The New Sudoku Players' Forum

Website · by **denis_berthier** » Sat Apr 02, 2022 4:49 am

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Here is a quick survey of the full list of 972 non-T&E(2) puzzles https://docs.google.com/spreadsheets/d/1t-PsJT-pKGQEWjSbbNBXzLcxb5Inmooszntu9ZVCW_M/edit#gid=0 introduced by mith here: http://forum.enjoysudoku.com/the-hardest-sudokus-new-thread-t6539-1231.html
It includes the previous list of 246.
Here, I'm using the new IDs given them by mith.

For completeness, let me first recall that all of them are in T&E(W2, 2), which has become the new frontier.

Using only (Naked + Hidden + Super-HIdden) Subsets + Finned Fish + the Tridagon elimination rule defined in the 1st post of this thread, 216 puzzles can be solved:

Code: Select all: 4 5 9 21 22 23 25 29 30 35 36 37 38 51 52 53 54 55 56 57 58 71 73 83 84 85 86 87 88 91 92 101 106 107 108 109 112 121 122 123 124 130 134 135 136 139 140 141 154 157 163 166 167 171 177 178 179 180 192 193 194 195 198 199 200 201 204 205 206 207 208 209 228 233 247 248 249 250 251 252 253 272 273 274 300 301 302 303 304 305 306 307 320 323 324 325 326 327 328 329 330 331 332 344 345 346 347 348 349 372 375 376 388 389 390 391 392 401 403 404 405 406 438 439 443 453 455 461 463 464 465 466 467 468 469 470 501 502 507 508 525 526 527 530 531 561 568 571 572 573 598 599 631 632 633 636 644 664 665 667 668 670 671 673 679 681 697 698 699 700 701 705 707 714 716 720 721 722 723 728 729 736 737 742 743 751 752 790 795 861 862 864 865 866 867 873 874 875 876 877 895 902 904 905 906 907 908 918 921 929 941 946 949 950 954 955

Adding whips of length ≤ 12, 505 more puzzles can be solved. Notice that the Tridagon elimination rule may apply before any whip is actually used or after.

Code: Select all: 1 2 3 6 7 10 11 13 14 15 16 17 18 19 24 26 27 28 31 32 33 34 39 40 41 42 43 44 45 46 47 48 49 50 59 60 61 62 63 64 65 66 67 68 69 70 72 74 75 78 79 81 82 93 94 95 96 97 98 99 100 102 103 104 105 110 111 113 114 115 116 117 118 119 120 125 126 127 129 131 132 133 137 138 142 143 144 145 146 147 148 149 150 151 152 153 158 159 160 161 162 170 172 174 175 176 181 182 183 184 185 186 187 188 189 190 191 196 197 202 203 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 227 229 230 231 238 239 240 241 242 243 244 245 246 254 255 256 257 258 259 260 261 262 263 264 268 270 271 275 276 277 278 279 280 281 282 283 284 285 286 287 292 293 294 295 313 314 315 316 317 318 319 321 322 353 354 355 356 357 362 363 364 368 369 370 371 373 374 377 378 382 383 384 402 411 412 413 414 415 416 420 429 430 433 437 440 441 444 445 446 447 448 449 450 462 479 484 503 504 505 506 509 510 511 512 513 514 516 523 528 529 559 560 562 569 574 575 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 634 635 637 638 639 640 641 642 643 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 666 669 672 674 675 676 677 678 680 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 702 703 704 706 708 709 710 711 712 713 715 717 718 719 724 725 726 727 730 731 732 733 734 735 738 739 740 741 744 745 746 747 748 749 750 753 754 755 756 757 758 759 760 761 762 763 765 767 768 769 770 771 772 773 774 775 776 777 778 782 783 784 785 786 787 788 789 791 792 793 794 796 797 798 799 800 801 802 803 809 810 811 813 814 816 817 819 820 821 822 823 824 830 831 833 835 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 863 868 869 870 871 872 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 896 897 898 899 900 901 903 909 910 911 912 913 914 916 917 919 920 922 923 924 925 926 927 928 930 931 934 935 936 938 939 940 943 944 951 961

Until now, that makes a total of 721 puzzles solved without using Tridagon links.

Adding tridagon-links and Tridagon-Forcing-Whips of length ≤ 15, 41 more puzzles can be solved, pushing the total to 762:

Code: Select all: 8 12 20 76 77 80 89 90 155 156 164 165 168 169 173 226 232 234 235 236 237 265 266 267 442 459 518 522 524 563 564 565 567 570 595 596 597 932 933 945 948

Having a Tridagon-link and being solvable by Tridagon-Forcing-Whips is not equivalent: among the 972-721=251 puzzles that are not solved with the Tridagon elimination rule, only 3 puzzles don't have a tridagon-link:

Code: Select all: #827: +-------------------+-------------------+-------------------+ ! 1 23 348 ! 248 5 6 ! 7 248 9 ! ! 2478 25 4578 ! 1248 248 9 ! 1248 3 6 ! ! 6 29 489 ! 3 7 1248 ! 1248 5 248 ! +-------------------+-------------------+-------------------+ ! 23489 2359 34589 ! 59 6 248 ! 248 7 1 ! ! 248 16 1468 ! 12478 3 12478 ! 9 248 5 ! ! 2489 7 14589 ! 59 248 1248 ! 3 6 248 ! +-------------------+-------------------+-------------------+ ! 379 16 13679 ! 24678 248 2478 ! 5 12489 23478 ! ! 5 8 167 ! 2467 9 3 ! 246 124 247 ! ! 379 4 2 ! 678 1 5 ! 68 89 378 ! +-------------------+-------------------+-------------------+ #828 +-------------------+-------------------+-------------------+ ! 1 248 3 ! 248 5 6 ! 24789 278 49 ! ! 248 5 7 ! 1 248 9 ! 23468 238 346 ! ! 6 9 248 ! 3 2478 2478 ! 5 128 14 ! +-------------------+-------------------+-------------------+ ! 248 6 1 ! 5 3 248 ! 248 9 7 ! ! 59 23478 59 ! 248 6 2478 ! 12348 1238 134 ! ! 2478 23478 248 ! 9 2478 1 ! 2348 6 5 ! +-------------------+-------------------+-------------------+ ! 3 17 2569 ! 26 29 25 ! 17 4 8 ! ! 45789 1478 45689 ! 468 489 3458 ! 13679 137 2 ! ! 2489 248 24689 ! 7 1 2348 ! 369 5 369 ! +-------------------+-------------------+-------------------+ #836 +-------------------+-------------------+-------------------+ ! 1 248 3 ! 248 5 6 ! 24789 278 49 ! ! 248 5 7 ! 1 248 9 ! 23468 238 346 ! ! 6 9 248 ! 3 2478 2478 ! 5 128 14 ! +-------------------+-------------------+-------------------+ ! 248 6 1 ! 5 3 248 ! 248 9 7 ! ! 59 23478 59 ! 248 6 12478 ! 12348 1238 134 ! ! 2478 23478 248 ! 9 2478 12478 ! 12348 6 5 ! +-------------------+-------------------+-------------------+ ! 3 17 2569 ! 26 29 25 ! 17 4 8 ! ! 45789 1478 45689 ! 468 489 3458 ! 13679 137 2 ! ! 2489 248 24689 ! 7 1 2348 ! 369 5 369 ! +-------------------+-------------------+-------------------+

It is easy to see that these 3 puzzles have a more general pattern, with 1 (or even 2) additional candidate(s) in 3 cells. I'm curious to see if anyone can use such OR3 or OR4 relations in any smart way.

(Note. I'm making no statistical claim: due to the way it is built, the collection can only be strongly biased.)

by **marek stefanik** » Sat Apr 02, 2022 12:52 pm

All three grids are closely related – after altering 69r12c69 in #827, it is isomorphic to #828 and deleting one of the clues in #828 gives us #836.

In #836 we can use the TH with a short kraken:

Code: Select all: +-------------------+-------------------+-------------------+ ! 1 #248 3 !#248 5 6 ! 24789 278 49 ! !#248 5 7 ! 1 #248 9 ! 23468 238 346 ! ! 6 9 #248 ! 3 2478 #248+7 ! 5 128 14 ! +-------------------+-------------------+-------------------+ !#248 6 1 ! 5 3 #248 ! 248 9 7 ! ! 59 #37–248 59 !#248 6 *12478 ! 12348 1238 134 ! ! 2478 23478#248 ! 9 #248+7 12478 ! 12348 6 5 ! +-------------------+-------------------+-------------------+ ! 3 17 2569 ! 26 29 25 ! 17 4 8 ! ! 45789 1478 45689 ! 468 489 3458 ! 13679 137 2 ! ! 2489 248 24689 ! 7 1 2348 ! 369 5 369 ! +-------------------+-------------------+-------------------+

37r5c2 == [7r3c6 | 7r6c5] – 7r5c6 = 7r5c2 => –248r5c2

At this point I struggled to find an elegant way through, and eliminated 7r8c1 with a long B2B net.
The rest should be in T&E(1) and can be solved easily with the parity argument:

Code: Select all: .------------------.----------------.---------------. | 1 xz248 3 |xz248 5 6 | 7–248 278 9 | | 248 5 7 | 1 248 9 | 2468 238 36 | | 6 9 248 | 3 7 248 | 5 128 14 | :------------------+----------------+---------------: |x248 6 1 | 5 3 z248 |y248 9 7 | | 59 3 59 |y248 6 7 | 1248 128 14 | | 7 y248 z248 | 9 x248 1 | 3 6 5 | :------------------+----------------+---------------: | 3 17 2569 | 26 29 25 | 17 4 8 | | 4589 17 45689 | 468 489 3458 | 1679 137 2 | | 2489 248 24689 | 7 1 2348 | 69 5 36 | '------------------'----------------'---------------'

Fixing the permutation in b4, then chaining through b125 gives us the permutation in b5.
NP xzr1c24, HS yb6 => –248r1c7

Code: Select all: .------------------.---------------.--------------. | 1 248 3 | 248 5 6 | 7 y28 9 | | 248 5 7 | 1 248 9 | 248 3 6 | | 6 9 bxy248 | 3 7 xy248 | 5 128 14 | :------------------+---------------+--------------: |x248 6 1 | 5 3 z248 | 248 9 7 | | 59 3 59 |y248 6 7 | 248 128 14 | | 7 y248 z248 | 9 x248 1 | 3 6 5 | :------------------+---------------+--------------: | 3 7 269 | 26 29 5 | 1 4 8 | |c4589 1 45689 | 468 489 3 | 69 7 2 | | 2489 248 24689 | 7 1 a248 | 69 5 3 | '------------------'---------------'--------------'

HS yr1 => y is not 4
NP xyr3c36 (remote triple with r4c6) => whichever digit appears in r9c6 is forced into r3c3, then in b7 into r8c1 and cannot be x (r4c1) => it must be y => y is not 2
y = 8, stte

Marek

Website · by **denis_berthier** » Sat Apr 02, 2022 2:43 pm

marek stefanik wrote:All three grids are closely related – after altering 69r12c69 in #827, it is isomorphic to #828 and deleting one of the clues in #828 gives us #836.

I've also noticed bigger clusters of puzzles closely related.
There remains too much redundancy in the database:
- normally, a puzzle isomorphic to an existing one shouldn't enter the database; the same should be true for the expansions;
- there should be a way of extracting only the smallest puzzles

by **mith** » Mon Apr 04, 2022 4:46 pm

Denis, on the first point, 827 and 828 are not isomorphic themselves, but they are closely related by flipping the completed rectangle mentioned. It's necessary to keep track of these cases because, while they will solve identically from the expanded forms, they do not have the same minimals (and of course in some cases the minimals will turn out to be harder than the original puzzle).

I will be working on some tools to relate puzzles of both the "digit swapped" type (found by applying the algorithm suggested by 999_Springs earlier in the thread) and the "subset" type (found by enumerating all minimals and then singles-expanding those minimals - not all minimals will singles-expand to the original puzzle). Still not sure what the best way to notate those relationships in the database itself would be, though - we may just have to rely on the solution-minlex form and simple scripts to do that work.

836 is pretty interesting. The trivalue oddagon elimination is similar to some of the others we've looked at, where some guardian(s) force another guardian, but this one remains tough after the elimination. I don't see an elegant way through either, but I'll have to look at it some more later.

by **mith** » Mon Apr 04, 2022 4:56 pm

It's worth noting that there *is* another ALS technique to get a digit in this puzzle, this time after the trivalue oddagon elims and using that new bivalue cell as one of the links:

9r9c1 =ALSc1= 7r6c1 - (7=3)r5c2 - 3r5c9 =ALSc9= 9r1c9 => -9r9c9 (and then +9r1c9)

Given the number of trivalue cells in these puzzles, the existence of ALS links involving the trivalue oddagon boxes (such as the one in c1 here) isn't too surprising; worth keeping in mind for the harder versions of the pattern.

Website · by **denis_berthier** » Tue Apr 05, 2022 4:51 am

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Hi mith,
No doubt your last .xl version of the database is a great progress wrt to what you previously published directly on the forum.
After using it as reported in this thread, I have a few more remarks:
- it's a good thing to keep it independent of the older T&E(2) "hardest" collection; the search for hardest in the old SER sense and the search for non-T&E(2) seem to be neatly separated.
- I don't know how many different seeds you used at the start (I think not so many), but it'd be interesting to have each puzzle marked with its original seed;
- what's useful for extending the database may not be useful for analysing it: I understand you need to keep expanded forms that are not "minimal" (not a good word here, but I can't find any better one; I mean expanded forms that have no superset in the expanded database)
- a possible solution to the above would be to have a 3rd database of "minimal" expanded puzzles (or a script for easily extracting it); it should keep the same IDs as the full database of expanded forms (no problem with missing IDs)

mith wrote:I will be working on some tools to relate puzzles of both the "digit swapped" type (found by applying the algorithm suggested by 999_Springs earlier in the thread) and the "subset" type (found by enumerating all minimals and then singles-expanding those minimals - not all minimals will singles-expand to the original puzzle). Still not sure what the best way to notate those relationships in the database itself would be, though - we may just have to rely on the solution-minlex form and simple scripts to do that work.

If more than the "minimal" expanded forms are needed by anyone (3rd database solution), I think solution-minlex and scripts would be the most versatile way of proceeding.

mith wrote:836 is pretty interesting. The trivalue oddagon elimination is similar to some of the others we've looked at, where some guardian(s) force another guardian, but this one remains tough after the elimination. I don't see an elegant way through either, but I'll have to look at it some more later.

At this point, I'm not much into looking at cases with more than 2 additional candidates in different cells. A general analysis of the possibilities (along the lines I used for Tridagon-links) seems quite difficult. Moreover, the possibility for a human solver to identify such a pattern is questionable, unless he is specifically warned that there is one; but solving and solving with an oracle are two different things.
It's the usual problem of the simple cases of a pattern versus variants.

by **mith** » Tue Apr 05, 2022 10:51 pm

I would disagree with the human solver part of that. Once you know about the technique (not in the puzzle specifically, just as a concept), the patterns are not hard to spot. I'd say this is a big advantage such a large scale pattern has over other types of patterns - even the complex versions with krakens on the guardians have lot of trivalue cells on the same three digits in such an arrangement that begs for further investigation.

Whether it is the case that the relevant deductions using the trivalue oddagon will always be accessible for a human solver is still to be determined, but none of the examples so far are all that complex.

(That said, I have finally just started the first new script looking for more puzzles outside T&E(singles,2), so should have some new data to look at soon.)

Website · by **denis_berthier** » Wed Apr 06, 2022 3:55 pm

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A quick analysis of Hendrik's collection of 70 11.8s here: http://forum.enjoysudoku.com/the-hardest-sudokus-new-thread-t6539-1240.html

As I mentioned in the "hardest" thread:
- 44 of these puzzles (3 4 13 14 15 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 39 40 44 45 46 47 48 49 50 51 52 52 54 55 56 57 58 59 70) are in T&E(2)
- the rest (26)is in T&E(W2, 2).
- all of these puzzles have a Tridagon elimination rule of the type defined in the first post, available before the application of any whip.

Here are more results in the same vein as what I did with mith's collection. They show (not unexpectedly) some difference in the global distribution.
- None of these puzzles can be solved with only Subsets + FinnedFish + the Tridagon elimination rule
- If we add whips[≤12], all of them can be solved, except 3: #48, #50 and #51
- The length of the whips necessary to solve a puzzle seems unrelated to whether is is in T&E(2) or not.
- Tridagon-links are not needed.

As for the 3 not solved, I've checked #48:
98.76.5..7.54.98....6......69.8.7....57.46.8...45.....4.8...93......46.........2. 11.8/1.2/1.2 #48

Code: Select all: Resolution state after Singles and whips[1]: +-------------------------+-------------------------+-------------------------+ ! 9 8 123 ! 7 6 123 ! 5 14 1234 ! ! 7 123 5 ! 4 123 9 ! 8 16 1236 ! ! 123 4 6 ! 123 12358 12358 ! 1237 179 12379 ! +-------------------------+-------------------------+-------------------------+ ! 6 9 123 ! 8 123 7 ! 1234 145 12345 ! ! 123 5 7 ! 1239 4 6 ! 123 8 1239 ! ! 8 123 4 ! 5 1239 123 ! 1237 1679 123679 ! +-------------------------+-------------------------+-------------------------+ ! 4 1267 8 ! 126 1257 125 ! 9 3 157 ! ! 1235 1237 1239 ! 1239 1235789 4 ! 6 157 1578 ! ! 135 1367 139 ! 1369 135789 1358 ! 147 2 14578 ! +-------------------------+-------------------------+-------------------------+ 187 candidates.

Code: Select all: hidden-pairs-in-a-row: r3{n5 n8}{c5 c6} ==> r3c6≠3, r3c6≠2, r3c6≠1, r3c5≠3, r3c5≠2, r3c5≠1 whip[5]: r3n9{c8 c9} - b6n9{r6c9 r6c8} - c8n7{r6 r8} - c9n7{r9 r6} - r6n6{c9 .} ==> r3c8≠1 +-------------------------+-------------------------+-------------------------+ ! 9 8 123 ! 7 6 123 ! 5 14 1234 ! ! 7 123 5 ! 4 123 9 ! 8 16 1236 ! ! 123 4 6 ! 123 58 58 ! 1237 79 12379 ! +-------------------------+-------------------------+-------------------------+ ! 6 9 123 ! 8 123 7 ! 1234 145 12345 ! ! 123 5 7 ! 1239 4 6 ! 123 8 1239 ! ! 8 123 4 ! 5 1239 123 ! 1237 1679 123679 ! +-------------------------+-------------------------+-------------------------+ ! 4 1267 8 ! 126 1257 125 ! 9 3 157 ! ! 1235 1237 1239 ! 1239 1235789 4 ! 6 157 1578 ! ! 135 1367 139 ! 1369 135789 1358 ! 147 2 14578 ! +-------------------------+-------------------------+-------------------------+ tridagon type diag for digits 1, 2 and 3 in blocks: b5, with cells: r5c4 (target cell), r4c5, r6c6 b4, with cells: r5c1, r4c3, r6c2 b2, with cells: r3c4, r2c5, r1c6 b1, with cells: r3c1, r2c2, r1c3 ==> r5c4≠1,2,3 naked-single ==> r5c4=9 hidden-pairs-in-a-block: b6{n6 n9}{r6c8 r6c9} ==> r6c9≠7, r6c9≠3, r6c9≠2, r6c9≠1, r6c8≠7, r6c8≠1 hidden-single-in-a-block ==> r6c7=7 hidden-pairs-in-a-block: b3{n7 n9}{r3c8 r3c9} ==> r3c9≠3, r3c9≠2, r3c9≠1 naked-triplets-in-a-column: c5{r2 r4 r6}{n1 n2 n3} ==> r9c5≠3, r9c5≠1, r8c5≠3, r8c5≠2, r8c5≠1, r7c5≠2, r7c5≠1 Final resolution state: +-------------------+-------------------+-------------------+ ! 9 8 123 ! 7 6 123 ! 5 14 1234 ! ! 7 123 5 ! 4 123 9 ! 8 16 1236 ! ! 123 4 6 ! 123 58 58 ! 123 79 79 ! +-------------------+-------------------+-------------------+ ! 6 9 123 ! 8 123 7 ! 1234 145 12345 ! ! 123 5 7 ! 9 4 6 ! 123 8 123 ! ! 8 123 4 ! 5 123 123 ! 7 69 69 ! +-------------------+-------------------+-------------------+ ! 4 1267 8 ! 126 57 125 ! 9 3 157 ! ! 1235 1237 1239 ! 123 5789 4 ! 6 157 1578 ! ! 135 1367 139 ! 136 5789 1358 ! 14 2 14578 ! +-------------------+-------------------+-------------------+

After applying the Tridagon elimination rule, it remains in T&E(BRT, 2) - more precisely in T&E(W2, 1).

by **marek stefanik** » Wed Apr 06, 2022 7:22 pm

denis_berthier wrote:After applying the Tridagon elimination rule, it remains in T&E(W2, 2)

If that is true, it might be the first sub-10-SER not in T&E(2) (it is 9.7 skfr).

My first instinct was to relabel in b5, after which only finned X-wings and short XY-chains are needed, but seeing that relabeling in b4 simplifies the puzzle even further, I came up with the following:

Code: Select all: .------------------.-----------------.------------------. | 9 8 123 | 7 6 123 | 5 14 1234 | | 7 123 5 | 4 123 9 | 8 z16 z1236 | | 123 4 6 | 123 58 58 | 123 79 79 | :------------------+-----------------+------------------: | 6 9 x123 | 8 z123 7 | 1234 145 12345 | |y123 5 7 | 9 4 6 | 123 8 123 | | 8 z123 4 | 5 123 123 | 7 69 69 | :------------------+-----------------+------------------: | 4 1267 8 | 126 57 125 | 9 3 157 | | 1235 1237 1239 | 123 5789 4 | 6 157 1578 | | 135 1367 139 | 136 5789 1358 | 14 2 14578 | '------------------'-----------------'------------------'

Let xyz be the digit in b4p348, in that order. HS zb5, HP z6r2c89

Code: Select all: .------------------.-----------------.------------------. | 9 8 ayz | 7 6 123 | 5 14-y 1234-y| | 7 123 5 | 4 123 9 | 8 z6 z6 | |bxz 4 6 | 123 58 58 |cxy 79 79 | :------------------+-----------------+------------------: | 6 9 x | 8 z 7 | 1234 145 12345 | | y 5 7 | 9 4 6 | 123 8 123 | | 8 z 4 | 5 123 123 | 7 69 69 | :------------------+-----------------+------------------: | 4 1267 8 | 126 57 125 | 9 3 157 | | 1235 1237 1239 | 123 5789 4 | 6 157 1578 | | 135 1367 139 | 136 5789 1358 | 14 2 14578 | '------------------'-----------------'------------------'

(y=z)r1c3 – (z=x)r3c1 – (x=y)r3c7 => –yr1c89, HS yb3

Code: Select all: .------------------.-----------------.------------------. | 9 8 yz | 7 6 123 | 5 14 1234 | | 7 123 5 | 4 123 9 | 8 z6 z6 | | xz 4 6 | 123 58 58 | y 79 79 | :------------------+-----------------+------------------: | 6 9 x | 8 z123 7 | 1234 145 12345 | | y 5 7 | 9 4 6 | 123 8 xz1–23 | | 8 z 4 | 5 123 123 | 7 69 69 | :------------------+-----------------+------------------: | 4 1267 8 | 126 57 125 | 9 3 157 | | 1235 1237 1239 | 123 5789 4 | 6 157 1578 | | 135 1367 139 | 136 5789 1358 |xz1–4 2 14578 | '------------------'-----------------'------------------'

whichever digit appears in r5c9 is forced into r9c7 in c7 => –4r9c7, –23r5c9; then 4.2 skfr

Note that the extra cell (here r3c7) is the same as in these patterns found by eleven (in fact, they can be used to eliminate 123r4c7 directly, reducing the puzzle to 8.4 skfr).

I think the other puzzles should also be studied, I am currently trying to make sense of this Xsudo's deduction in the second puzzle on the 11.7 list:

Code: Select all: 987......6..95.....4.......3..5.261.2..31........96.32...26.5.9...1....3....3.12. .----------------------.-----------------.-------------------. | 9 8 7 |#46 #24 134 | 234 #456 1456 | | 6 123 123 | 9 5 13478 | 23478 478 1478 | | 15 4 1235 |#678 #278 1378 | 23789 #6789 1678 | :----------------------+-----------------+-------------------: | 3 79 489 | 5 #478 2 | 6 1 #478 | | 2 67 468 | 3 1 478 | 4789 #59 4578 | | 14578 157 1458 |#478 9 6 |#478 3 2 | :----------------------+-----------------+-------------------: | 1478 137 1348 | 2 6 #478 | 5 #478 9 | | 4578 25679 245689 | 1 #478 59 |#478 #4678 3 | | 4578 5679 45689 |#478 3 59 | 1 2 6–478| '----------------------'-----------------'-------------------'

Added: Suppose 478r9c9. In c8, one of 478 must appear in r13c8. Let's call that digit x.

Code: Select all: .----------------------.-----------------.-------------------. | 9 8 7 | 46 24 134 | 234 x45 1456 | | 6 123 123 | 9 5 13478 | 23478 478 1478 | | 15 4 1235 | 678 278 1378 | 23789 x789 1678 | :----------------------+-----------------+-------------------: | 3 79 489 | 5 a478 2 | 6 1 478 | | 2 67 468 | 3 1 478 | 4789 59 4578 | | 14578 157 1458 |b478 9 6 | 478 3 2 | :----------------------+-----------------+-------------------: | 1478 137 1348 | 2 6 478 | 5 478 9 | | 4578 25679 245689 | 1 478 59 |a478 6 3 | | 4578 5679 45689 | 478 3 59 | 1 2 b478 | '----------------------'-----------------'-------------------'

x must take one of the marked cells in b9 and then one of the marked cells either (b) in c4 or (a) in c5.
If the other two cells contain the same digit, b6 is broken, if they contain different digits, (after filling in r5c6 and r7c8) b8 is broken, ie. contra.
Therefore -478r9c9, reducing the puzzle to 4.2 skfr.

Marek

Website · by **denis_berthier** » Thu Apr 07, 2022 2:39 am

marek stefanik wrote:
denis_berthier wrote:After applying the Tridagon elimination rule, it remains in T&E(W2, 2)
If that is true, it might be the first sub-10-SER not in T&E(2) (it is 9.7 skfr).

I've corrected this typo. I meant "T&E(BRT, 2), more precisely T&E(W2, 1)", also named B2B.

marek stefanik wrote:My first instinct was to relabel in b5,

Eleven's digit relabelling is a smart solving method, but it doesn't allow to conclude anything about classifications. See my new post here: http://forum.enjoysudoku.com/eleven-s-variable-replacement-method-and-its-complexity-t39277-6.html

Website · by **denis_berthier** » Wed Apr 20, 2022 12:14 pm

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In a previous post (http://forum.enjoysudoku.com/the-tridagon-rule-t39859-39.html), I analysed mitt's full list of 972 non-T&E(2) puzzles https://docs.google.com/spreadsheets/d/1t-PsJT-pKGQEWjSbbNBXzLcxb5Inmooszntu9ZVCW_M/edit#gid=0 introduced by mith here: http://forum.enjoysudoku.com/the-hardest-sudokus-new-thread-t6539-1231.html.

I showed that:
- using only (Naked + Hidden + Super-HIdden) Subsets + Finned Fish + the Tridagon elimination rule defined in the 1st post of this thread, 216 puzzles can be solved;
- adding whips of length ≤ 12, 505 more puzzles can be solved;
- adding tridagon-links and Tridagon-Forcing-Whips of length ≤ 15, 41 more puzzles can be solved, pushing the total to 762.

That left 210 unsolved by the above-mentioned resolution rules.

Since then, I've added eleven's replacement technique to SudoRules arsenal in cases there is some trivalue oddagon pattern with additional clues in at most 3 blocks. (I'll say more on this soon in the CSP-Rules thread (http://forum.enjoysudoku.com/csp-rules-sudorules-kakurules-t38200.html).
Eleven's replacement technique, based on the relevant blocks of the tridagon pattern allows to solve all the 210 remaining puzzles.

Take a random example in the 210 remaining list, #269:

Code: Select all: +-------+-------+-------+ ! . 2 3 ! . . . ! . 8 9 ! ! 4 . 6 ! . . . ! . . . ! ! 7 8 . ! . . . ! . . . ! +-------+-------+-------+ ! . . . ! . . 5 ! . 9 . ! ! . . . ! 9 2 . ! . 5 1 ! ! . . . ! 3 . 1 ! 2 . 8 ! +-------+-------+-------+ ! 3 . . ! 5 . . ! . 1 . ! ! . . . ! 8 9 . ! . 2 3 ! ! . . . ! . 1 3 ! . . 5 ! +-------+-------+-------+ .23....894.6......78............5.9....92..51...3.12.83..5...1....89..23....13..5;11,7;10,6;2,6;28;269;;;;;;;;;;;;;;

The resolution path starts as:

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 15 2 3 ! 1467 4567 467 ! 14567 8 9 ! ! 4 159 6 ! 127 3578 2789 ! 157 37 27 ! ! 7 8 159 ! 1246 3456 2469 ! 1456 346 246 ! +-------------------+-------------------+-------------------+ ! 1268 13467 12478 ! 467 4678 5 ! 3467 9 467 ! ! 68 3467 478 ! 9 2 4678 ! 3467 5 1 ! ! 569 45679 4579 ! 3 467 1 ! 2 467 8 ! +-------------------+-------------------+-------------------+ ! 3 4679 24789 ! 5 467 2467 ! 46789 1 467 ! ! 156 14567 1457 ! 8 9 467 ! 467 2 3 ! ! 2689 4679 24789 ! 2467 1 3 ! 46789 467 5 ! +-------------------+-------------------+-------------------+ 194 candidates. hidden-pairs-in-a-column: c7{n8 n9}{r7 r9} ==> r9c7≠7, r9c7≠6, r9c7≠4, r7c7≠7, r7c7≠6, r7c7≠4 whip[4]: r7n2{c6 c3} - r7n8{c3 c7} - r7n9{c7 c2} - r2n9{c2 .} ==> r2c6≠2 whip[5]: c1n9{r6 r9} - c1n2{r9 r4} - c1n8{r4 r5} - c6n8{r5 r2} - r2n9{c6 .} ==> r6c2≠9 whip[5]: r2n9{c2 c6} - r2n8{c6 c5} - c5n3{r2 r3} - c5n5{r3 r1} - r1c1{n5 .} ==> r2c2≠1 whip[8]: r5c1{n6 n8} - c6n8{r5 r2} - r2n9{c6 c2} - r3n9{c3 c6} - c6n2{r3 r7} - r9n2{c4 c3} - r9n9{c3 c7} - r9n8{c7 .} ==> r9c1≠6

Then eleven's technique is chosen:

Code: Select all: ***** STARTING ELEVEN''S REPLACEMENT TECHNIQUE in resolution state: ***** +-------------------+-------------------+-------------------+ ! 15 2 3 ! 1467 4567 467 ! 14567 8 9 ! ! 4 59 6 ! 127 3578 789 ! 157 37 27 ! ! 7 8 159 ! 1246 3456 2469 ! 1456 346 246 ! +-------------------+-------------------+-------------------+ ! 1268 13467 12478 ! 467 4678 5 ! 3467 9 467 ! ! 68 3467 478 ! 9 2 4678 ! 3467 5 1 ! ! 569 4567 4579 ! 3 467 1 ! 2 467 8 ! +-------------------+-------------------+-------------------+ ! 3 4679 24789 ! 5 467 2467 ! 89 1 467 ! ! 156 14567 1457 ! 8 9 467 ! 467 2 3 ! ! 289 4679 24789 ! 2467 1 3 ! 89 467 5 ! +-------------------+-------------------+-------------------+ AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to digits 4, 6 and 7 in cells r9c8, r8c7 and r7c9, the resolution state is: +----------------------+----------------------+----------------------+ ! 15 2 3 ! 1467 4675 467 ! 14675 8 9 ! ! 467 59 467 ! 12467 354678 46789 ! 15467 3467 2467 ! ! 467 8 159 ! 12467 34675 24679 ! 14675 3467 2467 ! +----------------------+----------------------+----------------------+ ! 124678 13467 124678 ! 467 4678 5 ! 3467 9 467 ! ! 4678 3467 4678 ! 9 2 4678 ! 3467 5 1 ! ! 54679 4675 46759 ! 3 467 1 ! 2 467 8 ! +----------------------+----------------------+----------------------+ ! 3 4679 246789 ! 5 467 2467 ! 89 1 7 ! ! 15467 14675 14675 ! 8 9 467 ! 6 2 3 ! ! 289 4679 246789 ! 2467 1 3 ! 89 4 5 ! +----------------------+----------------------+----------------------+ THIS IS THE PUZZLE THAT WILL NOW BE SOLVED. DON''T FORGET TO DO THE RELEVANT DIGIT REPLACEMENTS AT THE END, based on the original givens.

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 15 2 3 ! 1467 4567 467 ! 1457 8 9 ! ! 467 59 467 ! 1247 34578 4789 ! 1457 367 246 ! ! 467 8 159 ! 1247 3457 2479 ! 1457 367 246 ! +----------------------+----------------------+----------------------+ ! 124678 13467 124678 ! 467 4678 5 ! 347 9 46 ! ! 4678 3467 4678 ! 9 2 4678 ! 347 5 1 ! ! 45679 4567 45679 ! 3 467 1 ! 2 67 8 ! +----------------------+----------------------+----------------------+ ! 3 469 24689 ! 5 46 246 ! 89 1 7 ! ! 1457 1457 1457 ! 8 9 47 ! 6 2 3 ! ! 289 679 26789 ! 267 1 3 ! 89 4 5 ! +----------------------+----------------------+----------------------+ 188 candidates. z-chain[3]: b1n4{r2c3 r3c1} - c9n4{r3 r4} - c4n4{r4 .} ==> r2c5≠4, r2c6≠4 z-chain[3]: r1n4{c6 c7} - c9n4{r2 r4} - c4n4{r4 .} ==> r3c5≠4, r3c6≠4 z-chain[4]: r4c9{n4 n6} - r4c4{n6 n7} - r6c5{n7 n6} - r7c5{n6 .} ==> r4c5≠4 z-chain[4]: r4c9{n6 n4} - r4c4{n4 n7} - r6c5{n7 n4} - r7c5{n4 .} ==> r4c5≠6 z-chain[5]: r7c5{n4 n6} - r6c5{n6 n7} - r6c8{n7 n6} - r4c9{n6 n4} - c4n4{r4 .} ==> r1c5≠4 whip[5]: b8n7{r8c6 r9c4} - b8n2{r9c4 r7c6} - c6n6{r7 r5} - r4c4{n6 n4} - b2n4{r1c4 .} ==> r1c6≠7 whip[4]: r5n6{c3 c6} - r1c6{n6 n4} - c4n4{r1 r4} - r4c9{n4 .} ==> r4c3≠6 whip[4]: r5n6{c3 c6} - r1c6{n6 n4} - c4n4{r1 r4} - r4c9{n4 .} ==> r4c2≠6 whip[4]: r5n6{c3 c6} - r1c6{n6 n4} - c4n4{r1 r4} - r4c9{n4 .} ==> r4c1≠6 whip[5]: r1c6{n6 n4} - c4n4{r1 r4} - r6c5{n4 n7} - r6c8{n7 n6} - r4c9{n6 .} ==> r1c5≠6 hidden-pairs-in-a-column: c5{n4 n6}{r6 r7} ==> r6c5≠7 whip[3]: r1c6{n6 n4} - b8n4{r7c6 r7c5} - r6c5{n4 .} ==> r5c6≠6 whip[1]: r5n6{c3 .} ==> r6c1≠6, r6c2≠6, r6c3≠6 z-chain[3]: r6n4{c3 c5} - b5n6{r6c5 r4c4} - r4c9{n6 .} ==> r4c3≠4, r4c2≠4, r4c1≠4 z-chain[5]: b5n7{r4c5 r5c6} - r8c6{n7 n4} - c5n4{r7 r6} - r6n6{c5 c8} - r6n7{c8 .} ==> r4c3≠7, r4c2≠7, r4c1≠7 z-chain[5]: b1n7{r2c3 r3c1} - c8n7{r3 r6} - r6n6{c8 c5} - c5n4{r6 r7} - r8c6{n4 .} ==> r2c6≠7 z-chain[5]: r1n7{c5 c7} - c8n7{r2 r6} - r6n6{c8 c5} - c5n4{r6 r7} - r8c6{n4 .} ==> r3c6≠7 whip[4]: c6n7{r8 r5} - r4n7{c4 c7} - r4n3{c7 c2} - c2n1{r4 .} ==> r8c2≠7 biv-chain[5]: r9c7{n9 n8} - r7n8{c7 c3} - r7n2{c3 c6} - r3c6{n2 n9} - b1n9{r3c3 r2c2} ==> r9c2≠9 biv-chain[5]: b5n8{r4c5 r5c6} - r2c6{n8 n9} - r3c6{n9 n2} - r7n2{c6 c3} - b4n2{r4c3 r4c1} ==> r4c1≠8 z-chain[5]: r7c5{n6 n4} - r7c2{n4 n9} - r2n9{c2 c6} - r3c6{n9 n2} - r7n2{c6 .} ==> r7c3≠6 z-chain[5]: r7c5{n4 n6} - r7c2{n6 n9} - r2n9{c2 c6} - r3c6{n9 n2} - r7n2{c6 .} ==> r7c3≠4 whip[6]: c2n1{r8 r4} - r4c1{n1 n2} - r4c3{n2 n8} - c5n8{r4 r2} - r2c6{n8 n9} - r2c2{n9 .} ==> r8c2≠5 biv-chain[3]: c2n5{r6 r2} - b1n9{r2c2 r3c3} - b4n9{r6c3 r6c1} ==> r6c1≠5 z-chain[4]: c6n7{r8 r5} - c2n7{r5 r6} - c2n5{r6 r2} - c1n5{r1 .} ==> r8c1≠7 z-chain[5]: b8n7{r8c6 r9c4} - c2n7{r9 r6} - c2n5{r6 r2} - r2n9{c2 c6} - c6n8{r2 .} ==> r5c6≠7 hidden-single-in-a-column ==> r8c6=7 whip[1]: r8n4{c3 .} ==> r7c2≠4 whip[1]: b5n7{r4c5 .} ==> r4c7≠7 biv-chain[3]: r2n2{c9 c4} - r9c4{n2 n6} - r4n6{c4 c9} ==> r2c9≠6 biv-chain[4]: r6c8{n7 n6} - c5n6{r6 r7} - r7c2{n6 n9} - c1n9{r9 r6} ==> r6c1≠7 z-chain[4]: r6n4{c3 c5} - r6n6{c5 c8} - b6n7{r6c8 r5c7} - r5n3{c7 .} ==> r5c2≠4 biv-chain[3]: c2n4{r8 r6} - c2n5{r6 r2} - c1n5{r1 r8} ==> r8c1≠4 naked-pairs-in-a-column: c1{r1 r8}{n1 n5} ==> r4c1≠1 naked-single ==> r4c1=2 naked-pairs-in-a-row: r9{c1 c7}{n8 n9} ==> r9c3≠9, r9c3≠8 biv-chain[4]: r1c5{n5 n7} - r4c5{n7 n8} - r4c3{n8 n1} - b1n1{r3c3 r1c1} ==> r1c1≠5 singles ==> r1c1=1, r8c1=5 biv-chain[4]: b1n5{r3c3 r2c2} - r2n9{c2 c6} - r2n8{c6 c5} - b2n3{r2c5 r3c5} ==> r3c5≠5 z-chain[3]: r2n3{c8 c5} - r3c5{n3 n7} - b1n7{r3c1 .} ==> r2c8≠7 biv-chain[5]: c2n5{r2 r6} - c2n4{r6 r8} - c2n1{r8 r4} - r4c3{n1 n8} - c5n8{r4 r2} ==> r2c5≠5 hidden-single-in-a-block ==> r1c5=5 hidden-pairs-in-a-block: b3{n1 n5}{r2c7 r3c7} ==> r3c7≠7, r3c7≠4, r2c7≠7, r2c7≠4 biv-chain[4]: r3c6{n2 n9} - r3c3{n9 n5} - c7n5{r3 r2} - r2n1{c7 c4} ==> r2c4≠2 hidden-single-in-a-row ==> r2c9=2 naked-quads-in-a-row: r3{c1 c5 c8 c9}{n4 n7 n3 n6} ==> r3c4≠7, r3c4≠4 z-chain[4]: r3n4{c1 c9} - c9n6{r3 r4} - b5n6{r4c4 r6c5} - r6n4{c5 .} ==> r5c1≠4 biv-chain[5]: r6c1{n9 n4} - c2n4{r6 r8} - c2n1{r8 r4} - r4c3{n1 n8} - b7n8{r7c3 r9c1} ==> r9c1≠9 singles ==> r9c1=8, r9c7=9, r7c7=8, r6c1=9 whip[1]: c1n4{r3 .} ==> r2c3≠4 biv-chain[4]: r5c1{n6 n7} - c7n7{r5 r1} - b3n4{r1c7 r3c9} - c1n4{r3 r2} ==> r2c1≠6 biv-chain[4]: r2n6{c3 c8} - r2n3{c8 c5} - r2n8{c5 c6} - r5n8{c6 c3} ==> r5c3≠6 biv-chain[5]: r6c8{n7 n6} - c5n6{r6 r7} - r7c2{n6 n9} - c3n9{r7 r3} - c3n5{r3 r6} ==> r6c3≠7 biv-chain[5]: r2n6{c3 c8} - r2n3{c8 c5} - b2n8{r2c5 r2c6} - r2n9{c6 c2} - r7c2{n9 n6} ==> r9c3≠6 singles ==> r2c3=6, r2c8=3, r3c5=3, r5c1=6 biv-chain[5]: r5n8{c3 c6} - r2c6{n8 n9} - c2n9{r2 r7} - c2n6{r7 r9} - r9n7{c2 c3} ==> r5c3≠7 stte 123756489 756489132 489132576 218675394 634928751 975341268 392564817 541897623 867213945

(In the present case, no final digit permutation is needed.)

The last part could probably be shortened, but this is not my point.

Website · by **denis_berthier** » Fri Apr 22, 2022 11:43 am

.
Consider the classical "trivalue oddagon" impossible pattern in four blocks forming a rectangle:

Code: Select all: +-------------------------------+-------------------------------+ ! 123 . . ! 123 . . ! ! . 123 . ! . 123 . ! ! . . 123 ! . . 123 ! +-------------------------------+-------------------------------+ ! 123 . . ! . . 123 ! ! . 123 . ! . 123 . ! ! . . 123 ! 123 . . ! +-------------------------------+-------------------------------+

where "." can be anything.

Without making any other assumption, the most general possible resolution state is:

Code: Select all: +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+

Theorem 1: the trivalue oddagon pattern can be proven contradictory in T&E(Singles, 4).
Proof: whichever way one choses values for r1c1, r2c2, r1c4 and r4c1, one of the 123-cells in b4 has no possible value.
[Edit:] corrected the typo.

Theorem 2: after using eleven's replacement technique, the trivalue oddagon pattern can be proven contradictory in T&E(Singles, 1)

More precisely and somehow surprisingly:

Theorem 3: after using eleven's replacement technique, the trivalue oddagon pattern can be proven contradictory in Z5
Proof:
Applying eleven's replacement to block b1 gives the following RS:

Code: Select all: +-------------------------------+-------------------------------+-------------------------------+ ! 1 123456789 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 2 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 3 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123456789 123 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+

Select resolution theory Z5 and apply function solve-sukaku-grid to it (using SudoRules as an assistant theorem prover):

Code: Select all: biv-chain[3]: r1c4{n2 n3} - r2c5{n3 n1} - r3c6{n1 n2} ==> r1c5≠2, r1c6≠2, r3c4≠2, r3c5≠2 biv-chain[3]: r1c4{n3 n2} - r3c6{n2 n1} - r2c5{n1 n3} ==> r1c5≠3, r1c6≠3, r2c4≠3, r2c6≠3 biv-chain[3]: r2c5{n1 n3} - r1c4{n3 n2} - r3c6{n2 n1} ==> r2c4≠1, r2c6≠1, r3c4≠1, r3c5≠1 biv-chain[3]: r4c1{n2 n3} - r5c2{n3 n1} - r6c3{n1 n2} ==> r4c3≠2, r5c1≠2, r5c3≠2, r6c1≠2 biv-chain[3]: r4c1{n3 n2} - r6c3{n2 n1} - r5c2{n1 n3} ==> r4c2≠3, r5c1≠3, r6c1≠3, r6c2≠3 biv-chain[3]: r5c2{n1 n3} - r4c1{n3 n2} - r6c3{n2 n1} ==> r4c2≠1, r4c3≠1, r5c3≠1, r6c2≠1 z-chain[4]: b4n1{r6c3 r5c2} - b4n3{r5c2 r4c1} - r4c6{n3 n2} - r3c6{n2 .} ==> r6c6≠1 z-chain[4]: b4n3{r4c1 r5c2} - b4n1{r5c2 r6c3} - r6c4{n1 n2} - r1c4{n2 .} ==> r4c4≠3 z-chain[4]: r4c1{n2 n3} - r4c6{n3 n1} - r6c4{n1 n3} - r1c4{n3 .} ==> r4c4≠2 z-chain[4]: r6c3{n2 n1} - r6c4{n1 n3} - r4c6{n3 n1} - r3c6{n1 .} ==> r6c6≠2 z-chain[5]: b4n1{r6c3 r5c2} - b4n3{r5c2 r4c1} - r4c6{n3 n2} - r5c5{n2 n3} - r2c5{n3 .} ==> r6c5≠1 z-chain[5]: b4n3{r4c1 r5c2} - b4n1{r5c2 r6c3} - r6c4{n1 n2} - r5c5{n2 n1} - r2c5{n1 .} ==> r4c5≠3 z-chain[5]: b4n2{r6c3 r4c1} - b4n3{r4c1 r5c2} - r5c5{n3 n1} - b2n1{r2c5 r3c6} - b2n2{r3c6 .} ==> r6c4≠2 biv-chain[3]: r6c4{n1 n3} - r1c4{n3 n2} - r3c6{n2 n1} ==> r4c6≠1, r5c6≠1 biv-chain[2]: r4c6{n3 n2} - r4c1{n2 n3} ==> r4c7≠3, r4c8≠3, r4c9≠3 biv-chain[2]: r4c1{n2 n3} - r4c6{n3 n2} ==> r4c7≠2, r4c8≠2, r4c9≠2, r4c5≠2 biv-chain[3]: b4n2{r6c3 r4c1} - r4c6{n2 n3} - r6c4{n3 n1} ==> r6c3≠1 singles ==> r6c3=2, r4c1=3, r4c6=2, r3c6=1, r2c5=3, r1c4=2, r5c5=1 GRID 0 HAS NO SOLUTION : NO CANDIDATE FOR FOR BN-CELL b4n1

Note: a simpler resolution path can be obtained if Subsets are allowed, but this is not my point here, and anyway, it still requires Z5.

I think this is the most powerful example of eleven's replacement technique as of now: from T&E(4) to Z5.

Website · by **denis_berthier** » Sat Apr 23, 2022 7:23 am

.
Theorem 4 in my previous post has a theoretical interest, but does it have any practical one?

A priori, it doesn't imply anything on non contradictory puzzles, as they can't have the trivalue-oddagon impossible pattern. Any real puzzle will have additional candidates in at least one cell of the pattern.
One may however suppose that it generally implies some restrictions to the possible complexity of a puzzle that has the trivalue-oddagon impossible pattern plus a few candidates.
The only way to test this is to use existing puzzles and as of today the largest such collection is the mith's 972 one already studied in my previous posts.

So, the question in this post will be: in this 972 database, if not using any tridagon related resolution rule, but using eleven's replacement technique, can all the puzzles be solved in Z5 and otherwise how far must one go?

78 (i.e. 8%) are not solved in Z5: 8 12 20 76 77 80 89 90 128 129 155 156 164 165 168 169 173 226 232 234 235 236 237 265 266 267 269 271 396 442 459 517 518 522 524 563 564 565 567 570 595 596 597 677 776 827 828 836 837 846 884 915 932 933 937 942 944 945 947 948 952 953 956 957 958 959 960 962 963 964 965 966 967 968 969 970 971 972

Of those 78:
- 67 are solved in W6: 20 76 77 80 89 90 129 155 164 165 168 169 173 226 232 234 235 236 237 266 267 269 271 442 518 522 524 563 564 565 567 570 595 596 597 677 776 837 846 884 915 932 933 937 942 944 945 947 948 952 953 956 957 958 959 960 962 963 964 965 966 967 968 969 970 971 972

- 2 are solved in W7: 156 265

- 8 are solved in W8: 8 12 128 459 517 827 828 836

- the last one is solved in W9: 396

Notice that in these quick calculations, eleven's replacement is (automatically) tried only in the blocks of the possible trivalue-oddagons that have the 3 candidates and only them in each of their 3 cells.

by **mith** » Sun Apr 24, 2022 11:17 pm

I'll be curious to see if any of the new ones defeat the replacement technique. I have at least one example of a puzzle with guardians in all four boxes after singles (or basics even), but after some short chains it's down to just two.

Code: Select all: ........1.....234..35.1.......4..65....6.12.36....5.1...7.4.....89.5....21.....3. ED=10.4/7.2/2.6

I'll have to code up a systematic search for these, it would be somewhat surprising if this is the only one found so far.

Website · by **denis_berthier** » Sun Apr 24, 2022 11:55 pm

mith wrote:I'll be curious to see if any of the new ones defeat the replacement technique. I have at least one example of a puzzle with guardians in all four boxes after singles (or basics even), but after some short chains it's down to just two.
Code: Select all
........1.....234..35.1.......4..65....6.12.36....5.1...7.4.....89.5....21.....3. ED=10.4/7.2/2.6
.

The replacement technique works - somehow. With it (and no tridagon rule), the puzzle is solved in W6.

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 4789 24679 2468 ! 35789 36789 346789 ! 5789 6789 1 ! ! 1789 679 168 ! 5789 6789 2 ! 3 4 56789 ! ! 4789 3 5 ! 789 1 46789 ! 789 26789 26789 ! +----------------------+----------------------+----------------------+ ! 1789 279 1238 ! 4 23789 3789 ! 6 5 789 ! ! 45789 4579 48 ! 6 789 1 ! 2 789 3 ! ! 6 279 238 ! 3789 23789 5 ! 4789 1 4789 ! +----------------------+----------------------+----------------------+ ! 35 56 7 ! 12389 4 3689 ! 189 2689 2689 ! ! 34 8 9 ! 1237 5 367 ! 147 267 2467 ! ! 2 1 46 ! 789 6789 6789 ! 45789 3 456789 ! +----------------------+----------------------+----------------------+ 215 candidates

hidden-pairs-in-a-column: c4{n1 n2}{r7 r8} ==> r8c4≠7, r8c4≠3, r7c4≠9, r7c4≠8, r7c4≠3
whip[1]: b8n3{r8c6 .} ==> r1c6≠3, r4c6≠3
biv-chain[3]: r5c3{n8 n4} - c2n4{r5 r1} - b1n2{r1c2 r1c3} ==> r1c3≠8
biv-chain[4]: r5c3{n8 n4} - b7n4{r9c3 r8c1} - b7n3{r8c1 r7c1} - c1n5{r7 r5} ==> r5c1≠8
biv-chain[4]: r8c1{n4 n3} - r7c1{n3 n5} - b4n5{r5c1 r5c2} - c2n4{r5 r1} ==> r1c1≠4, r3c1≠4
hidden-single-in-a-row ==> r3c6=4
whip[1]: r3n6{c9 .} ==> r1c8≠6, r2c9≠6
hidden-pairs-in-a-block: b3{n2 n6}{r3c8 r3c9} ==> r3c9≠9, r3c9≠8, r3c9≠7, r3c8≠9, r3c8≠8, r3c8≠7
hidden-pairs-in-a-row: r1{n2 n4}{c2 c3} ==> r1c3≠6, r1c2≠9, r1c2≠7, r1c2≠6
whip[1]: r1n6{c6 .} ==> r2c5≠6
hidden-triplets-in-a-column: c1{n3 n4 n5}{r7 r8 r5} ==> r5c1≠9, r5c1≠7
biv-chain[4]: r4n3{c5 c3} - c3n1{r4 r2} - c3n6{r2 r9} - c5n6{r9 r1} ==> r1c5≠3
singles ==> r1c4=3, r2c4=5, r1c7=5, r9c9=5
hidden-pairs-in-a-block: b5{n2 n3}{r4c5 r6c5} ==> r6c5≠9, r6c5≠8, r6c5≠7, r4c5≠9, r4c5≠8, r4c5≠7
z-chain[3]: r9n6{c6 c3} - b7n4{r9c3 r8c1} - r8n3{c1 .} ==> r8c6≠6
whip[1]: r8n6{c9 .} ==> r7c8≠6, r7c9≠6
hidden-triplets-in-a-row: r7{n3 n5 n6}{c6 c1 c2} ==> r7c6≠9, r7c6≠8
whip[1]: r7n8{c9 .} ==> r9c7≠8
whip[1]: r7n9{c9 .} ==> r9c7≠9

Code: Select all: Resolution state: +----------------+----------------+----------------+ ! 789 24 24 ! 3 6789 6789 ! 5 789 1 ! ! 1789 679 168 ! 5 789 2 ! 3 4 789 ! ! 789 3 5 ! 789 1 4 ! 789 26 26 ! +----------------+----------------+----------------+ ! 1789 279 1238 ! 4 23 789 ! 6 5 789 ! ! 45 4579 48 ! 6 789 1 ! 2 789 3 ! ! 6 279 238 ! 789 23 5 ! 4789 1 4789 ! +----------------+----------------+----------------+ ! 35 56 7 ! 12 4 36 ! 189 289 289 ! ! 34 8 9 ! 12 5 37 ! 147 267 2467 ! ! 2 1 46 ! 789 6789 6789 ! 47 3 5 ! +----------------+----------------+----------------+

AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to digits 7, 8 and 9 in cells r1c8, r2c9 and r3c7,
the resolution state is:

Code: Select all: +----------------------+----------------------+----------------------+ ! 789 24 24 ! 3 6789 6789 ! 5 7 1 ! ! 1789 6789 16789 ! 5 789 2 ! 3 4 8 ! ! 789 3 5 ! 789 1 4 ! 9 26 26 ! +----------------------+----------------------+----------------------+ ! 1789 2789 123789 ! 4 23 789 ! 6 5 789 ! ! 45 45789 4789 ! 6 789 1 ! 2 789 3 ! ! 6 2789 23789 ! 789 23 5 ! 4789 1 4789 ! +----------------------+----------------------+----------------------+ ! 35 56 789 ! 12 4 36 ! 1789 2789 2789 ! ! 34 789 789 ! 12 5 3789 ! 14789 26789 246789 ! ! 2 1 46 ! 789 6789 6789 ! 4789 3 5 ! +----------------------+----------------------+----------------------+

whip[1]: r9n9{c6 .} ==> r8c6≠9
whip[1]: b1n8{r3c1 .} ==> r4c1≠8
z-chain[5]: c5n6{r9 r1} - c5n8{r1 r5} - r5c8{n8 n9} - r4c9{n9 n7} - c6n7{r4 .} ==> r9c5≠7
z-chain[5]: b8n7{r9c6 r9c4} - r3c4{n7 n8} - b5n8{r6c4 r5c5} - r5c8{n8 n9} - r4c9{n9 .} ==> r4c6≠7
whip[1]: c6n7{r9 .} ==> r9c4≠7
biv-chain[3]: r9c4{n9 n8} - r3c4{n8 n7} - r2c5{n7 n9} ==> r9c5≠9
z-chain[5]: r9n7{c7 c6} - r9n9{c6 c4} - r6c4{n9 n8} - r4c6{n8 n9} - r4c9{n9 .} ==> r6c7≠7
whip[1]: c7n7{r9 .} ==> r7c9≠7, r8c9≠7
biv-chain[3]: r7c9{n9 n2} - r3c9{n2 n6} - b9n6{r8c9 r8c8} ==> r8c8≠9
z-chain[3]: r7n8{c8 c3} - r7n7{c3 c7} - c7n1{r7 .} ==> r8c7≠8
biv-chain[5]: r5n5{c2 c1} - c1n4{r5 r8} - c9n4{r8 r6} - r6c7{n4 n8} - r5c8{n8 n9} ==> r5c2≠9
biv-chain[5]: c9n7{r4 r6} - r6n4{c9 c7} - r9n4{c7 c3} - c3n6{r9 r2} - c3n1{r2 r4} ==> r4c3≠7
biv-chain[5]: r6c7{n8 n4} - r9n4{c7 c3} - c3n6{r9 r2} - c3n1{r2 r4} - b4n3{r4c3 r6c3} ==> r6c3≠8
z-chain[5]: c3n6{r2 r9} - r9n4{c3 c7} - c9n4{r8 r6} - c9n7{r6 r4} - c1n7{r4 .} ==> r2c3≠7

Code: Select all: +-------------------+-------------------+-------------------+ ! 89 24 24 ! 3 689 689 ! 5 7 1 ! ! 179 679 169 ! 5 79 2 ! 3 4 8 ! ! 78 3 5 ! 78 1 4 ! 9 26 26 ! +-------------------+-------------------+-------------------+ ! 179 2789 12389 ! 4 23 89 ! 6 5 79 ! ! 45 4578 4789 ! 6 789 1 ! 2 89 3 ! ! 6 2789 2379 ! 789 23 5 ! 48 1 479 ! +-------------------+-------------------+-------------------+ ! 35 56 789 ! 12 4 36 ! 178 289 29 ! ! 34 789 789 ! 12 5 378 ! 147 268 2469 ! ! 2 1 46 ! 89 68 6789 ! 478 3 5 ! +-------------------+-------------------+-------------------+

AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to digits 7, 8 and 9 in cells r6c4, r5c5 and r4c6,
the resolution state is:

Code: Select all: +----------------------+----------------------+----------------------+ ! 789 24 24 ! 3 6789 6789 ! 5 789 1 ! ! 1789 6789 16789 ! 5 789 2 ! 3 4 789 ! ! 789 3 5 ! 789 1 4 ! 789 26 26 ! +----------------------+----------------------+----------------------+ ! 1789 2789 123789 ! 4 23 9 ! 6 5 789 ! ! 45 45789 4789 ! 6 8 1 ! 2 789 3 ! ! 6 2789 23789 ! 7 23 5 ! 4789 1 4789 ! +----------------------+----------------------+----------------------+ ! 35 56 789 ! 12 4 36 ! 1789 2789 2789 ! ! 34 789 789 ! 12 5 3789 ! 14789 26789 246789 ! ! 2 1 46 ! 789 6789 6789 ! 4789 3 5 ! +----------------------+----------------------+----------------------+

whip[1]: c1n9{r3 .} ==> r2c3≠9, r2c2≠9
whip[1]: b8n9{r9c5 .} ==> r9c7≠9
z-chain[4]: r2c5{n7 n9} - r2c9{n9 n8} - r4c9{n8 n7} - c1n7{r4 .} ==> r2c2≠7, r2c3≠7
whip[1]: b1n7{r3c1 .} ==> r4c1≠7
whip[5]: r2n8{c3 c9} - r4c9{n8 n7} - r5c8{n7 n9} - r1c8{n9 n7} - r3n7{c7 .} ==> r3c1≠8
whip[6]: r3c1{n9 n7} - r1c1{n7 n8} - r2c2{n8 n6} - r7n6{c2 c6} - r1c6{n6 n7} - r2c5{n7 .} ==> r2c1≠9
z-chain[5]: r4c9{n8 n7} - r5c8{n7 n9} - b3n9{r1c8 r3c7} - c1n9{r3 r1} - b1n8{r1c1 .} ==> r2c9≠8
whip[1]: r2n8{c3 .} ==> r1c1≠8
naked-pairs-in-a-block: b1{r1c1 r3c1}{n7 n9} ==> r2c1≠7
biv-chain[3]: r1n8{c8 c6} - r3c4{n8 n9} - b1n9{r3c1 r1c1} ==> r1c8≠9
z-chain[3]: c8n9{r8 r5} - b6n7{r5c8 r4c9} - r2c9{n7 .} ==> r7c9≠9, r8c9≠9
z-chain[5]: r1c8{n8 n7} - b6n7{r5c8 r4c9} - r7c9{n7 n2} - r3n2{c9 c8} - c8n6{r3 .} ==> r8c8≠8
biv-chain[6]: r2n7{c5 c9} - r4c9{n7 n8} - r4c1{n8 n1} - b1n1{r2c1 r2c3} - c3n6{r2 r9} - c5n6{r9 r1} ==> r1c5≠7
whip[6]: c7n1{r8 r7} - r7c4{n1 n2} - r7c9{n2 n8} - r7c8{n8 n9} - r5c8{n9 n7} - r4c9{n7 .} ==> r8c7≠7
whip[6]: r6n4{c9 c7} - b6n8{r6c7 r4c9} - c1n8{r4 r2} - r2c2{n8 n6} - b7n6{r7c2 r9c3} - r9n4{c3 .} ==> r6c9≠9
singles ==> r2c9=9, r2c5=7
biv-chain[3]: r1n8{c6 c8} - b3n7{r1c8 r3c7} - r9n7{c7 c6} ==> r9c6≠8
x-wing-in-rows: n8{r3 r9}{c4 c7} ==> r8c7≠8, r7c7≠8, r6c7≠8
whip[1]: b6n8{r6c9 .} ==> r7c9≠8, r8c9≠8
biv-chain[3]: r7c9{n7 n2} - r3c9{n2 n6} - b9n6{r8c9 r8c8} ==> r8c8≠7
biv-chain[4]: b6n8{r6c9 r4c9} - b6n7{r4c9 r5c8} - r1c8{n7 n8} - r7n8{c8 c3} ==> r6c3≠8
biv-chain[4]: c7n8{r9 r3} - b3n7{r3c7 r1c8} - r5c8{n7 n9} - r6c7{n9 n4} ==> r9c7≠4
stte
742396581
186572349
935814726
821439657
457681293
693725418
568143972
379258164
214967835
Permute 7 and 8.

I've applied the technique twice (manual choice here because I wanted to see what happened) - but in my previous analysis of your 972 database it was sometimes automatically applied twice to some of the puzzles.

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