The New Sudoku Players' Forum

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

.
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

.
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.

by **mith** » Mon Apr 25, 2022 3:46 pm

Try this one:

Code: Select all: .......12.....34.5..514.6......6.3...2.35..6..6.4.1.5..1.23....7.8......932....4. ED=10.7/10.7/2.6

YZF gets to here before needing brute force:

Code: Select all: .------------------.-------------------.------------------. | 3468 4789 347 | 568 789 56 | 789 1 2 | | 126 789 16 | 6789 29 3 | 4 789 5 | | 28 789 5 | 1 4 2789 | 6 37 3789 | :------------------+-------------------+------------------: | 1458 4578 1479 | 789 6 29 | 3 2789 14 | | 148 2 1479 | 3 5 789 | 789 6 14 | | 38 6 379 | 4 2789 1 | 28 5 789 | :------------------+-------------------+------------------: | 456 1 46 | 2 3 45789 | 5789 789 789 | | 7 45 8 | 569 19 4569 | 1259 23 36 | | 9 3 2 | 578 178 5678 | 1578 4 67 | '------------------'-------------------'------------------'

And here's SE up to the 11.7 step (with uniqueness disabled):

Code: Select all: +----------------+----------------+----------------+ | 346 4789 34 | 568 789 56 | 789 1 2 | | 26 789 1 | 6789 29 3 | 4 789 5 | | 28 789 5 | 1 4 2789 | 6 37 3789 | +----------------+----------------+----------------+ | 1458 58 479 | 789 6 29 | 3 2789 14 | | 148 2 479 | 3 5 789 | 79 6 14 | | 38 6 379 | 4 2789 1 | 28 5 789 | +----------------+----------------+----------------+ | 45 1 6 | 2 3 47 | 5789 789 789 | | 7 45 8 | 569 19 4569 | 125 23 36 | | 9 3 2 | 578 178 5678 | 15 4 67 | +----------------+----------------+----------------+ 11.7, Contradiction Forcing Chain: r5c7.8 on ==> r4c8.9 both on & off

This isn't particularly hard with the trivalue oddagon (all guardians lead to 2r3c6, and stte from there), but it should be harder to get to a point where you can apply replacement.

(I found this one by pulling all expanded forms with 28c or fewer and EP of at least 4.5, then finding the max count of digits per box - ignoring whichever box contains the singletons. There are quite a few with a max of 4 per box, but all of these with high EP have a hidden pair/naked triple in at least one of those boxes. This 10.7 is the highest EP with max 5 per box. I haven't checked all of the 10+ EP puzzles yet, but this is the only one I've seen so far that keeps guardians in all boxes up to BF. There could also be some with lower EP which are being lowered by uniqueness, of course. I'll script up a more systematic check later.)

Website · by **denis_berthier** » Mon Apr 25, 2022 4:34 pm

mith wrote:Try this one:
Code: Select all
.......12.....34.5..514.6......6.3...2.35..6..6.4.1.5..1.23....7.8......932....4. ED=10.7/10.7/2.6

This isn't particularly hard with the trivalue oddagon (all guardians lead to 2r3c6, and stte from there), but it should be harder to get to a point where you can apply replacement.

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 3468 4789 34679 ! 56789 789 56789 ! 789 1 2 ! ! 1268 789 1679 ! 6789 2789 3 ! 4 789 5 ! ! 28 789 5 ! 1 4 2789 ! 6 3789 3789 ! +-------------------+-------------------+-------------------+ ! 1458 45789 1479 ! 789 6 2789 ! 3 2789 14789 ! ! 148 2 1479 ! 3 5 789 ! 1789 6 14789 ! ! 38 6 379 ! 4 2789 1 ! 2789 5 789 ! +-------------------+-------------------+-------------------+ ! 456 1 46 ! 2 3 45789 ! 5789 789 789 ! ! 7 45 8 ! 569 19 4569 ! 1259 239 1369 ! ! 9 3 2 ! 5678 178 5678 ! 1578 4 1678 ! +-------------------+-------------------+-------------------+ 193 candidates.

If you apply some form of T&E on the guardians, I can apply it also. Suppose r3c9≠3, then

Code: Select all: (solve-sukaku-grid-by-eleven-replacement 7 8 9 1 7 2 8 3 9 +-------------------+-------------------+-------------------+ ! 3468 4789 34679 ! 56789 789 56789 ! 789 1 2 ! ! 1268 789 1679 ! 6789 2789 3 ! 4 789 5 ! ! 28 789 5 ! 1 4 2789 ! 6 3789 3789 ! +-------------------+-------------------+-------------------+ ! 1458 45789 1479 ! 789 6 2789 ! 3 2789 14789 ! ! 148 2 1479 ! 3 5 789 ! 1789 6 14789 ! ! 38 6 379 ! 4 2789 1 ! 2789 5 789 ! +-------------------+-------------------+-------------------+ ! 456 1 46 ! 2 3 45789 ! 5789 789 789 ! ! 7 45 8 ! 569 19 4569 ! 1259 239 1369 ! ! 9 3 2 ! 5678 178 5678 ! 1578 4 1678 ! +-------------------+-------------------+-------------------+ )

AFTER APPLYING ELEVEN''S REPLACEMENT METHOD to digits 7, 8 and 9 in cells r1c7, r2c8 and r3c9,
the resolution state is:

Code: Select all: +----------------------+----------------------+----------------------+ ! 346789 4789 346789 ! 56789 789 56789 ! 7 1 2 ! ! 126789 789 16789 ! 6789 2789 3 ! 4 8 5 ! ! 2789 789 5 ! 1 4 2789 ! 6 3789 9 ! +----------------------+----------------------+----------------------+ ! 145789 45789 14789 ! 789 6 2789 ! 3 2789 14789 ! ! 14789 2 14789 ! 3 5 789 ! 1789 6 14789 ! ! 3789 6 3789 ! 4 2789 1 ! 2789 5 789 ! +----------------------+----------------------+----------------------+ ! 456 1 46 ! 2 3 45789 ! 5789 789 789 ! ! 789 45 789 ! 56789 1789 456789 ! 125789 23789 136789 ! ! 789 3 2 ! 56789 1789 56789 ! 15789 4 16789 ! +----------------------+----------------------+----------------------+

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.

singles ==> r3c8=3, r8c9=3, r9c9=6
whip[1]: c9n1{r5 .} ==> r5c7≠1

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 34689 489 34689 ! 5689 89 5689 ! 7 1 2 ! ! 12679 79 1679 ! 679 279 3 ! 4 8 5 ! ! 278 78 5 ! 1 4 278 ! 6 3 9 ! +----------------------+----------------------+----------------------+ ! 145789 45789 14789 ! 789 6 2789 ! 3 279 1478 ! ! 14789 2 14789 ! 3 5 789 ! 89 6 1478 ! ! 3789 6 3789 ! 4 2789 1 ! 289 5 78 ! +----------------------+----------------------+----------------------+ ! 456 1 46 ! 2 3 45789 ! 589 79 78 ! ! 789 45 789 ! 56789 1789 456789 ! 12589 279 3 ! ! 789 3 2 ! 5789 1789 5789 ! 1589 4 6 ! +----------------------+----------------------+----------------------+ 181 candidates.

naked-pairs-in-a-column: c9{r6 r7}{n7 n8} ==> r5c9≠8, r5c9≠7, r4c9≠8, r4c9≠7
biv-chain[3]: r7c3{n6 n4} - b8n4{r7c6 r8c6} - c6n6{r8 r1} ==> r1c3≠6
z-chain[3]: b2n2{r2c5 r3c6} - b2n7{r3c6 r2c4} - r2c2{n7 .} ==> r2c5≠9
z-chain[3]: b7n7{r9c1 r8c3} - r5n7{c3 c6} - r3n7{c6 .} ==> r2c1≠7
z-chain[3]: c2n7{r3 r4} - r5n7{c1 c6} - r3n7{c6 .} ==> r2c3≠7
z-chain[4]: r1n3{c1 c3} - r1n4{c3 c2} - r8n4{c2 c6} - c6n6{r8 .} ==> r1c1≠6
whip[1]: r1n6{c6 .} ==> r2c4≠6
naked-pairs-in-a-row: r2{c2 c4}{n7 n9} ==> r2c5≠7, r2c3≠9, r2c1≠9
singles ==> r2c5=2, r3c1=2, r4c6=2, r6c7=2, r8c8=2, r7c6≠7, r4c2≠7
hidden-pairs-in-a-block: b2{n5 n6}{r1c4 r1c6} ==> r1c6≠9, r1c6≠8, r1c4≠9, r1c4≠8
z-chain[4]: r3c6{n8 n7} - r5c6{n7 n9} - r5c7{n9 n8} - c9n8{r6 .} ==> r7c6≠8
whip[1]: r7n8{c9 .} ==> r8c7≠8, r9c7≠8
z-chain[5]: r5c7{n9 n8} - r5c6{n8 n7} - b2n7{r3c6 r2c4} - b2n9{r2c4 r1c5} - r6n9{c5 .} ==> r5c3≠9, r5c1≠9
finned-x-wing-in-rows: n9{r5 r7}{c6 c7} ==> r9c7≠9, r8c7≠9
whip[1]: b9n9{r7c8 .} ==> r7c6≠9
naked-pairs-in-a-block: b9{r8c7 r9c7}{n1 n5} ==> r7c7≠5
z-chain[3]: r5n7{c3 c6} - r5n9{c6 c7} - r4c8{n9 .} ==> r4c3≠7, r4c1≠7
z-chain[3]: r4n8{c3 c4} - r4n7{c4 c8} - r6c9{n7 .} ==> r6c3≠8, r6c1≠8
biv-chain[4]: r2c4{n7 n9} - r1c5{n9 n8} - r6n8{c5 c9} - b6n7{r6c9 r4c8} ==> r4c4≠7
singles ==> r4c8=7, r6c9=8, r5c7=9, r7c7=8, r7c9=7, r7c8=9
whip[1]: c6n9{r9 .} ==> r8c4≠9, r8c5≠9, r9c4≠9, r9c5≠9
naked-pairs-in-a-column: c6{r3 r5}{n7 n8} ==> r9c6≠8, r9c6≠7, r8c6≠8, r8c6≠7
finned-x-wing-in-rows: n8{r3 r5}{c6 c2} ==> r4c2≠8
whip[1]: c2n8{r3 .} ==> r1c1≠8, r1c3≠8
biv-chain[3]: b2n8{r3c6 r1c5} - c5n9{r1 r6} - b5n7{r6c5 r5c6} ==> r5c6≠8, r3c6≠7
singles ==> r3c6=8, r1c5=9, r2c4=7, r2c2=9, r6c5=7
GRID 0 HAS NO SOLUTION : NO CANDIDATE FOR FOR CN-CELL c6n7

Conclusion: in Z5, we have shown that r3c9≠7,8,9 and that r3c9 = 3

The end is obtained by Singles.

For me, there are so many "guardians" that this puzzle doesn't have much to do with the trivalue oddagon impossible pattern.

by **mith** » Mon Apr 25, 2022 5:14 pm

I was actually overlooking one guardian (1r5c7) because it's missing from the pencilmark grids - it can be eliminated by a rather nasty looking region or cell forcing chain.

I disagree with the general sentiment that an approach of guardians leading to other guardians is trial and error, though. The 2s here are trivially connected (any one of them implies the other two immediately) and connecting the 3 to the 2s isn't much more effort (2r4c8 => 3r8c8 => 3r3c9). The 6 is the only one that even involves another digit (2r3c6 => 2r2c1 => 1r2c3 => 6r2c4; the other direction is a bit nastier).

by **mith** » Mon Apr 25, 2022 5:19 pm

Actually, the 1 isn't so bad either:

after locked candidates:

Code: Select all: .--------------------.--------------------.-------------------. | 3468 4789 34679 | 56789 789 56789 | 789 1 2 | | 1268 789 1679 | 6789 2789 3 | 4 789 5 | | 28 789 5 | 1 4 2789 | 6 3789 3789 | :--------------------+--------------------+-------------------: | 1458 45789 1479 | 789 6 2789 | 3 2789 14789 | | 148 2 1479 | 3 5 789 | 1789 6 14789 | | 38 6 379 | 4 2789 1 | 2789 5 789 | :--------------------+--------------------+-------------------: | 456 1 46 | 2 3 45789 | 5789 789 789 | | 7 45 8 | 569 19 4569 | 1259 239 1369 | | 9 3 2 | 5678 178 5678 | 1578 4 1678 | '--------------------'--------------------'-------------------'

1r5c7 => 5789b9p1237 => 2r8c7 => 2r4c8 etc.

by **mith** » Mon Apr 25, 2022 5:58 pm

Anyway, the point of this exercise was not whether this is easier to solve with a trivalue oddagon or with replacement, but whether replacement could be applied directly. The T&E+replacement approach is pretty interesting.

Website · by **denis_berthier** » Tue Apr 26, 2022 6:52 am

.
Hi mith,
I agree it's not a competition between tridagon related rules and replacement (even if considering only solving*). I don't see any reason for applying replacement (which requires much paperwork if not mechanised) as long as tridagon related rules can be applied.
As usual, my point in not using tridagons in my last global study of your 972 database was only to compare the resolution power of various techniques.

It's not a surprise that any pattern close enough to the impossible trivalue oddagon is also a good target for replacement. The previous example shows that replacement can even be applied in a block when one cell in it has more than the 3 digits.

(*) In generation, I wonder how the hardest puzzles would resist replacement. My current implementation is too restrictive for trying it as such.
Eleven, did you try anything like that?

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