The New Sudoku Players' Forum

by **dobrichev** » Mon Mar 07, 2011 1:17 am

There are 416 essentially different bands.
Creating a valid puzzle, all unavoidable sets (UA) in the solution grid must be hit with at least one clue.
Each of the bands has UA sets located entirely in its 27 cells. Fortunately all these UA have only 2 valid permutations.
So, each valid puzzle must hit all UA of the 6 solution grid's bands, plus all cross-band UA.
The number of UA in each valid solution grid is a huge number and is specific to the grid. The in-band UA are not too much and variety is limited from the limited number of different bands.

Let start with a non-minimal multiple solution puzzle setting all 54 cells within bands 2 and 3 given as clues. The problem is to find the minimal amount of the clues within the rest 27 cells.

Code: Select all: +-----+-----+-----+ |. . .|. . .|. . .| # non-givens |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ |x x x|x x x|x x x| # givens |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+

This simplification eliminates all UA but those within band 1. Also the clues we are searching for depend only from band 1 (one from 416) and not from the specific completion.
Solving such pseudo-puzzles we can find for all 416 bands:
- the number of solutions (column "Sol" in the table below);
- all minimal UA sets within the band (column "UA" is the number of UA);
- the minimal number of the additional clues required to hit all in-band UA (MinClues);
- all valid permutations of N additional clues that hit all UA, for some reasonable small N (54+N).

Code: Select all: Band Sol UA MinClues 54+2 54+3 54+4 54+5 54+6 54+7 BF7 In17 BF17 BF17/7 1 1728 27 6 0 0 0 0 729 17496 0.04 13 0.02 0.489 2 576 33 5 0 0 0 324 10179 107406 0.24 412 0.62 2.526 3 192 63 4 0 0 108 4194 41895 260689 0.59 285 0.43 0.720 4 864 21 5 0 0 0 252 7977 91579 0.21 129 0.19 0.928 5 192 55 4 0 0 108 5328 55883 381253 0.87 1054 1.58 1.820 6 288 27 4 0 0 101 4354 47881 346825 0.79 2250 3.37 4.272 7 192 35 3 0 6 912 15274 120750 736608 1.68 418 0.63 0.374 8 516 15 5 0 0 0 1107 22952 221705 0.50 78 0.12 0.232 9 864 31 4 0 0 54 2664 34263 233862 0.53 291 0.44 0.819 10 288 33 4 0 0 92 4054 46351 316983 0.72 1232 1.85 2.559 11 288 53 3 0 12 960 13192 88853 535487 1.22 1798 2.69 2.211 12 228 21 4 0 0 224 5935 61867 472411 1.07 1506 2.26 2.099 13 228 25 4 0 0 312 8028 75284 518732 1.18 494 0.74 0.627 14 168 29 4 0 0 462 9778 85113 583989 1.33 1450 2.17 1.635 15 276 27 4 0 0 415 9387 79647 597967 1.36 1684 2.52 1.854 16 120 15 5 0 0 0 972 20641 189421 0.43 25 0.04 0.087 17 192 41 3 0 72 2694 23527 159902 1024372 2.33 690 1.03 0.444 18 192 55 4 0 0 108 5166 55112 384903 0.88 959 1.44 1.641 19 192 51 4 0 0 189 7326 67765 371705 0.85 1210 1.81 2.144 20 288 59 3 0 21 1359 17286 117162 724311 1.65 3330 4.99 3.027 21 192 55 4 0 0 108 5166 54316 344907 0.78 1127 1.69 2.152 22 168 41 3 0 5 1062 16972 117373 744199 1.69 3139 4.70 2.778 23 144 30 4 0 0 520 11338 91307 610253 1.39 1232 1.85 1.329 24 168 41 3 0 5 1062 17028 117384 743366 1.69 3237 4.85 2.867 25 276 47 3 0 30 1772 21288 163331 1034043 2.35 1818 2.72 1.158 26 228 53 3 0 110 3165 26847 236927 1452984 3.30 2061 3.09 0.934 27 96 56 4 0 0 1053 20029 135601 883221 2.01 137 0.21 0.102 28 144 52 3 0 38 1987 23238 142501 875244 1.99 1590 2.38 1.196 29 516 81 2 9 543 6758 45132 258360 1063414 2.42 286 0.43 0.177 30 96 56 4 0 0 1053 20067 131888 848535 1.93 151 0.23 0.117 31 228 63 3 0 64 3407 36798 251688 1326145 3.02 749 1.12 0.372 32 192 11 5 0 0 0 546 11851 103634 0.24 124 0.19 0.788 33 288 15 5 0 0 0 63 3723 48568 0.11 110 0.16 1.491 34 192 15 4 0 0 200 4976 47232 362371 0.82 493 0.74 0.896 35 288 19 4 0 0 38 2012 22797 170257 0.39 355 0.53 1.373 36 276 17 4 0 0 39 2389 29560 228094 0.52 198 0.30 0.572 37 276 16 4 0 0 273 5359 47641 356989 0.81 433 0.65 0.799 38 180 14 4 0 0 182 4653 43864 336284 0.76 555 0.83 1.087 39 264 16 4 0 0 22 1592 21573 182921 0.42 354 0.53 1.274 40 180 24 3 0 3 656 11043 86216 586161 1.33 747 1.12 0.839 41 264 16 4 0 0 74 4058 44734 299811 0.68 365 0.55 0.802 42 288 17 4 0 0 126 3592 36014 255826 0.58 337 0.50 0.867 43 288 18 4 0 0 145 3903 34938 229335 0.52 514 0.77 1.476 44 168 17 4 0 0 34 2410 29318 210942 0.48 278 0.42 0.868 45 288 15 5 0 0 0 90 4902 60141 0.14 82 0.12 0.898 46 168 13 4 0 0 37 1822 20906 175135 0.40 259 0.39 0.974 47 288 23 4 0 0 29 1829 25396 184170 0.42 307 0.46 1.098 48 288 15 5 0 0 0 63 3723 46430 0.11 157 0.24 2.227 49 288 19 4 0 0 38 2012 22689 170514 0.39 327 0.49 1.263 50 156 31 3 0 3 737 11998 91726 608049 1.38 804 1.20 0.871 51 168 16 4 0 0 35 2252 29904 225022 0.51 435 0.65 1.273 52 156 14 4 0 0 95 4070 42351 300960 0.68 503 0.75 1.101 53 168 29 4 0 0 267 7535 65102 428968 0.98 344 0.52 0.528 54 288 11 5 0 0 0 378 9407 94815 0.22 96 0.14 0.667 55 288 18 4 0 0 161 5035 43556 271002 0.62 555 0.83 1.349 56 168 30 3 0 2 555 9686 74369 474611 1.08 845 1.27 1.172 57 288 22 4 0 0 56 2441 30383 231602 0.53 423 0.63 1.203 58 168 17 4 0 0 40 2690 31498 241326 0.55 378 0.57 1.031 59 288 20 4 0 0 29 1865 24619 203105 0.46 323 0.48 1.047 60 288 21 4 0 0 65 3020 35444 253833 0.58 680 1.02 1.764 61 180 22 4 0 0 304 7036 65832 471352 1.07 855 1.28 1.194 62 288 20 4 0 0 29 1865 24727 201610 0.46 366 0.55 1.195 63 180 17 4 0 0 187 4270 41681 335005 0.76 529 0.79 1.040 64 228 19 4 0 0 195 5094 51288 391839 0.89 524 0.78 0.881 65 228 26 3 0 7 661 10051 87642 645733 1.47 871 1.30 0.888 66 168 17 4 0 0 34 2410 29479 215230 0.49 351 0.53 1.074 67 288 15 5 0 0 0 90 4902 59984 0.14 99 0.15 1.087 68 168 30 3 0 2 555 9697 74781 465200 1.06 697 1.04 0.987 69 168 15 4 0 0 41 2448 30690 228088 0.52 349 0.52 1.008 70 168 29 4 0 0 349 8134 68263 442845 1.01 753 1.13 1.120 71 228 15 4 0 0 122 3238 28838 226575 0.52 186 0.28 0.541 72 288 11 5 0 0 0 312 7929 79187 0.18 257 0.38 2.137 73 228 18 4 0 0 166 5006 43738 274527 0.62 536 0.80 1.286 74 276 9 5 0 0 0 490 10393 91816 0.21 156 0.23 1.119 75 276 16 4 0 0 88 3428 30659 201843 0.46 144 0.22 0.470 76 228 15 4 0 0 79 3671 40068 301247 0.69 479 0.72 1.047 77 228 17 4 0 0 45 2765 34491 252665 0.57 604 0.90 1.574 78 228 15 4 0 0 90 3533 36870 281733 0.64 493 0.74 1.152 79 264 16 4 0 0 51 3440 40611 292579 0.67 587 0.88 1.321 80 288 19 4 0 0 146 4632 39196 259898 0.59 528 0.79 1.338 81 288 16 4 0 0 110 3142 27891 222431 0.51 497 0.74 1.471 82 288 16 4 0 0 122 3091 27558 220649 0.50 388 0.58 1.158 83 516 9 5 0 0 0 301 6427 62715 0.14 93 0.14 0.976 84 516 11 5 0 0 0 527 11425 97548 0.22 76 0.11 0.513 85 216 16 4 0 0 254 6705 61028 411256 0.94 787 1.18 1.260 86 276 16 4 0 0 159 4132 42618 305664 0.70 634 0.95 1.366 87 216 14 4 0 0 53 2736 32029 259150 0.59 377 0.56 0.958 88 168 16 4 0 0 17 1902 28010 218112 0.50 361 0.54 1.090 89 192 31 3 0 3 636 9676 76643 514925 1.17 758 1.14 0.969 90 288 22 4 0 0 56 2441 29269 224980 0.51 421 0.63 1.232 91 192 18 4 0 0 32 2603 32367 235305 0.54 360 0.54 1.007 92 264 16 4 0 0 14 1563 24016 203319 0.46 284 0.43 0.920 93 276 9 5 0 0 0 410 8796 85880 0.20 173 0.26 1.326 94 216 13 4 0 0 64 2691 29766 237553 0.54 274 0.41 0.760 95 228 13 4 0 0 139 3027 31245 238409 0.54 393 0.59 1.085 96 180 16 4 0 0 59 3565 40470 311357 0.71 550 0.82 1.163 97 228 12 4 0 0 80 3101 32795 261912 0.60 130 0.19 0.327 98 228 15 4 0 0 122 3216 28799 210956 0.48 129 0.19 0.403 99 180 16 4 0 0 46 2897 34666 252071 0.57 569 0.85 1.486 100 288 10 5 0 0 0 300 6848 63160 0.14 134 0.20 1.397 101 192 13 5 0 0 0 78 4681 55693 0.13 62 0.09 0.733 102 192 16 4 0 0 61 3006 32675 253046 0.58 348 0.52 0.906 103 156 14 4 0 0 59 3121 34526 266204 0.61 380 0.57 0.940 104 192 12 5 0 0 0 168 6428 69637 0.16 14 0.02 0.132 105 288 10 5 0 0 0 258 6050 62778 0.14 80 0.12 0.839 106 276 16 4 0 0 196 5418 49669 353801 0.80 578 0.87 1.076 107 192 27 3 0 10 889 11844 92071 637172 1.45 828 1.24 0.856 108 168 17 4 0 0 40 2690 32257 237902 0.54 376 0.56 1.041 109 288 12 5 0 0 0 499 12460 113898 0.26 258 0.39 1.492 110 168 17 4 0 0 40 2690 32343 245609 0.56 364 0.55 0.976 111 276 16 4 0 0 120 3521 37592 258768 0.59 509 0.76 1.295 112 168 17 4 0 0 34 2410 29243 212759 0.48 349 0.52 1.080 113 216 12 4 0 0 182 4004 39406 300779 0.68 584 0.87 1.279 114 228 12 4 0 0 80 3101 32477 253149 0.58 189 0.28 0.492 115 180 10 5 0 0 0 543 12040 111128 0.25 259 0.39 1.535 116 228 9 5 0 0 0 421 8831 83965 0.19 224 0.34 1.757 117 288 19 4 0 0 210 6625 64110 479612 1.09 958 1.43 1.315 118 180 30 3 0 7 803 11476 83861 597383 1.36 1103 1.65 1.216 119 192 15 4 0 0 16 1765 23357 177600 0.40 267 0.40 0.990 120 192 27 3 0 10 998 12416 94119 657702 1.50 669 1.00 0.670 121 156 18 4 0 0 325 7374 66836 475501 1.08 1154 1.73 1.598 122 288 11 5 0 0 0 374 9244 85661 0.19 94 0.14 0.723 123 192 15 4 0 0 200 4980 44517 347201 0.79 461 0.69 0.874 124 168 13 4 0 0 37 1822 21054 176108 0.40 132 0.20 0.494 125 168 30 3 0 2 555 9476 75390 480178 1.09 913 1.37 1.252 126 228 15 4 0 0 90 3533 37695 281976 0.64 594 0.89 1.387 127 288 23 4 0 0 29 1829 23850 179294 0.41 355 0.53 1.304 128 168 28 4 0 0 280 6953 61822 409399 0.93 494 0.74 0.795 129 276 16 4 0 0 88 3428 30209 195484 0.44 131 0.20 0.441 130 228 26 3 0 7 661 10201 81765 583184 1.33 1047 1.57 1.182 131 264 15 4 0 0 39 2575 29767 229019 0.52 314 0.47 0.903 132 288 16 4 0 0 87 3144 30638 218799 0.50 291 0.44 0.876 133 516 11 5 0 0 0 309 7054 67568 0.15 9 0.01 0.088 134 276 19 4 0 0 379 8032 72714 509083 1.16 913 1.37 1.181 135 288 19 4 0 0 38 2012 22841 191177 0.43 328 0.49 1.130 136 288 19 4 0 0 38 2012 22797 172529 0.39 324 0.49 1.237 137 264 15 4 0 0 35 1954 23389 181889 0.41 104 0.16 0.377 138 288 21 4 0 0 65 3020 36574 257241 0.59 849 1.27 2.173 139 168 26 4 0 0 360 8422 70185 447080 1.02 493 0.74 0.726 140 276 11 5 0 0 0 326 8130 83342 0.19 225 0.34 1.778 141 276 17 4 0 0 174 5177 47505 326223 0.74 559 0.84 1.128 142 168 28 4 0 0 513 10056 78006 485106 1.10 322 0.48 0.437 143 192 32 3 0 13 1078 14019 98407 535684 1.22 733 1.10 0.901 144 144 27 4 0 0 266 7271 62534 425027 0.97 225 0.34 0.349 145 168 33 4 0 0 694 13364 100736 555065 1.26 894 1.34 1.061 146 144 30 4 0 0 340 8978 81031 546965 1.24 577 0.86 0.695 147 192 33 3 0 4 727 10671 82917 522104 1.19 521 0.78 0.657 148 192 17 4 0 0 22 2190 27141 213745 0.49 150 0.22 0.462 149 288 31 4 0 0 42 2412 30462 212986 0.48 143 0.21 0.442 150 168 28 4 0 0 513 10041 78300 486640 1.11 514 0.77 0.696 151 288 23 4 0 0 33 2697 32683 237740 0.54 334 0.50 0.925 152 288 19 4 0 0 20 1205 16554 142165 0.32 238 0.36 1.102 153 156 24 4 0 0 440 10045 87170 567239 1.29 889 1.33 1.032 154 168 32 4 0 0 477 11257 92113 535596 1.22 744 1.11 0.915 155 156 29 4 0 0 1032 17024 128991 716836 1.63 1239 1.86 1.138 156 168 23 4 0 0 388 8481 71978 452669 1.03 597 0.89 0.868 157 288 18 4 0 0 221 6337 60866 462908 1.05 763 1.14 1.085 158 144 23 4 0 0 314 7341 65664 444925 1.01 422 0.63 0.625 159 192 50 4 0 0 336 8211 75728 484470 1.10 557 0.83 0.757 160 192 22 5 0 0 0 1131 21291 163632 0.37 278 0.42 1.119 161 228 17 4 0 0 185 5159 53151 398216 0.91 625 0.94 1.034 162 144 30 3 0 5 1004 14780 112969 706682 1.61 884 1.32 0.824 163 228 17 4 0 0 185 5159 53246 401111 0.91 631 0.95 1.036 164 144 26 4 0 0 246 7821 67490 413020 0.94 490 0.73 0.781 165 180 29 3 0 21 1624 20169 142762 755494 1.72 1174 1.76 1.023 166 180 24 4 0 0 464 9947 83113 501820 1.14 692 1.04 0.908 167 168 17 4 0 0 62 3556 39076 279146 0.63 316 0.47 0.745 168 228 29 3 0 14 1099 15714 113030 589540 1.34 1530 2.29 1.709 169 168 30 3 0 4 766 13758 100116 548470 1.25 1203 1.80 1.444 170 156 30 3 0 11 1110 16446 124865 719932 1.64 1289 1.93 1.179 171 216 14 4 0 0 28 1755 22702 178467 0.41 382 0.57 1.409 172 216 16 4 0 0 192 5725 54327 360888 0.82 552 0.83 1.007 173 180 23 4 0 0 573 10162 86116 658592 1.50 720 1.08 0.720 174 216 14 4 0 0 229 5001 46509 349891 0.80 383 0.57 0.721 175 168 17 4 0 0 62 3556 38397 263117 0.60 297 0.44 0.743 176 168 39 3 0 21 1488 18392 128644 685512 1.56 1082 1.62 1.039 177 168 30 3 0 4 766 13731 100095 543643 1.24 1253 1.88 1.518 178 168 26 4 0 0 484 10459 87108 492180 1.12 1023 1.53 1.369 179 168 30 4 0 0 639 13011 100366 547835 1.25 1146 1.72 1.377 180 180 33 3 0 3 813 12860 90807 523488 1.19 779 1.17 0.980 181 180 27 3 0 7 741 10684 85711 590741 1.34 787 1.18 0.877 182 276 15 4 0 0 108 3509 37615 273933 0.62 388 0.58 0.933 183 276 11 5 0 0 0 244 6740 71605 0.16 144 0.22 1.324 184 276 25 3 0 13 832 11029 90995 536490 1.22 945 1.42 1.160 185 180 34 3 0 6 868 13974 104161 655735 1.49 1266 1.90 1.271 186 180 20 4 0 0 75 3979 47603 366912 0.83 700 1.05 1.256 187 228 16 4 0 0 70 2516 23591 186668 0.42 454 0.68 1.602 188 228 14 4 0 0 110 3726 40022 317993 0.72 409 0.61 0.847 189 180 16 4 0 0 63 3091 34537 282186 0.64 356 0.53 0.831 190 168 24 4 0 0 481 9421 74788 468599 1.07 214 0.32 0.301 191 180 16 4 0 0 157 4578 44326 326097 0.74 407 0.61 0.822 192 288 16 4 0 0 74 2533 23145 185331 0.42 385 0.58 1.368 193 264 15 4 0 0 37 2195 25806 201357 0.46 259 0.39 0.847 194 156 25 4 0 0 731 12303 89141 538625 1.23 611 0.92 0.747 195 228 14 4 0 0 110 3726 36688 251530 0.57 381 0.57 0.997 196 276 13 4 0 0 44 2020 21077 167341 0.38 399 0.60 1.570 197 216 26 3 0 9 699 10533 86209 543747 1.24 978 1.46 1.184 198 156 14 4 0 0 94 4318 46980 339630 0.77 728 1.09 1.411 199 156 26 3 0 7 1089 16498 115061 634121 1.44 1324 1.98 1.375 200 180 29 3 0 3 760 13455 102433 599933 1.36 1180 1.77 1.295 201 180 32 3 0 25 1578 18834 128923 712227 1.62 1355 2.03 1.253 202 180 16 4 0 0 59 3326 39764 291369 0.66 653 0.98 1.476 203 180 18 4 0 0 190 5613 56115 376209 0.86 781 1.17 1.367 204 168 20 4 0 0 321 7500 68662 488940 1.11 629 0.94 0.847 205 168 25 4 0 0 360 7998 71096 483451 1.10 655 0.98 0.892 206 168 32 3 0 15 1197 16620 127687 707538 1.61 1492 2.23 1.389 207 156 29 3 0 3 835 14867 113776 671631 1.53 1338 2.00 1.312 208 156 22 4 0 0 417 9155 79413 513173 1.17 552 0.83 0.708 209 120 28 4 0 0 953 16154 122029 679945 1.55 1307 1.96 1.266 210 192 26 3 0 12 1038 13012 91359 520387 1.18 517 0.77 0.654 211 120 32 3 0 4 1231 18330 127685 673827 1.53 1568 2.35 1.532 212 192 28 3 0 4 788 12781 99319 586828 1.33 1211 1.81 1.359 213 192 17 4 0 0 34 2068 27499 199422 0.45 143 0.21 0.472 214 168 15 4 0 0 110 3634 33809 249184 0.57 208 0.31 0.550 215 144 34 4 0 0 447 10679 92358 588041 1.34 1078 1.61 1.207 216 192 39 4 0 0 399 9635 75935 405325 0.92 996 1.49 1.618 217 192 23 5 0 0 0 1275 21432 161470 0.37 289 0.43 1.179 218 288 19 4 0 0 20 1205 14879 124127 0.28 207 0.31 1.098 219 516 11 5 0 0 0 177 4497 48340 0.11 86 0.13 1.172 220 516 9 5 0 0 0 485 10127 94312 0.21 238 0.36 1.662 221 276 16 4 0 0 78 2850 29042 222378 0.51 485 0.73 1.436 222 288 22 4 0 0 20 1511 21395 177460 0.40 404 0.61 1.499 223 516 9 5 0 0 0 529 10847 102200 0.23 297 0.44 1.914 224 864 17 6 0 0 0 0 162 6480 0.01 8 0.01 0.813 225 864 17 5 0 0 0 90 3540 43428 0.10 163 0.24 2.472 226 288 19 4 0 0 20 1205 14879 121129 0.28 224 0.34 1.218 227 216 16 4 0 0 94 3182 33079 245063 0.56 535 0.80 1.438 228 276 13 4 0 0 44 2020 21568 174563 0.40 308 0.46 1.162 229 288 22 4 0 0 20 1511 19733 157146 0.36 450 0.67 1.886 230 288 19 4 0 0 20 1205 16554 140309 0.32 220 0.33 1.033 231 216 14 4 0 0 74 2747 30111 232045 0.53 630 0.94 1.788 232 216 9 5 0 0 0 539 11652 108640 0.25 285 0.43 1.727 233 276 11 5 0 0 0 540 11416 110118 0.25 269 0.40 1.609 234 276 16 4 0 0 78 2850 28546 223418 0.51 582 0.87 1.715 235 228 16 4 0 0 114 3268 34367 239100 0.54 278 0.42 0.766 236 288 19 4 0 0 113 3544 35959 244786 0.56 1057 1.58 2.843 237 288 11 6 0 0 0 0 606 14930 0.03 31 0.05 1.367 238 288 11 5 0 0 0 331 7672 72566 0.17 161 0.24 1.461 239 288 15 4 0 0 84 2677 28090 200030 0.45 434 0.65 1.429 240 228 9 5 0 0 0 213 4558 47544 0.11 12 0.02 0.166 241 516 13 5 0 0 0 777 17018 156147 0.36 176 0.26 0.742 242 288 23 4 0 0 33 2697 34679 253583 0.58 423 0.63 1.098 243 264 17 4 0 0 84 4377 51042 367343 0.84 747 1.12 1.339 244 192 41 4 0 0 264 6569 55897 326581 0.74 687 1.03 1.385 245 264 16 4 0 0 14 1836 26831 217159 0.49 138 0.21 0.418 246 288 23 4 0 0 33 2697 32996 249722 0.57 432 0.65 1.139 247 864 19 5 0 0 0 108 5490 72468 0.16 132 0.20 1.199 248 180 32 3 0 6 856 13614 102956 648507 1.47 1236 1.85 1.255 249 288 27 4 0 0 60 3759 45018 286410 0.65 821 1.23 1.888 250 228 16 4 0 0 94 3827 42619 328103 0.75 269 0.40 0.540 251 192 43 4 0 0 300 7164 58398 386408 0.88 317 0.47 0.540 252 576 25 5 0 0 0 864 17568 160965 0.37 265 0.40 1.084 253 192 21 5 0 0 0 852 16388 135373 0.31 83 0.12 0.404 254 576 19 6 0 0 0 0 2133 37806 0.09 16 0.02 0.279 255 192 39 4 0 0 228 5989 58303 397284 0.90 250 0.37 0.414 256 192 17 5 0 0 0 1626 28241 209253 0.48 222 0.33 0.699 257 288 22 4 0 0 301 7567 72797 524494 1.19 1004 1.50 1.261 258 192 39 4 0 0 417 9639 74158 414850 0.94 973 1.46 1.544 259 288 13 5 0 0 0 692 15050 141320 0.32 184 0.28 0.857 260 192 23 5 0 0 0 1275 21354 159938 0.36 314 0.47 1.293 261 156 19 4 0 0 388 9157 82661 525066 1.19 985 1.48 1.235 262 156 26 4 0 0 545 11515 90541 537600 1.22 905 1.36 1.109 263 276 23 4 0 0 407 9298 82136 548114 1.25 1290 1.93 1.550 264 228 18 4 0 0 165 4921 52137 389936 0.89 841 1.26 1.420 265 156 25 3 0 5 869 13105 98204 613119 1.39 1206 1.81 1.295 266 216 15 4 0 0 191 4757 47939 314604 0.72 699 1.05 1.463 267 216 16 4 0 0 225 5624 53268 397842 0.90 707 1.06 1.170 268 228 18 4 0 0 165 4997 52165 379046 0.86 851 1.27 1.478 269 288 27 4 0 0 60 3759 47587 334253 0.76 873 1.31 1.720 270 180 29 3 0 3 733 12793 103748 692941 1.58 1060 1.59 1.007 271 180 17 4 0 0 273 6370 56051 399605 0.91 786 1.18 1.295 272 276 17 4 0 0 152 4373 46769 316351 0.72 804 1.20 1.674 273 288 41 3 0 54 2024 18611 128768 761337 1.73 876 1.31 0.758 274 288 23 4 0 0 286 7301 71806 489184 1.11 572 0.86 0.770 275 288 19 4 0 0 178 4809 46071 312285 0.71 251 0.38 0.529 276 168 31 3 0 16 1346 16075 118554 747977 1.70 617 0.92 0.543 277 168 25 4 0 0 348 8756 77366 500700 1.14 701 1.05 0.922 278 288 19 4 0 0 178 5806 52265 347439 0.79 759 1.14 1.439 279 192 29 3 0 2 648 11290 87084 576036 1.31 548 0.82 0.626 280 144 17 4 0 0 92 5030 55060 369227 0.84 429 0.64 0.765 281 144 25 4 0 0 460 9751 81645 495237 1.13 280 0.42 0.372 282 288 23 4 0 0 33 2697 32395 235671 0.54 592 0.89 1.654 283 156 16 4 0 0 268 6349 61369 445045 1.01 798 1.20 1.181 284 228 16 4 0 0 94 3827 42302 319082 0.73 391 0.59 0.807 285 192 35 3 0 8 934 12340 93644 609328 1.39 714 1.07 0.772 286 144 15 4 0 0 80 4116 46550 322633 0.73 462 0.69 0.943 287 192 15 4 0 0 42 2498 30970 241037 0.55 173 0.26 0.473 288 144 29 4 0 0 400 9298 75923 451055 1.03 371 0.56 0.542 289 192 13 5 0 0 0 156 7486 85681 0.19 41 0.06 0.315 290 192 15 4 0 0 96 4848 50042 392656 0.89 531 0.80 0.891 291 264 17 4 0 0 60 3737 45795 364798 0.83 785 1.18 1.417 292 192 41 4 0 0 264 6527 62539 420878 0.96 787 1.18 1.231 293 264 16 4 0 0 38 2168 28571 221547 0.50 457 0.68 1.358 294 192 22 5 0 0 0 1131 21165 162900 0.37 331 0.50 1.338 295 264 14 4 0 0 62 3374 40208 307716 0.70 501 0.75 1.072 296 180 30 3 0 18 1407 16449 118631 764121 1.74 1420 2.13 1.224 297 216 18 4 0 0 358 8397 76682 508108 1.16 1134 1.70 1.470 298 216 16 4 0 0 130 4006 41408 292587 0.67 231 0.35 0.520 299 156 19 4 0 0 356 8409 74389 498713 1.13 1308 1.96 1.727 300 192 39 4 0 0 417 9598 81885 517471 1.18 1145 1.71 1.457 301 156 29 4 0 0 537 11343 92433 567041 1.29 1148 1.72 1.333 302 168 11 5 0 0 0 594 13660 122148 0.28 73 0.11 0.394 303 288 21 4 0 0 128 4243 41308 265847 0.60 298 0.45 0.738 304 192 39 4 0 0 228 5989 58346 396310 0.90 383 0.57 0.636 305 576 25 5 0 0 0 864 17568 159958 0.36 298 0.45 1.227 306 192 17 5 0 0 0 1626 28128 208247 0.47 286 0.43 0.904 307 576 19 6 0 0 0 0 2133 36738 0.08 21 0.03 0.376 308 192 43 4 0 0 300 7164 58359 385191 0.88 268 0.40 0.458 309 192 21 5 0 0 0 852 16348 135814 0.31 103 0.15 0.499 310 276 25 4 0 0 261 5627 49797 351377 0.80 274 0.41 0.513 311 180 26 3 0 12 1119 14931 109094 616683 1.40 1187 1.78 1.267 312 168 32 4 0 0 504 11905 98188 591348 1.34 1042 1.56 1.160 313 180 29 3 0 7 787 11777 85404 490499 1.12 642 0.96 0.862 314 288 31 4 0 0 42 2412 30684 215116 0.49 201 0.30 0.615 315 228 31 3 0 13 1204 17421 130509 721533 1.64 1309 1.96 1.195 316 144 28 3 0 14 1124 14467 105710 610251 1.39 976 1.46 1.053 317 192 39 4 0 0 399 9519 83354 525233 1.19 1052 1.58 1.319 318 144 34 4 0 0 447 10679 92786 588456 1.34 818 1.23 0.915 319 288 45 3 0 69 2254 19786 127494 712916 1.62 1250 1.87 1.155 320 192 37 4 0 0 507 10794 87858 507201 1.15 1014 1.52 1.316 321 156 34 3 0 22 1624 20132 145972 773428 1.76 1184 1.77 1.008 322 156 31 4 0 0 858 15394 121320 669103 1.52 1139 1.71 1.121 323 144 27 4 0 0 218 7630 71126 483569 1.10 703 1.05 0.957 324 288 21 4 0 0 42 3519 40867 313276 0.71 626 0.94 1.316 325 168 29 4 0 0 391 10029 84194 535032 1.22 676 1.01 0.832 326 168 24 4 0 0 481 9413 75135 472735 1.08 714 1.07 0.995 327 288 21 4 0 0 42 3519 41027 315316 0.72 662 0.99 1.382 328 276 37 3 0 19 1261 16391 118483 648732 1.48 1033 1.55 1.049 329 216 25 4 0 0 686 12709 98797 560555 1.27 1006 1.51 1.182 330 192 34 3 0 3 717 10795 82977 516118 1.17 764 1.14 0.975 331 192 39 4 0 0 336 8400 70175 469761 1.07 770 1.15 1.079 332 144 27 4 0 0 317 8650 77421 493434 1.12 616 0.92 0.822 333 156 26 3 0 2 792 13982 107608 628478 1.43 1204 1.80 1.262 334 192 27 3 0 8 980 12422 93849 651507 1.48 563 0.84 0.569 335 144 27 4 0 0 218 7627 70001 464857 1.06 772 1.16 1.094 336 144 23 4 0 0 314 7389 65334 435651 0.99 386 0.58 0.583 337 192 39 4 0 0 336 8400 68527 432772 0.98 797 1.19 1.213 338 168 32 3 0 18 1183 14868 108692 673585 1.53 1005 1.51 0.982 339 168 34 4 0 0 580 12017 89838 485993 1.11 843 1.26 1.142 340 192 37 3 0 18 1269 16060 110188 595680 1.35 797 1.19 0.881 341 192 37 4 0 0 507 10498 87212 546857 1.24 1003 1.50 1.208 342 120 29 4 0 0 789 14294 104326 586764 1.33 995 1.49 1.117 343 120 26 4 0 0 703 13238 97742 557610 1.27 706 1.06 0.834 344 168 28 4 0 0 478 11242 92291 546297 1.24 994 1.49 1.198 345 216 38 3 0 73 2407 23003 149863 723232 1.64 1442 2.16 1.313 346 216 28 4 0 0 452 9582 78574 476497 1.08 360 0.54 0.498 347 192 50 4 0 0 336 8211 68701 459791 1.05 760 1.14 1.088 348 96 56 4 0 0 1053 20070 131172 849222 1.93 21 0.03 0.016 349 144 49 3 0 8 1416 20140 142503 918051 2.09 394 0.59 0.283 350 168 32 4 0 0 790 16308 121460 753032 1.71 740 1.11 0.647 351 120 42 3 0 20 2522 31303 230998 1402402 3.19 2081 3.12 0.977 352 120 49 3 0 56 3202 34897 234126 1483378 3.37 906 1.36 0.402 353 120 37 3 0 20 1743 24193 178493 1059306 2.41 957 1.43 0.595 354 168 38 3 0 30 2157 26088 187771 1098434 2.50 805 1.21 0.483 355 144 48 3 0 37 2525 27408 181272 1132063 2.57 1498 2.24 0.871 356 156 30 3 0 8 1209 19597 144827 850748 1.93 1568 2.35 1.214 357 120 37 3 0 4 1428 22010 168295 1031219 2.35 1883 2.82 1.202 358 156 44 3 0 65 3033 32383 213895 1175134 2.67 1899 2.84 1.064 359 120 53 3 0 24 1884 21370 136444 841445 1.91 287 0.43 0.225 360 144 25 4 0 0 310 9925 87669 577258 1.31 504 0.75 0.575 361 168 40 3 0 32 1789 22343 155121 937260 2.13 1454 2.18 1.022 362 192 36 4 0 0 408 10001 88696 553928 1.26 942 1.41 1.120 363 156 31 4 0 0 1018 18583 148536 940158 2.14 1751 2.62 1.226 364 156 44 3 0 39 2413 26968 211226 1339231 3.05 1845 2.76 0.907 365 144 47 3 0 10 1449 20947 149531 920405 2.09 1703 2.55 1.218 366 180 39 3 0 24 1831 23146 185532 1177901 2.68 1728 2.59 0.966 367 192 43 4 0 0 588 11620 92646 614770 1.40 1068 1.60 1.144 368 168 39 3 0 16 1311 19241 127671 740987 1.69 1354 2.03 1.203 369 180 39 3 0 79 2895 28337 224645 1328033 3.02 1915 2.87 0.950 370 192 43 4 0 0 588 11827 92441 574091 1.31 1043 1.56 1.196 371 168 34 3 0 11 1273 18277 125812 748572 1.70 1516 2.27 1.334 372 192 36 4 0 0 408 9810 85038 566941 1.29 1019 1.53 1.184 373 144 29 4 0 0 601 13298 103152 622612 1.42 930 1.39 0.984 374 168 39 3 0 16 1311 19256 130583 757311 1.72 1277 1.91 1.110 375 168 29 4 0 0 486 11362 92721 626115 1.42 818 1.23 0.860 376 168 39 3 0 16 1311 19045 126849 799258 1.82 1299 1.95 1.070 377 228 29 4 0 0 418 10279 89381 613362 1.40 609 0.91 0.654 378 228 45 3 0 32 2101 25632 193298 1079158 2.45 1709 2.56 1.043 379 144 35 4 0 0 425 11386 94645 596926 1.36 1037 1.55 1.144 380 168 30 4 0 0 1329 21214 169394 1116408 2.54 286 0.43 0.169 381 120 45 3 0 18 1987 26545 176071 1099116 2.50 1492 2.23 0.894 382 156 52 3 0 92 3627 35559 229119 1261861 2.87 1896 2.84 0.989 383 216 52 3 0 158 4212 35340 271590 1557792 3.54 929 1.39 0.393 384 216 28 3 0 16 1237 18583 136796 797782 1.81 768 1.15 0.634 385 180 35 3 0 24 1748 22725 178683 1134608 2.58 1651 2.47 0.958 386 168 45 3 0 36 2351 28689 190898 1117303 2.54 1636 2.45 0.964 387 168 35 4 0 0 751 15057 113050 708002 1.61 1276 1.91 1.187 388 96 47 4 0 0 1098 19403 134602 883875 2.01 1003 1.50 0.747 389 120 46 3 0 12 2145 28339 197553 1272143 2.89 423 0.63 0.219 390 264 57 3 0 138 3501 29047 210398 1199576 2.73 729 1.09 0.400 391 168 39 4 0 0 726 14697 112714 763616 1.74 623 0.93 0.537 392 192 23 3 0 2 742 14044 113970 716614 1.63 636 0.95 0.584 393 192 55 4 0 0 108 5328 54752 349122 0.79 347 0.52 0.654 394 144 51 3 0 24 1750 22610 148195 942082 2.14 632 0.95 0.442 395 192 51 4 0 0 189 7326 68345 406856 0.93 541 0.81 0.876 396 168 45 3 0 36 2351 28381 181697 1125400 2.56 755 1.13 0.442 397 192 55 4 0 0 108 5166 56261 369840 0.84 363 0.54 0.646 398 192 63 4 0 0 108 4194 42767 277152 0.63 236 0.35 0.561 399 288 59 3 0 21 1359 17312 113355 633438 1.44 1218 1.82 1.266 400 192 55 4 0 0 108 5166 56078 357258 0.81 337 0.50 0.621 401 144 52 3 0 38 1987 23346 150723 920404 2.09 616 0.92 0.441 402 192 55 4 0 0 108 5166 56401 377544 0.86 516 0.77 0.900 403 192 55 4 0 0 108 5328 53928 355536 0.81 679 1.02 1.258 404 96 56 4 0 0 1053 20029 135407 877001 1.99 138 0.21 0.104 405 168 41 3 0 5 1062 16973 113654 753307 1.71 1184 1.77 1.035 406 192 55 4 0 0 108 5166 54621 360422 0.82 690 1.03 1.261 407 288 53 3 0 12 960 13192 96378 625206 1.42 633 0.95 0.667 408 144 30 4 0 0 520 11338 90595 561432 1.28 403 0.60 0.473 409 192 51 4 0 0 189 7326 69615 463482 1.05 866 1.30 1.230 410 288 27 4 0 0 101 4354 48942 342966 0.78 698 1.05 1.340 411 288 33 4 0 0 92 4054 46789 336527 0.77 394 0.59 0.771 412 576 33 5 0 0 0 324 10179 105075 0.24 187 0.28 1.172 413 1728 27 6 0 0 0 0 729 17496 0.04 4 0.01 0.151 414 576 33 5 0 0 0 324 10179 107189 0.24 61 0.09 0.375 415 864 21 5 0 0 0 252 7977 90857 0.21 30 0.04 0.217 416 864 31 4 0 0 54 2664 34263 232386 0.53 71 0.11 0.201 Sum= 9 2739 213032 3483430 28422K 183M 416 277764 416.00 =6*46294 min=0.016 max=4.272

Column "54+7" contains the number of valid completions with up to 7 additional clues (including all combinations of 6 clues + one redundant), and so on.
Column "BF7" (Band Factor for 7 clues) is the weight of the band normalized to 1. If 7 random clue values are set in 7 random cells within the 27 cells of the band, and this clue combination resulted in a valid grid, then the probability a specific band to be formed is not flat, and this factor is some approximation for band's weight.
Column "In17" is the number of occurrences of the band within the known 46294 grids having 17-clue puzzles.
"BF17" is the actual weight of the band within grids with 17s.
"BF17/7" is the ratio between BF17 and BF7, just to present that some correlation exists but is not stable.

The number of non-minimal valid completions in the above table is incorrect. It is fixed below in this table.
There is similar thread Minimal clues to complete a grid examining k-templates.

by **dobrichev** » Mon Mar 07, 2011 2:29 am

The bands ordered by frequency in 46294 grids with 17-clue puzzles is in this table below.

Hidden Text: Show

Code: Select all: Occur Band 4 413 8 224 9 133 12 240 13 001 14 104 16 254 21 307 21 348 25 016 30 415 31 237 41 289 61 414 62 101 71 416 73 302 76 084 78 008 80 105 82 045 83 253 86 219 93 083 94 122 96 054 99 067 103 309 104 137 110 033 124 032 129 004 129 098 130 097 131 129 132 124 132 247 134 100 137 027 138 245 138 404 143 149 143 213 144 075 144 183 150 148 151 030 156 074 157 048 161 238 163 225 173 093 173 287 176 241 184 259 186 071 187 412 189 114 198 036 201 314 207 218 208 214 214 190 220 230 222 256 224 116 224 226 225 140 225 144 231 298 236 398 238 152 238 220 250 255 251 275 257 072 258 109 259 046 259 115 259 193 265 252 267 119 268 308 269 233 269 250 274 094 274 310 278 044 278 160 278 235 280 281 284 092 285 003 285 232 286 029 286 306 286 380 287 359 289 217 291 009 291 132 297 175 297 223 298 303 298 305 307 047 308 228 314 131 314 260 316 167 317 251 322 142 323 059 324 136 327 049 328 135 331 294 334 151 337 042 337 400 344 053 347 393 348 102 349 069 349 112 351 066 354 039 355 035 355 127 356 189 360 091 360 346 361 088 363 397 364 110 365 041 366 062 371 288 376 108 377 087 378 058 380 103 381 195 382 171 383 174 383 304 385 192 386 336 388 082 388 182 391 284 393 095 394 349 394 411 399 196 403 408 404 222 407 191 409 188 412 002 418 007 421 090 422 158 423 057 423 242 423 389 429 280 432 246 433 037 434 239 435 051 450 229 454 187 457 293 461 123 462 286 479 076 485 221 490 164 493 034 493 078 493 139 494 013 494 128 497 081 501 295 503 052 504 360 509 111 514 043 514 150 516 402 517 210 521 147 524 064 528 080 529 063 531 290 535 227 536 073 541 395 548 279 550 096 552 172 552 208 555 038 555 055 557 159 559 141 563 334 569 099 572 274 577 146 578 106 582 234 584 113 587 079 592 282 594 126 597 156 604 077 609 377 611 194 616 332 616 401 617 276 623 391 625 161 626 324 629 204 630 231 631 163 632 394 633 407 634 086 636 392 642 313 653 202 655 205 662 327 669 120 676 325 679 403 680 060 687 244 690 017 690 406 692 166 697 068 698 410 699 266 700 186 701 277 703 323 706 343 707 267 714 285 714 326 720 173 728 198 729 390 733 143 740 350 744 154 747 040 747 243 749 031 753 070 755 396 758 089 759 278 760 347 763 157 764 330 768 384 770 331 772 335 779 180 781 203 785 291 786 271 787 085 787 181 787 292 797 337 797 340 798 283 804 050 804 272 805 354 818 318 818 375 821 249 828 107 841 264 843 339 845 056 849 138 851 268 855 061 866 409 871 065 873 269 876 273 884 162 889 153 894 145 905 262 906 352 913 125 913 134 929 383 930 373 942 362 945 184 957 353 958 117 959 018 973 258 976 316 978 197 985 261 994 344 995 342 996 216 1003 341 1003 388 1004 257 1005 338 1006 329 1014 320 1019 372 1023 178 1033 328 1037 379 1042 312 1043 370 1047 130 1052 317 1054 005 1057 236 1060 270 1068 367 1078 215 1082 176 1103 118 1127 021 1134 297 1139 322 1145 300 1146 179 1148 301 1154 121 1174 165 1180 200 1184 321 1184 405 1187 311 1203 169 1204 333 1206 265 1210 019 1211 212 1218 399 1232 010 1232 023 1236 248 1239 155 1250 319 1253 177 1266 185 1276 387 1277 374 1289 170 1290 263 1299 376 1307 209 1308 299 1309 315 1324 199 1338 207 1354 368 1355 201 1420 296 1442 345 1450 014 1454 361 1492 206 1492 381 1498 355 1506 012 1516 371 1530 168 1568 211 1568 356 1590 028 1636 386 1651 385 1684 015 1703 365 1709 378 1728 366 1751 363 1798 011 1818 025 1845 364 1883 357 1896 382 1899 358 1915 369 2061 026 2081 351 2250 006 3139 022 3237 024 3330 020

Next, for the known 17-clue puzzles we can count the crossing bands (band B1 from 416 crosses stack B2 from 416).

For the 49151 17s (each consist of 9 crossings) there are 69343 different crossing combinations, not taking into account where two bands share a box.

Below is a summary for the combinations, grouped by number of occurrences in 17-clue puzzle grids. Here the puzzles are processed and solution grids having more than one 17-clue puzzle are counted more than once.
Column 1 is the group (class) size.
Columns 2 and 3 are the crossing bands for an exemplar of this group.
Column 4 is the number of occurrences of this crossing box.

Hidden Text: Show

Code: Select all: Gr.size Exemplar B1xB2 Num occurrences 1 020 024 144 1 022 024 137 1 020 022 112 1 020 357 111 1 006 024 107 1 024 351 106 1 022 378 99 1 020 364 94 1 024 357 88 2 022 381 86 3 020 382 85 1 020 358 83 1 020 351 82 1 024 369 80 1 020 378 79 1 024 211 78 2 022 382 77 3 024 025 75 4 022 369 73 5 024 382 72 2 024 381 71 2 024 385 70 3 022 366 69 7 024 026 68 4 026 351 67 2 024 363 66 2 024 028 65 5 022 357 64 3 022 212 63 7 022 385 62 3 022 365 61 8 168 378 60 7 024 371 59 10 026 358 58 11 024 364 57 9 024 168 56 7 168 381 55 6 025 382 54 8 024 329 53 8 351 382 52 14 024 386 51 14 024 368 50 11 024 317 49 17 369 382 48 24 356 357 47 16 201 351 46 18 366 369 45 26 358 363 44 24 363 378 43 27 364 387 42 20 357 365 41 30 382 386 40 46 374 382 39 31 381 386 38 29 364 369 37 57 378 386 36 50 368 378 35 52 369 378 34 57 381 382 33 84 382 385 32 95 378 385 31 96 378 381 30 110 371 378 29 130 386 387 28 150 382 399 27 180 385 386 26 217 385 409 25 214 387 399 24 284 386 405 23 293 386 399 22 378 383 405 21 394 383 386 20 504 387 405 19 534 403 409 18 654 399 405 17 726 386 410 16 844 399 409 15 1082 388 399 14 1275 405 410 13 1423 405 411 12 1704 407 410 11 2045 405 405 10 2433 401 406 9 2907 402 410 8 3499 409 411 7 4378 409 410 6 5256 406 411 5 6572 403 411 4 7999 410 410 3 9801 406 412 2 12372 411 416 1

Next, taking into account where the respective bands are crossed (box number of the minlex representation of the first band, and box number for the second band), there are 256191 distinct combinations for the 46294 grids.
Grouped by the number of occurrences, here is the summary.

Code: Select all: Gr.size Exemplar Num occurences 1 022 2 022 3 25 2 020 1 024 3 20 2 020 1 024 2 19 1 020 2 024 1 18 5 006 3 024 3 17 5 014 2 020 2 16 7 011 2 385 2 15 13 006 3 358 3 14 25 006 3 020 1 13 38 006 1 024 1 12 47 006 1 024 3 11 104 006 2 198 1 10 197 006 1 022 2 9 432 005 1 024 2 8 794 005 2 020 1 7 1637 002 2 382 2 6 3615 002 3 026 2 5 8516 002 1 020 2 4 21529 002 1 011 3 3 57179 001 1 299 1 2 162042 001 2 017 1 1

Columns 2,3,4,5 for the exemplar represent Band1 number, Box in band 1, Band 2, Box in band 2 respectively.

The same data for the puzzles, i.e. puzzle solutions weighted by puzzle count in the solution, give very different result since the occurrence count is in the same order as puzzle count in some grids.

Hidden Text: Show

Back to solution grids.
Here is the number of occurrences of the crossing bands for the top of the list (10+ grids)

Hidden Text: Show

Code: Select all: Band1 Box1 Band2 Box2 Num Grids with 17s 022 2 022 3 25 020 1 024 3 20 024 2 358 3 20 020 1 024 2 19 022 1 024 1 19 020 2 024 1 18 006 3 024 3 17 020 2 022 1 17 020 3 022 2 17 020 3 024 3 17 022 2 382 1 17 014 2 020 2 16 022 2 024 2 16 022 2 206 3 16 022 2 351 3 16 022 2 405 2 16 011 2 385 2 15 020 3 357 2 15 022 1 024 3 15 022 3 024 1 15 022 3 297 3 15 022 1 378 3 15 024 2 357 2 15 006 3 358 3 14 011 1 024 1 14 020 1 025 1 14 020 1 273 3 14 020 1 365 1 14 020 2 365 3 14 020 1 382 3 14 022 2 022 2 14 022 2 024 1 14 022 2 369 3 14 024 3 206 2 14 024 1 351 2 14 024 2 351 1 14 006 3 020 1 13 011 2 358 3 13 012 1 363 1 13 020 2 022 2 13 020 1 024 1 13 020 2 024 2 13 020 2 026 3 13 020 3 185 3 13 020 1 351 1 13 020 2 357 1 13 020 2 369 2 13 020 2 371 1 13 020 2 382 1 13 022 3 351 2 13 022 1 369 1 13 022 1 385 2 13 024 1 024 3 13 024 3 024 3 13 024 2 026 3 13 024 3 026 3 13 024 3 028 2 13 024 3 206 1 13 024 1 351 1 13 024 3 363 3 13 024 3 369 2 13 006 1 024 1 12 006 1 024 2 12 006 1 026 3 12 006 1 315 2 12 015 1 020 2 12 020 1 020 3 12 020 3 023 2 12 020 1 026 1 12 020 3 170 3 12 020 3 197 3 12 020 2 270 3 12 020 3 351 2 12 020 3 351 3 12 020 1 357 1 12 020 2 357 3 12 020 3 357 1 12 020 1 358 2 12 020 2 363 1 12 020 1 364 2 12 020 2 364 1 12 020 3 364 1 12 020 3 365 1 12 020 3 366 2 12 022 2 024 3 12 022 3 024 3 12 022 3 168 2 12 022 1 356 1 12 022 1 357 1 12 022 3 358 1 12 022 3 365 1 12 022 2 378 3 12 022 3 405 2 12 024 3 025 2 12 024 1 199 2 12 024 3 351 3 12 024 1 363 1 12 024 1 381 2 12 024 2 382 3 12 006 1 024 3 11 006 2 356 1 11 006 3 357 2 11 006 3 366 1 11 011 3 351 1 11 015 2 357 2 11 020 1 022 3 11 020 2 211 1 11 020 2 212 1 11 020 1 265 2 11 020 3 272 3 11 020 2 355 1 11 020 1 357 2 11 020 2 358 1 11 020 2 366 1 11 020 3 366 3 11 020 1 368 1 11 020 1 378 3 11 020 1 381 1 11 020 2 382 2 11 020 1 386 1 11 022 3 023 1 11 022 2 177 3 11 022 1 185 1 11 022 1 211 1 11 022 3 301 3 11 022 1 357 2 11 022 3 366 3 11 022 1 378 1 11 022 1 381 2 11 022 1 381 3 11 022 3 382 2 11 022 3 383 2 11 024 2 028 1 11 024 2 061 2 11 024 3 176 1 11 024 3 296 3 11 024 1 300 3 11 024 1 333 1 11 024 3 351 1 11 024 3 351 2 11 024 3 364 3 11 024 1 365 2 11 024 3 382 1 11 024 3 382 2 11 024 1 385 3 11 024 1 387 3 11 006 2 198 1 10 006 2 211 2 10 006 3 351 2 10 006 3 363 2 10 006 3 364 3 10 006 2 369 2 10 011 2 155 1 10 011 2 209 2 10 011 1 355 2 10 014 3 020 1 10 015 2 020 3 10 015 3 366 1 10 015 2 382 1 10 015 2 386 2 10 020 1 022 1 10 020 3 024 2 10 020 2 025 1 10 020 3 026 3 10 020 2 165 2 10 020 2 207 2 10 020 2 263 2 10 020 3 265 2 10 020 1 329 1 10 020 2 344 1 10 020 3 351 1 10 020 3 356 1 10 020 3 357 3 10 020 1 358 1 10 020 3 358 3 10 020 2 361 1 10 020 1 364 1 10 020 1 364 3 10 020 2 364 3 10 020 3 364 3 10 020 3 365 2 10 020 1 368 2 10 020 1 378 1 10 020 2 378 1 10 020 2 378 2 10 020 3 382 1 10 020 2 385 1 10 020 3 386 2 10 020 1 388 1 10 020 2 405 2 10 022 2 025 2 10 022 3 025 1 10 022 3 138 1 10 022 2 155 2 10 022 2 199 3 10 022 3 201 3 10 022 2 212 1 10 022 2 212 3 10 022 3 265 3 10 022 2 296 2 10 022 2 299 3 10 022 3 299 1 10 022 1 319 3 10 022 2 333 3 10 022 1 345 1 10 022 1 351 2 10 022 2 356 1 10 022 3 362 2 10 022 3 363 3 10 022 1 369 3 10 022 2 372 1 10 022 3 375 3 10 022 1 382 3 10 022 2 399 3 10 024 2 025 2 10 024 1 026 1 10 024 2 028 2 10 024 1 211 2 10 024 2 211 2 10 024 3 211 2 10 024 1 212 1 10 024 2 248 3 10 024 3 299 3 10 024 2 351 2 10 024 3 353 2 10 024 2 355 2 10 024 3 356 2 10 024 1 357 3 10 024 2 357 3 10 024 3 357 1 10 024 1 363 3 10 024 2 363 1 10 024 3 365 3 10 024 3 369 1 10 024 2 371 1 10 024 3 385 3 10 025 2 155 2 10 025 3 200 1 10 026 1 165 2 10 026 1 206 1 10 026 1 351 1 10 168 3 261 2 10 168 2 351 3 10 206 1 357 3 10 272 1 351 1 10 351 1 351 2 10 351 2 365 2 10 351 3 366 3 10 363 2 366 2 10 364 1 368 3 10

Note that different row permutations in the crossing bands result in different solution grids. The possible row permutations (6 for band1 * 6 for band2 = 36) are ignored in these statistics.

MD

by **dobrichev** » Mon Mar 07, 2011 3:11 am

17-clue puzzles occupy from 3+6+8 clues per band to 4+4+9 clues.

There are no puzzles with 3 or 9 clues simultaneously in a band and a stack.

The only puzzles which have 8 clues in a band in both directions are these three:

Code: Select all: ... ... ... ... ... ..1 ..2 .34 ... ... ..2 ... ... 1.5 ..6 ..3 ... ... .1. ... .4. .5. .78 .3. .6. 5.. ... ... ... ... ... ... .12 ... ..3 ..4 ... ..5 6.. ..1 ... ... ..2 ... ..7 ... .1. ... .6. .7. 8.. .8. 2.. 56. ... ... ... ... ..1 ..2 ..3 ... .4. ... ... 5.. ..2 .6. 3.. .5. .7. 18. ... 2.. .3. .1. ... ... 7.. ... 8..

MD

by **dobrichev** » Mon Mar 07, 2011 3:32 am

This is the distribution of 17s by band population.

Code: Select all: Count min max 1248 4 7 3 4 8 40375 5 6 7525 5 7

where
min = max(min(bands population), min(stacks population))
max = min(max(bands population), max(stacks population))

by **Serg** » Thu Mar 10, 2011 10:46 pm

Hi, dobrichev!
You published interesting results. I need some time to ponder them. If my understanding is right, you are trying to find regularities in 17-clues puzzles to limit efficiently space of search for new 17-clues puzzles. I have similar ideas concerning search for 18-clues horizontally symmetric puzzles. (I'll publish them in a while.)

Serg

by **champagne** » Thu Feb 18, 2016 9:40 pm

Hi mladen,
working on another topic, I fall back in that problem.

Being locked in the search described in
this post,

I have investigated what could be done with the last version of the brute force.
In the process applied, we use a top down process to the ED patterns 27 + X + Y

Code: Select all: selecting a gangster finding all permuttations of the pattern to apply to the gangster expanding the pattern with filters against redundancy Checking if the final puzzle is valid

For the time being, the bottleneck of the process is he brute force, costly when the puzzle has a multiple solution, which is the majority by far with 27+3+5.

The only way I see to filter the multiple solutions puzzles is to switch from a top down to a bottom up process. But let me tell more.

As you write in that thread, any valid band can be morphed to one of the 416 rowminlex bands.

Let's assume that we look for valid puzzles 27+5+3.
After magic filter, we have 8208 ED patterns to test.

The concept for each ED pattern would be

Code: Select all: - Find all valid Band 3 (3 clues) - for each valid band3 find all valid band2 (5 clues) again using pre tested bands check whether the final 2 bands is valid;

When band 2 and band 3 are defined, the gangster in band 1 is known. We are back in the top down process.

Following that idea, we need the collection of valid 3 clues in a band leading to

- a band with only one solution
- no redundant clue (although this is not that common with 3 clues)

Is it the definition of your column 54 + 3 ? (54+n)

I made my own test using the blue prints of Mc Gary and I am below your count
but this is without expansion of all 2 clues giving a multiple solution.
I need more code to find these
I'll cross check the results with yours.

I'll write a separate post to describe with more details what could be the bottum up process.

BTW, I have an oddity with the band (381)

123456789457289631698317254

I get the following UA's from the Mc Gary file

..1....1..11..1.1..11..1.1.
.1.1....1.1.1...1..1.1...11
.....11...1.1.111..1.1.111.
1.111..1.1.......1..111..11

Here you have one cell hitting all UA's. For sure, the corresponding band will have a multiple solution, but it has a good chance to be a source for a 2 or 3 clues valid band;
Good example any way to test the relevant code.

by **champagne** » Fri Feb 19, 2016 5:45 am

More on the 27 + X + Y bottom up search

I'll continue to use the 27 + X + 3 example, and more precisely 27 + 5 + 3 already processed.

Having applied the magic filter, we have to expand 8209 ED patterns.

The first remark is that, even if the use of the 416 was a necessary step to find valid bands, we are in a gangster vision.

If I consider my own partial set of "3 given" valid bands, not all gangsters have a valid band. Active gangsters are

Code: Select all: 5 to 13 16 to 18 21 to 30 32 to 36 41-42.

part of them have relabelling automorphisms, that should be used to reduce the count. Some of the 3 given band solutions are isomorph.

We want to start with the band having 3 given. Interesting in the split of my set of band solutions by "maxpat"

Code: Select all: 342 110 000 000 000 100 000 empty 16 100 100 000 100 000 000 empty 530 100 100 000 010 000 000 empty 356 100 100 000 000 000 100 empty 29 100 000 000 100 000 000 000 100 000 424 100 000 000 010 000 000 000 100 000 178 100 000 000 000 100 000 000 000 100 1875 (2739 in dobrichev's count, including 1+2 and 2+1 missing here)

Some possible patterns are missing, as all given in box 1, but these are likely excluded in the magic filter

So the number of starts, before automorphisms is a minimum of (0 to 29) and a maximum of 530

For the second band (here 5 given), we have to cross

gangsters authorised by the gangster in band 3
solution band authorised by the maxpat equivalent to the given 5 clues pattern.

Again, likely, isomorphisms can be applied to reduce the count, but the maximum of solutions is in average far below the 3483430 possibilities identified by Mladen Dobrichev.

I can say that compared to the count done in the top down process, this is peanut.

by **dobrichev** » Fri Feb 19, 2016 7:56 am

champagne wrote:Following that idea, we need the collection of valid 3 clues in a band leading to

- a band with only one solution
- no redundant clue (although this is not that common with 3 clues)

Is it the definition of your column 54 + 3 ? (54+n)

The definition from above is
"Column "54+7" contains the number of valid completions with up to 7 additional clues (including all combinations of 6 clues + one redundant), and so on."
In other words, uniqueness is guaranteed, but minimality is not.
The minimality criterion for 54+3 case is applicable only for band 29 which has 9 54+2 patterns - the 543 54+3 patterns must be tested whether they are supersets of the 9 54+2 ones.

I see no reason to work with Mc Gary's subset of UA since the complete sets of in-band UA per band is sufficiently short and is trackable.
Also, I see no reason to stick to minimality. The obtained patterns resolve only the in-band UA and this is found to be very weak condition. Cross-bands UA must be resolved too, and the enumerated set of all non-minimal clue combinations can be directly tested instead of be generated on the fly.

by **champagne** » Fri Feb 19, 2016 10:58 am

Hi Mladen,

clear and quick answers

dobrichev wrote:I see no reason to work with Mc Gary's subset of UA since the complete sets of in-band UA per band is sufficiently short and is trackable.

I have 2, but having the full set would be grate

a) I have the code to use the sets of UA's size 12 and less
b) It covers most of the cases for a small number of clues and I see what code I can write and use to fill the gap

dobrichev wrote:Also, I see no reason to stick to minimality. The obtained patterns resolve only the in-band UA and this is found to be very weak condition. Cross-bands UA must be resolved too, and the enumerated set of all non-minimal clue combinations can be directly tested instead of be generated on the fly.

I did not think about cross bands, but what I intend to do is to combine 2 "horizontal" bands.

If the partial solution in one band is not minimal, IMO the final result will not be a minimal sudoku.
Let me tell in another way.

Having the solution grid in the band 3, we have all mini columns established and this is what is used for the next steps in the process.
The redundant clue has no effect on that process. It can be cleared in the final grid.

,

by **blue** » Wed Feb 24, 2016 12:54 am

Hi Mladen,

dobrichev wrote:Solving such pseudo-puzzles we can find for all 416 bands:
- the number of solutions (column "Sol" in the table below);
- all minimal UA sets within the band (column "UA" is the number of UA);
- the minimal number of the additional clues required to hit all in-band UA (MinClues);
- all valid permutations of N additional clues that hit all UA, for some reasonable small N (54+N).

In working with champagne, and trying to produce the "54+N" numbers for the 44 "gangster" (PM) bands, I've come to the conclusion that either I've misunderstood what you were doing, or there are errors in your "54+N" columns, for N > MinClues(band).

At first, I thought I could calculate what I wanted, using your data and a map from "minlex bands" to "gangsters".
After a big mistake in the method, and then implementing the "fix", I wanted to verify my final results.
[ That's what finally lead me to making this post. ]

The way I understood your description, the numbers should match the number of 54+N clue subgrids that represent valid puzzles ... where the full grids have a (fixed) minlex band 1, and any (fixed/compatible) completion in bands 2 & 3.

For a while, I wondered if you were only counting N clues that had a "MinClues" size subset that was also a valid puzzle.
I don't think that was true, though, since some of the numbers that I'm getting are larger, and some are smaller than yours.

Have I misunderstood ?
Do I have an inevitable bug in my code ?

Below is a comparison for the first 10 bands.

Best Regards,
Blue.

Yours:

Code: Select all: band 54+2 54+3 54+4 54+5 54+6 54+7 1 0 0 0 0 729 17496 2 0 0 0 324 10179 107406 3 0 0 108 4194 41895 260689 4 0 0 0 252 7977 91579 5 0 0 108 5328 55883 381253 6 0 0 101 4354 47881 346825 7 0 6 912 15274 120750 736608 8 0 0 0 1107 22952 221705 9 0 0 54 2664 34263 233862

Mine:

Code: Select all: band 54+2 54+3 54+4 54+5 54+6 54+7 1 0 0 0 0 729 18225 2 0 0 0 324 10503 112491 3 0 0 108 4302 45333 257283 4 0 0 0 252 8229 92889 5 0 0 108 5436 58401 317241 6 0 0 101 4455 51291 292485 7 0 6 918 16164 117129 507843 8 0 0 0 1107 23139 179676 9 0 0 54 2718 36585 235521

by **dobrichev** » Wed Feb 24, 2016 6:25 am

Hi Blue,

I remember we had similar inconsistency in the number of 20-clue puzzles for the grid having 20 17s. From memory, I was wrong there because my code unexpectedly "optimized" some of the non-minimal subgrids.
Since the 54+N were generated in the same way and using the same code, it is much possible that my results are wrong due to the same error.

The case for band 3 col 54+7 is a big surprise, because you are finding less valid subgrids and if you are right, then my results contain invalid ones.

I will try to find the code I used, will check whether I can reproduce mine or yours results, and will dump the clues for some of the bands allowing row-by-row comparison of the clues.

For the purposes of 27+x+y approach, the efficiency of using the pre-calculated tables must be proven. My doubts are that the expansion of the tables of valid clue combinations to all possible column re-arrangements (or one-to-many pattern transformation) is costly and might cancel the reductions done previously by many-to-one pattern/band transformations. Unless it is done in a smart way on the right place of course, and such place could still exist.

by **blue** » Wed Feb 24, 2016 8:03 pm

dobrichev wrote:I will try to find the code I used, will check whether I can reproduce mine or yours results, and will dump the clues for some of the bands allowing row-by-row comparison of the clues.

Sounds good, but ...

The case for band 3 col 54+7 is a big surprise, because you are finding less valid subgrids and if you are right, then my results contain invalid ones.

If that's true, then you should be able to find the invalid results without comparing lists.
Then fixing bug(s) that lead to those results, might make the other numbers match too.

dobrichev wrote:For the purposes of 27+x+y approach, the efficiency of using the pre-calculated tables must be proven. My doubts are that the expansion of the tables of valid clue combinations to all possible column re-arrangements (or one-to-many pattern transformation) is costly and might cancel the reductions done previously by many-to-one pattern/band transformations. Unless it is done in a smart way on the right place of course, and such place could still exist.

All true. The "reductions done previously by many-to-one pattern/band transformations", used two ideas: exploit gangster automorphisms for the 27-clue band, and remove X+Y patterns that are known to not have valid puzzles ("magic patterns" filter). There is an easy way to use both ideas with the puzzle tables, in a "one-to-many" approach for generating test puzzles. The number of puzzles generated for a few X+Y cases, are close enough that I'm keeping an open mind about the whole approach. I'm close to giving up on the idea, though.

by **dobrichev** » Wed Feb 24, 2016 9:09 pm

Some progress here.

I found not all clue combinations are expanded to the 54+N size, so that the secondary non-zero column and later don't contain all minimal + non-minimal maps.
All my numbers now are <= than yours. Something has been fixed in the code within the last 5 years.

I have 3 questions now.
1) Are the mixed minimal+non-minimal clue combinations really needed? We can generate only minimal ones after all and easily expand them if/when needed.
2) What is the most suitable format? Bitmaps naked to min-lexed band, or compressed 27-bit bitmaps (integers), or 27-character strings, or 27+54 character strings? One file per band per size, or single file with columns for band and size, or something in between?
3) Do we need data for 54+8? It seems achievable (~1 hour of calculations), especially if keeping tables only for minimals.

My plan is to expand minimals in all possible ways (if we want mixture of minimals and non-minimals for a fixed size), and finally to run once more the solver to ensure there are no multiple-solution puzzles due to other errors in code.
Probably will do it tomorrow.

Cheers,
Mladen

by **blue** » Wed Feb 24, 2016 11:34 pm

dobrichev wrote:Some progress here.

I found not all clue combinations are expanded to the 54+N size, so that the secondary non-zero column and later don't contain all minimal + non-minimal maps.
All my numbers now are <= than yours. Something has been fixed in the code within the last 5 years.

Good news !

I have 3 questions now.
1) Are the mixed minimal+non-minimal clue combinations really needed? We can generate only minimal ones after all and easily expand them if/when needed.

Producing the non-minimals from the minimals is easy enough, but the obvious method would produce duplicates (if there were non-minimals with more than one minimal subset).

This is something I put in a PM to champagne, to show why the non-minimal cases are needed (at some point in the game):

(...) here is a 17 clue puzzle metioned in Mladen's thread, and a version with bands 2 and 3 filled in from the solution grid.
In the 54+4 puzzle, 2r2c9 is redundant, but (of course) it isn't redundant in the 17-clue puzzle.

Code: Select all
. . . | . . . | . . . . . . | . . 1 | . . 2 . . 3 | . . . | . 4 . ------+-------+------ . . . | . . . | 5 . . . . 2 | . 6 . | 3 . . . 5 . | . 7 . | 1 8 . ------+-------+------ . . . | 2 . . | . 3 . . 1 . | . . . | . . . 7 . . | . . . | 8 . . . . . | . . . | . . . . . . | . . 1 | . . 2 -> 2r2c9 is redundant . . 3 | . . . | . 4 . ------+-------+------ 6 3 1 | 4 9 8 | 5 2 7 8 7 2 | 1 6 5 | 3 9 4 4 5 9 | 3 7 2 | 1 8 6 ------+-------+------ 5 9 6 | 2 8 7 | 4 3 1 3 1 8 | 6 5 4 | 2 7 9 7 2 4 | 9 1 3 | 8 6 5

2) What is the most suitable format? Bitmaps naked to min-lexed band, or compressed 27-bit bitmaps (integers), or 27-character strings, or 27+54 character strings? One file per band per size, or single file with columns for band and size, or something in between?

If it's for sharing globally ... something I can't do ... then I'm not the one to ask. (I can generate them here).
If it's to use between us, for bug hunting, then (zip) compressed 27-bit bitmaps (integers), is fine. I would just need to know which bit is for (r1,c1), and so on. It would be nice to have the non-minimals included, too, and just a few files for (band,size) pairs where we still have differences.

3) Do we need data for 54+8? It seems achievable (~1 hour of calculations), especially if keeping tables only for minimals.

For the "27+x+y" problems (looking to make 17's out of them), they wouldn't be needed.
27+2+? results have been covered.
27+3+8's would reduce to 6+3+8, and would be better found as 8+3+6, etc.

Cheers,
Blue.

by **dobrichev** » Thu Feb 25, 2016 2:57 am

The revised results

Hidden Text: Show

Code: Select all: Band 54+2 54+3 54+4 54+5 54+6 54+7 1 0 0 0 0 729 18225 2 0 0 0 324 10503 112491 3 0 0 108 4302 45333 257283 4 0 0 0 252 8229 92889 5 0 0 108 5436 58401 317241 6 0 0 101 4455 51291 292485 7 0 6 918 16164 117129 507843 8 0 0 0 1107 23139 179676 9 0 0 54 2718 36585 235521 10 0 0 92 4146 49197 286869 11 0 12 972 14178 98031 430885 12 0 0 224 6161 59665 318689 13 0 0 312 8358 74694 372579 14 0 0 462 10212 82402 389844 15 0 0 415 9826 81688 389816 16 0 0 0 972 21060 169506 17 0 72 2766 26988 146553 557565 18 0 0 108 5274 58293 317223 19 0 0 189 7515 74232 376290 20 0 21 1380 18708 121704 504070 21 0 0 108 5274 58293 317223 22 0 5 1067 18207 124493 521971 23 0 0 520 11860 94865 437919 24 0 5 1067 18207 124493 521971 25 0 30 1802 24248 152198 603888 26 0 110 3275 31730 170911 633275 27 0 0 1053 21087 135486 542250 28 0 38 2025 26285 157845 610977 29 9 567 7398 51516 237762 803574 30 0 0 1053 21087 135486 542250 31 0 64 3471 40029 218658 781950 32 0 0 0 546 12291 101597 33 0 0 0 63 3786 48016 34 0 0 200 5382 47841 244398 35 0 0 38 2050 24981 158665 36 0 0 39 2428 30236 183823 37 0 0 273 5839 48964 245226 38 0 0 182 5027 46207 240106 39 0 0 22 1614 22296 147739 40 0 3 659 12051 86871 384624 41 0 0 74 4132 47369 267543 42 0 0 126 3746 36128 195645 43 0 0 145 4126 39350 213815 44 0 0 34 2444 31234 190371 45 0 0 0 90 4992 61645 46 0 0 37 1859 22537 144177 47 0 0 29 1858 25833 166396 48 0 0 0 63 3786 48016 49 0 0 38 2050 24981 158665 50 0 3 740 13176 92341 398974 51 0 0 35 2287 30924 193357 52 0 0 95 4182 44348 243667 53 0 0 267 7835 66523 322399 54 0 0 0 378 9978 87206 55 0 0 161 5230 48594 250985 56 0 2 557 10414 75983 343119 57 0 0 56 2497 30129 181591 58 0 0 40 2730 33702 202070 59 0 0 29 1894 25044 162880 60 0 0 65 3085 37824 224884 61 0 0 304 7481 64086 314897 62 0 0 29 1894 25044 162880 63 0 0 187 4570 42130 224742 64 0 0 195 5318 50101 261611 65 0 7 668 11527 84087 377130 66 0 0 34 2444 31234 190371 67 0 0 0 90 4992 61645 68 0 2 557 10414 75983 343119 69 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Bands and low-clue puzzles

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