordered primarily by row and then by column. Find the first available cell and consider all unused pentominoes and placements that can cover it. Try each one and continue recursively with backtracking. This is where it helps to have more rows than columns since it constrains early placement more. It also keeps the boundary pretty simple. Backtracking doesn't care if it left an unfillable hole. That will come up when it fails to complete the packing.

I wrote this in python code without trying to optimize any data structures. Still (unsurprisingly) it found all solutions in much less than 5 hours. It took 9 minutes 32 seconds on my MacBook. I thought it might help to memoize deadends, i.e. if a particular set of available pentominoes and cells does not lead to any solutions, then save it and never recurse on it again (some placements of pentominoes may cover the same partial solutions). It helped, but it was not a big win, bringing the time down to 2 minutes 17 seconds. I think the placement order is already pretty good for avoiding duplicate work.

Finally, here is my Python code. I hope it's fairly readable, though that was not my priority:

Code: Select all

import sys

PENTOMINOES = {
  'w': '.**!**!*!',
  'x': '.*!***!.*!',
  'f': '.**!**!.*!',
  'n': '.***!**!',
  'z': '.**!.*!**!',
  'v': '***!*!*!',
  'p': '***!**!',
  'u': '***!*.*!',
  'l': '****!*!',
  't': '***!.*!.*!',
  'y': '****!.*!',
  'i': '*****!'
}

def to_coordinates(pentomino):
  coordinates = []
  pos = (0, 0)
  for c in pentomino:
    if c == '*':
      coordinates.append(pos) 
   
    if c == '!':
      pos = (pos[0] + 1, 0) 
    else:
      pos = (pos[0], pos[1] + 1)
  return coordinates

def normalize_position(coordinates):
  imin = min(i for i, j in coordinates)
  jmin = min(j for i, j in coordinates)
  return [(i - imin, j - jmin) for i, j in coordinates]

def symmetries(coordinates):
  result = [coordinates]
  result += [[(-i, j) for i, j in coords] for coords in result]
  result += [[(i, -j) for i, j in coords] for coords in result]
  result += [[(j, i) for i, j in coords] for coords in result]
  return sorted(set(tuple(sorted(normalize_position(coords))) for coords in result))

def placements(coordinates, ibound, jbound):
  imax = max(i for i, j in coordinates)
  jmax = max(j for i, j in coordinates)
  for i in range(ibound - imax):
    for j in range(jbound - jmax):
      yield tuple((i + di, j + dj) for di, dj in coordinates)

def name(letter, cells):
  return letter + '_' + '_'.join('%d%d' % (i, j) for i, j in cells)

def coords_for_name(name):
  return name[2:].split('_')

NROWS = 10
NCOLS = 6
TOP_LEFT = '00'
TOP_RIGHT = '0%d' % (NCOLS - 1)
BOTTOM_LEFT = '%d0' % (NROWS - 1)
BOTTOM_RIGHT = '%d%d' % (NROWS - 1, NCOLS - 1)

def coord_to_pentomino(choices):
  res = {}
  for p in choices:
    for coord in p[2:].split('_'):
      res[coord] = p 
  return res

def state_key(pentominoes, cells):
  return (tuple(sorted(pentominoes)), tuple(sorted(cells)))

def is_canonical(choices):
    placement = coord_to_pentomino(choices)
    tl = placement[TOP_LEFT][0]
    tr = placement[TOP_RIGHT][0]
    bl = placement[BOTTOM_LEFT][0]
    br = placement[BOTTOM_RIGHT][0]
    return tl == min(tl, tr, bl, br)

# Note: only try to memoize deadends if a set is provided.
def place_pentominoes(pentominoes, cells, choices, min_coord_to_pentomino, deadends):
  if len(pentominoes) == 0:
    yield choices 
  elif deadends == None or state_key(pentominoes, cells) not in deadends:
    nextcoord = min(cells)
    found = False
    for p in min_coord_to_pentomino[nextcoord]:
      coords = coords_for_name(p)
      symbol = p[0]
      if (symbol in pentominoes
          and len(cells.intersection(coords)) == len(coords)):
        pentominoes.remove(symbol)
        cells.difference_update(coords)
        for res in place_pentominoes(pentominoes, cells,
                                     choices + [p], min_coord_to_pentomino, deadends):
          found = True
          yield res      
        cells.update(coords)
        pentominoes.add(symbol)
    if deadends != None and not found:
      deadends.add(state_key(pentominoes, cells)) 

pentomino_names = []
min_coord_to_pentomino = {}
for k in sorted(PENTOMINOES):
  names = []
  for sym in symmetries(to_coordinates(PENTOMINOES[k])):
    for p in placements(sym, NROWS, NCOLS):
      nm = name(k, p)
      pentomino_names.append(nm)
      min_coord_to_pentomino.setdefault(nm[2:4], []).append(nm)

all_cells = set() 
for p in pentomino_names:
  all_cells.update(coords_for_name(p))

deadends = set() if len(sys.argv) > 1 and sys.argv[1] == 'memoize' else None
ix = 0
for res in place_pentominoes(set(PENTOMINOES), all_cells, [],
                             min_coord_to_pentomino, deadends):
  if is_canonical(res):
    ix += 1
    print(ix, " ".join(res))
Update: When I reread this excerpt, I realize there's no reason to assume the computer is going as fast as possible. It could simply be delaying the solution for the human observer (but even then, it doesn't have to display them one at a time). My belief when I read it about 40 years ago was that it meant it would take 5 hours to find the solutions using a computer.

As further evidence taken this subsequent dialogue:
"Are you sure it can be done?" he demanded. "Absolutely. In fact, there are two solutions. Haven't you found even one? I thought you were good in mathematics."

"So I am. That's why I know how tough the job is. There are over a quadrillion possible arrangements to be checked."
"How do you work that out?" asked Duncan, delighted to discover something that had baffled his friend. Karl looked at a piece of paper covered with sketches and numbers.

"Well, excluding forbidden positions, and allowing for symmetry and rotation, it comes to factorial twelve times two to the twenty-first--you wouldn't understand why! That's quite a number; here it is."
Karl is supposed to be a math genius, but he's making a mistake I've seen from a lot of noobs to combinatorial search. You don't just plop down all 12! orderings and orientations and check if they work. As even my backtracking code does above, you set up the enumeration so it fails fast. If you do that effectively, you don't need to get anywhere near 12!.

However, even if Clarke had been thinking of a brute force search, you could still get a fast result using multiprocessing. Surely the Titan colonies in 2276 will have as much computing capacity as the Amazon or Google cloud in 2022.

Update 2: I also disagree with the first quote that
no human being has ever checked each one, or ever could
I imagine someone sufficiently obsessive had already done so by 1975. Let's say you have a list and can spend a minute looking over each, verifying no duplicate pentominoes. That's about 39 hours of work, no worse than careful proofreading.

A more interesting question is whether a human could learn to produce the enumeration on demand without anything written down. One approach would be like memorizing digits of pi, maybe using mnemonics. That's the least interesting way I can think of.

What would be better would be a generating rule from one solution to a successor, by changing just a few pieces. I'm not sure how many you have to remove before you can replace them differently.

Another thing that occurs to me is a divide and conquer approach. How many distinct boundaries are there between the first 6 pieces placed and the last 6 pieces (I could write a problem to find that now)? If there aren't too many, you could memorize just those boundaries, the subsets to either side and the expected solutions, and then combine them as cartesian products. You might be able to go down another level to 4 sets of 3 pieces. In that case, it seems well within reason that a human being could at least convince themselves that there are 2339 solutions and have a sense of each one.

Note: 1975 was a year before Appel and Haken published their computer proof of the four color theorem. That provides a cultural marker on what it means to "know the solutions" of a problem. However, pentominoes seem to be well in the grasp of the human brain, given enough work. You could definitely "check each" solution without a computer.
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pcallahan
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Re: Imperial Earth, Pentominoes, and SAT solvers

Post by pcallahan » March 8th, 2022, 1:31 pm

pcallahan wrote: ↑
March 6th, 2022, 2:42 pm
 I was able to get 168 solutions in an overnight run this way, so I'm still a long way from beating Clarke's prediction using a SAT solver.
Small update. I added some redundant constraints to the SAT representation. In addition to clauses that say some placement of pentomino α forces each of 5 cells to be covered by α, I add clauses that say each of the other 55 cells is not covered by α. Also, I added clauses that each cell position is covered by at least 1 pentomino. Running with cadical, it looks like it can enumerate about 1000 solutions in the same time as it was taking for 168. So that's a marginal improvement, but still a lot slower than backtracking.

It would be interesting to see if there is a SAT version that could at least beat the 5 hour bound (a few seconds per new solution). Also, though I don't know how to do it, there must be some way to tweak SAT solvers to output multiple solutions other than excluding existing ones and repeating the search. That might result in a much faster enumeration, possibly competitive with backtracking.

Smaller update: My backtracking code checks if the result is at a canonical orientation, and basically does 4x as much work as it needs to generate flips and rotations. A very simple fix for this is to use only one of the 4 orientations of V (W works too) since flips and rotations that preserve the 10x6 rectangle will generate all of these. It speeds up backtracking (though not by a factor of 4 in practice). It also means that I don't need to filter the SAT enumeration. I am running this out of curiosity just to see if I can get all the results by excluding previous and rerunning. There's no reason it shouldn't work.
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Macbi
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Re: Imperial Earth, Pentominoes, and SAT solvers

Post by Macbi » March 9th, 2022, 4:54 pm

  • For each letter, at least one of the placement variables must be true (one clause per letter).
  • For each placement and cell it covers, either the placement variable is false or the cell variable is true (5 clauses per placement).
These are known as 'cardinality constraints'. The trick is to know that there are multiple ways to encode cardinality constraints, and not all of them are equally effective. The SAT solvers work faster with some encodings than others. The pysat Python library contains several sophisticated encodings, numbered 0 to 8. I tried with encoding 8 since I remember that working quite well in previous projects. It got three and a half hours on the first try.

Code: Select all

import subprocess
import sys
import pysat.card

width = 10
height = 6

pentominos = {
    'F' : ((1,0),(2,0),(0,1),(1,1),(1,2)),
    'I' : ((0,0),(0,1),(0,2),(0,3),(0,4)),
    'L' : ((0,0),(0,1),(0,2),(0,3),(1,3)),
    'N' : ((0,0),(0,1),(0,2),(1,2),(1,3)),
    'P' : ((0,0),(0,1),(1,0),(1,1),(0,2)),
    'T' : ((0,0),(0,1),(0,2),(1,1),(2,1)),
    'U' : ((0,0),(0,2),(1,0),(1,1),(1,2)),
    'V' : ((0,0),(0,1),(0,2),(1,0),(2,0)),
    'W' : ((1,1),(0,1),(0,2),(1,0),(2,0)),
    'X' : ((1,0),(1,1),(1,2),(0,1),(2,1)),
    'Y' : ((1,0),(0,1),(1,1),(2,1),(3,1)),
    'Z' : ((0,0),(1,0),(1,1),(1,2),(2,2))
}

def canonical(pentomino):
    min_x = min(x for x,y in pentomino)
    min_y = min(y for x,y in pentomino)
    return tuple(sorted((x-min_x, y-min_y) for x,y in pentomino))

def rotate(pentomino):
    return canonical(tuple((-y,x) for x,y in pentomino))

def reflect(pentomino):
    return canonical(tuple((y,x) for x,y in pentomino))

pentomino_lists = {}
for letter, pentomino in pentominos.items():
    if letter == 'F':
        pentomino_lists[letter] = tuple({canonical(pentomino), rotate(pentomino)}) # Give fewer options to remove symmetries from overall solution
    else:
        pentomino_lists[letter] = tuple({
            canonical(pentomino), rotate(pentomino), rotate(rotate(pentomino)), rotate(rotate(rotate(pentomino))),
            reflect(pentomino), rotate(reflect(pentomino)), rotate(rotate(reflect(pentomino))), rotate(rotate(rotate(reflect(pentomino))))
        })
        
    
pentomino_variables = {letter:[] for letter in pentominos}
grid_variables = {(x,y):[] for x in range(width) for y in range(height)}

number_of_variables = 0
int_to_variable_name = {}
for letter in pentominos:
    pentomino_list = pentomino_lists[letter]
    for n, pentomino in enumerate(pentomino_list):
        for x_offset in range(width):
            for y_offset in range(height):
                if all((x+x_offset,y+y_offset) in grid_variables for x,y in pentomino):
                    number_of_variables += 1
                    int_to_variable_name[number_of_variables] = (letter, n, x_offset, y_offset) # Represents the nth orientation of the pentomino with that letter being present at those offsets
                    pentomino_variables[letter].append(number_of_variables)
                    for x,y in pentomino:
                        grid_variables[(x+x_offset,y+y_offset)].append(number_of_variables)

clauses = []

for letter in pentominos:
    new_clauses = pysat.card.CardEnc.equals( # Use each pentomino exactly once
        lits = pentomino_variables[letter],
        bound = 1,
        top_id = number_of_variables,
        encoding = 8
    )
    number_of_variables = max(number_of_variables,new_clauses.nv)
    clauses += new_clauses.clauses
    
for x,y in grid_variables:
    new_clauses = pysat.card.CardEnc.equals( # Cover each grid square exactly once
        lits = grid_variables[(x,y)],
        bound = 1,
        top_id = number_of_variables,
        encoding = 0
    )
    number_of_variables = max(number_of_variables,new_clauses.nv)
    clauses += new_clauses.clauses

solutions = []
while True:
    print(len(solutions))
    cnf_string = "p cnf " + str(number_of_variables) + " " + str(len(clauses)) + "\n"
    for clause in clauses:
        cnf_string += " ".join(str(literal) for literal in clause) + " 0\n"

    solver_process = subprocess.Popen(
            [sys.path[0] + "/kissat"],
            stdout=subprocess.PIPE, stdin=subprocess.PIPE, stderr=subprocess.PIPE)
    out, error = solver_process.communicate(bytes(cnf_string, 'utf-8'))
    error = error.decode('ascii')
    if error != "":
        print(error)

    solution = (out.decode('ascii')).split("\ns ")[1].split("\nc")[0].split("\nv ") # Format kissat ouput
    answers = [int(l) for l in " ".join(solution[1:]).split(" ")[:-1]] # looks like [-1, 2, 3,-4, 5, ...]
    sat = solution[0] # "SATISFIABLE" or "UNSATISFIABLE"
    if sat == "SATISFIABLE":
        solutions.append(answers)
        new_clauses = pysat.card.CardEnc.atleast(
            lits = [-literal for literal in answers if literal > 0],
            bound = 1,
            top_id = number_of_variables,
            encoding = 8
        )
        number_of_variables = max(number_of_variables,new_clauses.nv)
        clauses += new_clauses.clauses
    else:
        break
    
for answers in solutions:
    output_string = ''
    for y in range(height):
        for x in range(width):
            answer = [literal for literal in grid_variables[(x,y)] if literal in answers]
            if len(answer) != 1:
                print(answers)
                print(grid_variables[(x,y)])
                print(x, y, len(answer), [int_to_variable_name[literal] for literal in answer])
                raise Exception
            output_string += int_to_variable_name[answer[0]][0]
        output_string += '\n'
    print(output_string)
Further improvement could be obtained by tuning the choice of encodings and the parameters of the SAT solver. This can be done intelligently using a program like SATzilla.
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pcallahan
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Re: Imperial Earth, Pentominoes, and SAT solvers

Post by pcallahan » March 9th, 2022, 5:20 pm

Macbi wrote: ↑
March 9th, 2022, 4:54 pm
 These are known as 'cardinality constraints'. The trick is to know that there are multiple ways to encode cardinality constraints, and not all of them are equally effective. The SAT solvers work faster with some encodings than others.
Thanks. I will try your code when I get a chance.

I found when I was just getting into this last year (for tiling constraints) and using an ad hoc approach to cardinality, that "exactly one" usually worked fastest if I had one clause for "at least one" and clauses for each pair indicating they are not both true. I handled larger cardinalities with binary encodings and constraints like half-adder logic. These worked, but not as fast.

Does your code have a good way of enumerating all solutions other than introducing a new clause to exclude each as it's found? I think this is the major bottleneck for me, rather than cardinality constraints.
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Macbi
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Re: Imperial Earth, Pentominoes, and SAT solvers

Post by Macbi » March 9th, 2022, 5:24 pm

pcallahan wrote: ↑
March 9th, 2022, 5:20 pm
Does your code have a good way of enumerating all solutions other than introducing a new clause to exclude each as it's found?
No. You would think kissat would have a faster way to do this, but as far as I can tell there's no inbuilt option to find all solutions.
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pcallahan
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Re: Imperial Earth, Pentominoes, and SAT solvers

Post by pcallahan » March 10th, 2022, 2:54 pm

Macbi wrote: ↑
March 9th, 2022, 5:24 pm
 No. You would think kissat would have a faster way to do this, but as far as I can tell there's no inbuilt option to find all solutions.
I wonder if progress is being driven by a particular choice of benchmark. If there were contests on finding all solutions, you'd think they'd come up with something better. I can't really complain since I am impressed at how well these work, but that seems like an opportunity for improvement.
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