## The tridagon rule

Advanced methods and approaches for solving Sudoku puzzles

### Re: The tridagon rule

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.)
mith

Posts: 981
Joined: 14 July 2020

### Re: The tridagon rule

.
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.

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
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`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.`

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`hidden-pairs-in-a-row: r3{n5 n8}{c5 c6} ==> r3c6≠3, r3c6≠2, r3c6≠1, r3c5≠3, r3c5≠2, r3c5≠1whip[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≠1,2,3naked-single ==> r5c4=9hidden-pairs-in-a-block: b6{n6 n9}{r6c8 r6c9} ==> r6c9≠7, r6c9≠3, r6c9≠2, r6c9≠1, r6c8≠7, r6c8≠1hidden-single-in-a-block ==> r6c7=7hidden-pairs-in-a-block: b3{n7 n9}{r3c8 r3c9} ==> r3c9≠3, r3c9≠2, r3c9≠1naked-triplets-in-a-column: c5{r2 r4 r6}{n1 n2 n3} ==> r9c5≠3, r9c5≠1, r8c5≠3, r8c5≠2, r8c5≠1, r7c5≠2, r7c5≠1Final 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).
Last edited by denis_berthier on Thu Apr 07, 2022 2:36 am, edited 2 times in total.
denis_berthier
2010 Supporter

Posts: 4018
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Location: Paris

### Re: The tridagon rule

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:
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`.------------------.-----------------.------------------.| 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

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`.------------------.-----------------.------------------.| 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

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`.------------------.-----------------.------------------.| 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:
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`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.
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`.----------------------.-----------------.-------------------.| 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
marek stefanik

Posts: 358
Joined: 05 May 2021

### Re: The tridagon rule

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
denis_berthier
2010 Supporter

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Location: Paris

### Re: The tridagon rule

.
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:
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`     +-------+-------+-------+     ! . 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:
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`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≠4whip[4]: r7n2{c6 c3} - r7n8{c3 c7} - r7n9{c7 c2} - r2n9{c2 .} ==> r2c6≠2whip[5]: c1n9{r6 r9} - c1n2{r9 r4} - c1n8{r4 r5} - c6n8{r5 r2} - r2n9{c6 .} ==> r6c2≠9whip[5]: r2n9{c2 c6} - r2n8{c6 c5} - c5n3{r2 r3} - c5n5{r3 r1} - r1c1{n5 .} ==> r2c2≠1whip[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:
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`***** 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.`

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`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≠4z-chain[3]: r1n4{c6 c7} - c9n4{r2 r4} - c4n4{r4 .} ==> r3c5≠4, r3c6≠4z-chain[4]: r4c9{n4 n6} - r4c4{n6 n7} - r6c5{n7 n6} - r7c5{n6 .} ==> r4c5≠4z-chain[4]: r4c9{n6 n4} - r4c4{n4 n7} - r6c5{n7 n4} - r7c5{n4 .} ==> r4c5≠6z-chain[5]: r7c5{n4 n6} - r6c5{n6 n7} - r6c8{n7 n6} - r4c9{n6 n4} - c4n4{r4 .} ==> r1c5≠4whip[5]: b8n7{r8c6 r9c4} - b8n2{r9c4 r7c6} - c6n6{r7 r5} - r4c4{n6 n4} - b2n4{r1c4 .} ==> r1c6≠7whip[4]: r5n6{c3 c6} - r1c6{n6 n4} - c4n4{r1 r4} - r4c9{n4 .} ==> r4c3≠6whip[4]: r5n6{c3 c6} - r1c6{n6 n4} - c4n4{r1 r4} - r4c9{n4 .} ==> r4c2≠6whip[4]: r5n6{c3 c6} - r1c6{n6 n4} - c4n4{r1 r4} - r4c9{n4 .} ==> r4c1≠6whip[5]: r1c6{n6 n4} - c4n4{r1 r4} - r6c5{n4 n7} - r6c8{n7 n6} - r4c9{n6 .} ==> r1c5≠6hidden-pairs-in-a-column: c5{n4 n6}{r6 r7} ==> r6c5≠7whip[3]: r1c6{n6 n4} - b8n4{r7c6 r7c5} - r6c5{n4 .} ==> r5c6≠6whip[1]: r5n6{c3 .} ==> r6c1≠6, r6c2≠6, r6c3≠6z-chain[3]: r6n4{c3 c5} - b5n6{r6c5 r4c4} - r4c9{n6 .} ==> r4c3≠4, r4c2≠4, r4c1≠4z-chain[5]: b5n7{r4c5 r5c6} - r8c6{n7 n4} - c5n4{r7 r6} - r6n6{c5 c8} - r6n7{c8 .} ==> r4c3≠7, r4c2≠7, r4c1≠7z-chain[5]: b1n7{r2c3 r3c1} - c8n7{r3 r6} - r6n6{c8 c5} - c5n4{r6 r7} - r8c6{n4 .} ==> r2c6≠7z-chain[5]: r1n7{c5 c7} - c8n7{r2 r6} - r6n6{c8 c5} - c5n4{r6 r7} - r8c6{n4 .} ==> r3c6≠7whip[4]: c6n7{r8 r5} - r4n7{c4 c7} - r4n3{c7 c2} - c2n1{r4 .} ==> r8c2≠7biv-chain[5]: r9c7{n9 n8} - r7n8{c7 c3} - r7n2{c3 c6} - r3c6{n2 n9} - b1n9{r3c3 r2c2} ==> r9c2≠9biv-chain[5]: b5n8{r4c5 r5c6} - r2c6{n8 n9} - r3c6{n9 n2} - r7n2{c6 c3} - b4n2{r4c3 r4c1} ==> r4c1≠8z-chain[5]: r7c5{n6 n4} - r7c2{n4 n9} - r2n9{c2 c6} - r3c6{n9 n2} - r7n2{c6 .} ==> r7c3≠6z-chain[5]: r7c5{n4 n6} - r7c2{n6 n9} - r2n9{c2 c6} - r3c6{n9 n2} - r7n2{c6 .} ==> r7c3≠4whip[6]: c2n1{r8 r4} - r4c1{n1 n2} - r4c3{n2 n8} - c5n8{r4 r2} - r2c6{n8 n9} - r2c2{n9 .} ==> r8c2≠5biv-chain[3]: c2n5{r6 r2} - b1n9{r2c2 r3c3} - b4n9{r6c3 r6c1} ==> r6c1≠5z-chain[4]: c6n7{r8 r5} - c2n7{r5 r6} - c2n5{r6 r2} - c1n5{r1 .} ==> r8c1≠7z-chain[5]: b8n7{r8c6 r9c4} - c2n7{r9 r6} - c2n5{r6 r2} - r2n9{c2 c6} - c6n8{r2 .} ==> r5c6≠7hidden-single-in-a-column ==> r8c6=7whip[1]: r8n4{c3 .} ==> r7c2≠4whip[1]: b5n7{r4c5 .} ==> r4c7≠7biv-chain[3]: r2n2{c9 c4} - r9c4{n2 n6} - r4n6{c4 c9} ==> r2c9≠6biv-chain[4]: r6c8{n7 n6} - c5n6{r6 r7} - r7c2{n6 n9} - c1n9{r9 r6} ==> r6c1≠7z-chain[4]: r6n4{c3 c5} - r6n6{c5 c8} - b6n7{r6c8 r5c7} - r5n3{c7 .} ==> r5c2≠4biv-chain[3]: c2n4{r8 r6} - c2n5{r6 r2} - c1n5{r1 r8} ==> r8c1≠4naked-pairs-in-a-column: c1{r1 r8}{n1 n5} ==> r4c1≠1naked-single ==> r4c1=2naked-pairs-in-a-row: r9{c1 c7}{n8 n9} ==> r9c3≠9, r9c3≠8biv-chain[4]: r1c5{n5 n7} - r4c5{n7 n8} - r4c3{n8 n1} - b1n1{r3c3 r1c1} ==> r1c1≠5singles ==> r1c1=1, r8c1=5biv-chain[4]: b1n5{r3c3 r2c2} - r2n9{c2 c6} - r2n8{c6 c5} - b2n3{r2c5 r3c5} ==> r3c5≠5z-chain[3]: r2n3{c8 c5} - r3c5{n3 n7} - b1n7{r3c1 .} ==> r2c8≠7biv-chain[5]: c2n5{r2 r6} - c2n4{r6 r8} - c2n1{r8 r4} - r4c3{n1 n8} - c5n8{r4 r2} ==> r2c5≠5hidden-single-in-a-block ==> r1c5=5hidden-pairs-in-a-block: b3{n1 n5}{r2c7 r3c7} ==> r3c7≠7, r3c7≠4, r2c7≠7, r2c7≠4biv-chain[4]: r3c6{n2 n9} - r3c3{n9 n5} - c7n5{r3 r2} - r2n1{c7 c4} ==> r2c4≠2hidden-single-in-a-row ==> r2c9=2naked-quads-in-a-row: r3{c1 c5 c8 c9}{n4 n7 n3 n6} ==> r3c4≠7, r3c4≠4z-chain[4]: r3n4{c1 c9} - c9n6{r3 r4} - b5n6{r4c4 r6c5} - r6n4{c5 .} ==> r5c1≠4biv-chain[5]: r6c1{n9 n4} - c2n4{r6 r8} - c2n1{r8 r4} - r4c3{n1 n8} - b7n8{r7c3 r9c1} ==> r9c1≠9singles ==> r9c1=8, r9c7=9, r7c7=8, r6c1=9whip[1]: c1n4{r3 .} ==> r2c3≠4biv-chain[4]: r5c1{n6 n7} - c7n7{r5 r1} - b3n4{r1c7 r3c9} - c1n4{r3 r2} ==> r2c1≠6biv-chain[4]: r2n6{c3 c8} - r2n3{c8 c5} - r2n8{c5 c6} - r5n8{c6 c3} ==> r5c3≠6biv-chain[5]: r6c8{n7 n6} - c5n6{r6 r7} - r7c2{n6 n9} - c3n9{r7 r3} - c3n5{r3 r6} ==> r6c3≠7biv-chain[5]: r2n6{c3 c8} - r2n3{c8 c5} - b2n8{r2c5 r2c6} - r2n9{c6 c2} - r7c2{n9 n6} ==> r9c3≠6singles ==> r2c3=6, r2c8=3, r3c5=3, r5c1=6biv-chain[5]: r5n8{c3 c6} - r2c6{n8 n9} - c2n9{r2 r7} - c2n6{r7 r9} - r9n7{c2 c3} ==> r5c3≠7stte   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.
denis_berthier
2010 Supporter

Posts: 4018
Joined: 19 June 2007
Location: Paris

### Re: The tridagon rule

.
Consider the classical "trivalue oddagon" impossible pattern in four blocks forming a rectangle:
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`+-------------------------------+-------------------------------+! 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:
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`    +-------------------------------+-------------------------------+-------------------------------+    ! 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:
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`    +-------------------------------+-------------------------------+-------------------------------+    ! 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):

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`biv-chain[3]: r1c4{n2 n3} - r2c5{n3 n1} - r3c6{n1 n2} ==> r1c5≠2, r1c6≠2, r3c4≠2, r3c5≠2biv-chain[3]: r1c4{n3 n2} - r3c6{n2 n1} - r2c5{n1 n3} ==> r1c5≠3, r1c6≠3, r2c4≠3, r2c6≠3biv-chain[3]: r2c5{n1 n3} - r1c4{n3 n2} - r3c6{n2 n1} ==> r2c4≠1, r2c6≠1, r3c4≠1, r3c5≠1biv-chain[3]: r4c1{n2 n3} - r5c2{n3 n1} - r6c3{n1 n2} ==> r4c3≠2, r5c1≠2, r5c3≠2, r6c1≠2biv-chain[3]: r4c1{n3 n2} - r6c3{n2 n1} - r5c2{n1 n3} ==> r4c2≠3, r5c1≠3, r6c1≠3, r6c2≠3biv-chain[3]: r5c2{n1 n3} - r4c1{n3 n2} - r6c3{n2 n1} ==> r4c2≠1, r4c3≠1, r5c3≠1, r6c2≠1z-chain[4]: b4n1{r6c3 r5c2} - b4n3{r5c2 r4c1} - r4c6{n3 n2} - r3c6{n2 .} ==> r6c6≠1z-chain[4]: b4n3{r4c1 r5c2} - b4n1{r5c2 r6c3} - r6c4{n1 n2} - r1c4{n2 .} ==> r4c4≠3z-chain[4]: r4c1{n2 n3} - r4c6{n3 n1} - r6c4{n1 n3} - r1c4{n3 .} ==> r4c4≠2z-chain[4]: r6c3{n2 n1} - r6c4{n1 n3} - r4c6{n3 n1} - r3c6{n1 .} ==> r6c6≠2z-chain[5]: b4n1{r6c3 r5c2} - b4n3{r5c2 r4c1} - r4c6{n3 n2} - r5c5{n2 n3} - r2c5{n3 .} ==> r6c5≠1z-chain[5]: b4n3{r4c1 r5c2} - b4n1{r5c2 r6c3} - r6c4{n1 n2} - r5c5{n2 n1} - r2c5{n1 .} ==> r4c5≠3z-chain[5]: b4n2{r6c3 r4c1} - b4n3{r4c1 r5c2} - r5c5{n3 n1} - b2n1{r2c5 r3c6} - b2n2{r3c6 .} ==> r6c4≠2biv-chain[3]: r6c4{n1 n3} - r1c4{n3 n2} - r3c6{n2 n1} ==> r4c6≠1, r5c6≠1biv-chain[2]: r4c6{n3 n2} - r4c1{n2 n3} ==> r4c7≠3, r4c8≠3, r4c9≠3biv-chain[2]: r4c1{n2 n3} - r4c6{n3 n2} ==> r4c7≠2, r4c8≠2, r4c9≠2, r4c5≠2biv-chain[3]: b4n2{r6c3 r4c1} - r4c6{n2 n3} - r6c4{n3 n1} ==> r6c3≠1singles ==> r6c3=2, r4c1=3, r4c6=2, r3c6=1, r2c5=3, r1c4=2, r5c5=1GRID 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.
Last edited by denis_berthier on Sun Apr 24, 2022 11:22 pm, edited 1 time in total.
denis_berthier
2010 Supporter

Posts: 4018
Joined: 19 June 2007
Location: Paris

### Re: The tridagon rule

.
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.
denis_berthier
2010 Supporter

Posts: 4018
Joined: 19 June 2007
Location: Paris

### Re: The tridagon rule

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.
mith

Posts: 981
Joined: 14 July 2020

### Re: The tridagon rule

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.
denis_berthier
2010 Supporter

Posts: 4018
Joined: 19 June 2007
Location: Paris

### Re: The tridagon rule

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.)
mith

Posts: 981
Joined: 14 July 2020

### Re: The tridagon rule

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 91 72 83 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.
denis_berthier
2010 Supporter

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Location: Paris

### Re: The tridagon rule

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).
mith

Posts: 981
Joined: 14 July 2020

### Re: The tridagon rule

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.
mith

Posts: 981
Joined: 14 July 2020

### Re: The tridagon rule

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.
mith

Posts: 981
Joined: 14 July 2020

### Re: The tridagon rule

.
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?
denis_berthier
2010 Supporter

Posts: 4018
Joined: 19 June 2007
Location: Paris

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