## The tridagon rule

Advanced methods and approaches for solving Sudoku puzzles

### Re: The tridagon rule

mith wrote:I wanted to highlight this puzzle:
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`..34.6....5....1.27.92..........851........28......4.65.86....46.28.4.51..1..5...boxes [4, 5, 7, 8] cells [[0, 5, 7], [2, 3, 7], [2, 4, 6], [1, 3, 8]] trips [3, 7, 9] guard 6boxes [4, 5, 7, 8] cells [[1, 3, 8], [2, 3, 7], [2, 4, 6], [1, 3, 8]] trips [3, 7, 9] guard 941;137;6;28;..34.6....5....1.27.92..........851........28......4.65.86....46.28.4.51..1..5...;44;272`

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`Resolution state after Singles and whips[1]:   +----------------------+----------------------+----------------------+    ! 128    128    3      ! 4      15789  6      ! 789    789    579    !    ! 48     5      46     ! 379    3789   379    ! 1      346789 2      !    ! 7      1468   9      ! 2      1358   13     ! 368    3468   35     !    +----------------------+----------------------+----------------------+    ! 2349   23469  467    ! 379    234679 8      ! 5      1      379    !    ! 1349   13469  4567   ! 13579  34679  379    ! 379    2      8      !    ! 12389  12389  57     ! 13579  2379   2379   ! 4      379    6      !    +----------------------+----------------------+----------------------+    ! 5      379    8      ! 6      12379  12379  ! 2379   379    4      !    ! 6      379    2      ! 8      379    4      ! 379    5      1      !    ! 349    3479   1      ! 379    2379   5      ! 236789 36789  379    !    +----------------------+----------------------+----------------------+ 197 candidates.`

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`hidden-pairs-in-a-row: r9{n6 n8}{c7 c8} ==> r9c8≠9, r9c8≠7, r9c8≠3, r9c7≠9, r9c7≠7, r9c7≠3, r9c7≠2singles ==> r7c7=2, r9c5=2, r6c6=2hidden-pairs-in-a-column: c4{n1 n5}{r5 r6} ==> r6c4≠9, r6c4≠7, r6c4≠3, r5c4≠9, r5c4≠7, r5c4≠3hidden-pairs-in-a-column: c5{n4 n6}{r4 r5} ==> r5c5≠9, r5c5≠7, r5c5≠3, r4c5≠9, r4c5≠7, r4c5≠3biv-chain[3]: r3c6{n1 n3} - r3c9{n3 n5} - b2n5{r3c5 r1c5} ==> r1c5≠1whip[1]: b2n1{r3c6 .} ==> r3c2≠1hidden-pairs-in-a-block: b1{n1 n2}{r1c1 r1c2} ==> r1c2≠8, r1c1≠8   +----------------------+----------------------+----------------------+    ! 12     12     3      ! 4      5789   6      ! 789    789    579    !    ! 48     5      46     ! 379    3789   379    ! 1      346789 2      !    ! 7      468    9      ! 2      1358   13     ! 368    3468   35     !    +----------------------+----------------------+----------------------+    ! 2349   23469  467    ! 379    46     8      ! 5      1      379    !    ! 1349   13469  4567   ! 15     46     379    ! 379    2      8      !    ! 1389   1389   57     ! 15     379    2      ! 4      379    6      !    +----------------------+----------------------+----------------------+    ! 5      379    8      ! 6      1379   1379   ! 2      379    4      !    ! 6      379    2      ! 8      379    4      ! 379    5      1      !    ! 349    3479   1      ! 379    2      5      ! 68     68     379    !    +----------------------+----------------------+----------------------+ tridagon type diag for digits 3, 7 and 9 in blocks:         b8, with cells: r7c6 (target cell), r8c5, r9c4        b9, with cells: r7c8, r8c7, r9c9        b5, with cells: r5c6, r6c5, r4c4        b6, with cells: r5c7, r6c8, r4c9 ==> r7c6≠3,7,9stte`
denis_berthier
2010 Supporter

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Location: Paris

### Re: The tridagon rule

mith wrote:There are still some true ambiguous cases - for example:
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`...4.6...4.7.89.368..37...429.....6...1.........9.3.......946.7...8.739....63..48boxes [0, 1, 6, 7] cells [[0, 4, 8], [1, 3, 8], [0, 4, 8], [0, 4, 8]] trips [1, 2, 5] guard 8boxes [0, 1, 6, 7] cells [[0, 4, 8], [1, 3, 8], [1, 5, 6], [0, 4, 8]] trips [1, 2, 5] guard 10boxes [0, 1, 6, 7] cells [[0, 4, 8], [1, 3, 8], [2, 3, 7], [0, 4, 8]] trips [1, 2, 5] guard 8boxes [0, 2, 6, 8] cells [[0, 4, 8], [2, 3, 7], [0, 5, 7], [1, 5, 6]] trips [1, 2, 5] guard 9boxes [0, 2, 6, 8] cells [[0, 4, 8], [2, 3, 7], [1, 3, 8], [1, 5, 6]] trips [1, 2, 5] guard 9boxes [1, 2, 4, 5] cells [[1, 3, 8], [2, 3, 7], [2, 3, 7], [0, 4, 8]] trips [1, 2, 5] guard 10boxes [1, 2, 7, 8] cells [[1, 3, 8], [2, 3, 7], [0, 4, 8], [1, 5, 6]] trips [1, 2, 5] guard 1boxes [4, 5, 7, 8] cells [[0, 5, 7], [0, 4, 8], [0, 4, 8], [1, 5, 6]] trips [1, 2, 5] guard 952;32;1;27;...4.6...4.7.89.368..37...429.....6...1.........9.3.......946.7...8.739....63..48;71;823`

Some preliminary cleaning makes things clearer:

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`Resolution state after Singles and whips[1]:   +----------------------+----------------------+----------------------+    ! 1359   1235   2359   ! 4      125    6      ! 125789 12578  1259   !    ! 4      125    7      ! 125    8      9      ! 125    3      6      !    ! 8      1256   2569   ! 3      7      125    ! 1259   125    4      !    +----------------------+----------------------+----------------------+    ! 2      9      3458   ! 157    145    158    ! 14578  6      135    !    ! 3567   345678 1      ! 257    2456   258    ! 245789 2578   2359   !    ! 567    45678  4568   ! 9      12456  3      ! 124578 12578  125    !    +----------------------+----------------------+----------------------+    ! 135    12358  2358   ! 125    9      4      ! 6      125    7      !    ! 156    12456  2456   ! 8      125    7      ! 3      9      125    !    ! 1579   1257   259    ! 6      3      125    ! 125    4      8      !    +----------------------+----------------------+----------------------+ 196 candidates.`

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`hidden-pairs-in-a-row: r1{n7 n8}{c7 c8} ==> r1c8≠5, r1c8≠2, r1c8≠1, r1c7≠9, r1c7≠5, r1c7≠2, r1c7≠1z-chain[3]: r9n7{c2 c1} - c1n9{r9 r1} - c1n1{r1 .} ==> r9c2≠1   +----------------------+----------------------+----------------------+    ! 1359   1235   2359   ! 4      125    6      ! 78     78     1259   !    ! 4      125    7      ! 125    8      9      ! 125    3      6      !    ! 8      1256   2569   ! 3      7      125    ! 1259   125    4      !    +----------------------+----------------------+----------------------+    ! 2      9      3458   ! 157    145    158    ! 14578  6      135    !    ! 3567   345678 1      ! 257    2456   258    ! 245789 2578   2359   !    ! 567    45678  4568   ! 9      12456  3      ! 124578 12578  125    !    +----------------------+----------------------+----------------------+    ! 135    12358  2358   ! 125    9      4      ! 6      125    7      !    ! 156    12456  2456   ! 8      125    7      ! 3      9      125    !    ! 1579   257    259    ! 6      3      125    ! 125    4      8      !    +----------------------+----------------------+----------------------+ tridagon type diag for digits 1, 2 and 5 in blocks:         b3, with cells: r1c9 (target cell), r2c7, r3c8        b2, with cells: r1c5, r2c4, r3c6        b9, with cells: r8c9, r9c7, r7c8        b8, with cells: r8c5, r9c6, r7c4 ==> r1c9≠1,2,5singles ==> r1c9=9, r3c3=9, r3c2=6, r9c1=9, r9c2=7, r5c7=9finned-x-wing-in-rows: n1{r9 r3}{c6 c7} ==> r2c7≠1whip[1]: b3n1{r3c8 .} ==> r3c6≠1naked-triplets-in-a-column: c7{r2 r3 r9}{n2 n5 n1} ==> r6c7≠5, r6c7≠2, r6c7≠1, r4c7≠5, r4c7≠1biv-chain[4]: c3n6{r8 r6} - b5n6{r6c5 r5c5} - r5n4{c5 c2} - b7n4{r8c2 r8c3} ==> r8c3≠2, r8c3≠5biv-chain[4]: b7n4{r8c2 r8c3} - c3n6{r8 r6} - r5n6{c1 c5} - r5n4{c5 c2} ==> r6c2≠4biv-chain[4]: c3n6{r6 r8} - b7n4{r8c3 r8c2} - r5n4{c2 c5} - r5n6{c5 c1} ==> r6c1≠6biv-chain[4]: b5n6{r5c5 r6c5} - c3n6{r6 r8} - r8n4{c3 c2} - r5n4{c2 c5} ==> r5c5≠2, r5c5≠5whip[7]: r5n4{c2 c5} - r5n6{c5 c1} - b4n3{r5c1 r4c3} - b4n4{r4c3 r6c3} - b4n8{r6c3 r6c2} - r6c7{n8 n7} - r6c1{n7 .} ==> r5c2≠5whip[8]: b2n1{r1c5 r2c4} - b2n5{r2c4 r3c6} - b3n5{r3c7 r2c7} - r9n5{c7 c3} - c3n2{r9 r7} - r7c4{n2 n5} - b9n5{r7c8 r8c9} - r8n2{c9 .} ==> r1c5≠2whip[1]: r1n2{c3 .} ==> r2c2≠2finned-x-wing-in-columns: n2{c5 c9}{r8 r6} ==> r6c8≠2t-whip[5]: r1c5{n1 n5} - r3c6{n5 n2} - b3n2{r3c7 r2c7} - r9n2{c7 c3} - r1n2{c3 .} ==> r1c2≠1t-whip[5]: c1n6{r8 r5} - r6n6{c3 c5} - r6n2{c5 c9} - r6n1{c9 c8} - c9n1{r4 .} ==> r8c1≠1biv-chain[2]: c1n1{r7 r1} - b2n1{r1c5 r2c4} ==> r7c4≠1biv-chain[3]: b8n1{r9c6 r8c5} - b2n1{r1c5 r2c4} - b2n2{r2c4 r3c6} ==> r9c6≠2biv-chain[3]: c7n1{r3 r9} - r9c6{n1 n5} - r3c6{n5 n2} ==> r3c7≠2biv-chain[3]: r9n2{c3 c7} - r2n2{c7 c4} - b8n2{r7c4 r8c5} ==> r8c2≠2x-wing-in-rows: n2{r6 r8}{c5 c9} ==> r5c9≠2z-chain[4]: r7n8{c2 c3} - r7n3{c3 c1} - b7n1{r7c1 r8c2} - r2c2{n1 .} ==> r7c2≠5z-chain[4]: r1c5{n5 n1} - r2n1{c4 c2} - r8n1{c2 c9} - r8n2{c9 .} ==> r8c5≠5biv-chain[3]: r1c5{n5 n1} - r8c5{n1 n2} - r7c4{n2 n5} ==> r2c4≠5biv-chain[3]: r2n5{c2 c7} - c7n2{r2 r9} - r9c3{n2 n5} ==> r1c3≠5, r8c2≠5biv-chain[3]: c1n6{r5 r8} - r8n5{c1 c9} - r5c9{n5 n3} ==> r5c1≠3hidden-pairs-in-a-column: c1{n1 n3}{r1 r7} ==> r7c1≠5, r1c1≠5whip[1]: b1n5{r2c2 .} ==> r6c2≠5singles ==> r6c2=8, r7c3=8biv-chain[3]: c1n3{r7 r1} - r1c3{n3 n2} - b7n2{r9c3 r7c2} ==> r7c2≠3stte`
denis_berthier
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Location: Paris

### Re: The tridagon rule

Yeah, the current code is only looking at the raw min-expands (many of which have basics available), and singles expansions for ph2010. Once the desktop is back up and running I'll incorporate some of my other code, check for basics, and run a full analysis of the min-expands (and perhaps the 11.6+ puzzles in the local fork of ph).
mith

Posts: 988
Joined: 14 July 2020

### Re: The tridagon rule

.
Continuing the analysis started here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859-60.html
Now concentrating only on the min-expands.
1) (reminder) How many puzzles can be solved with only Subsets + FinnedFish + the tridagon rule defined in the first post this thread:
8,196 = 13%

2) How many puzzles can be solved with only Subsets + FinnedFish + the tridagon rule + whips of length ≤ 5:
25,728 = 40.7%

So, 40% of all the min-expand puzzles can be solved with the most elementary tridagon rule provided that short whips are used. At this point, no Tridagon-Forcing-Whips and no Forcing-anything based on several guardians has been used.

3) How many puzzles can be solved with only Subsets + FinnedFish + the tridagon rule + whips of length ≤ 5 + Tridagon-Forcing-Whips of length ≤ 18
32,900 = 52.1%

4) How many puzzles can be solved with only Subsets + FinnedFish + the tridagon rule + whips of length ≤ 7 + Tridagon-Forcing-Whips of length ≤ 22
38,445 = 60,9%
Last edited by denis_berthier on Fri Jul 22, 2022 10:10 am, edited 1 time in total.
denis_berthier
2010 Supporter

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Location: Paris

### Re: The tridagon rule

.
Here is an example requiring larger chains, #7799 in the list of min-expands, i.e.:
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`+-------+-------+-------+! . . . ! . . . ! 7 8 . !! . . 6 ! 7 8 . ! . 3 2 !! . . . ! 2 . 3 ! 6 . 5 !+-------+-------+-------+! 2 . . ! . . . ! . . 3 !! 3 9 1 ! . . 7 ! . . . !! . 4 8 ! . . . ! . . . !+-------+-------+-------+! 5 . . ! . . . ! . 6 8 !! . . . ! 6 . . ! 5 2 . !! . . . ! 5 . 8 ! 3 . 7 !+-------+-------+-------+......78...678..32...2.36.52.......3391..7....48......5......68...6..52....5.83.7;1750;293903SER = 10.4`

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`Resolution state after Singles and whips[1]:   +----------------------+----------------------+----------------------+   ! 149    1235   23459  ! 149    14569  14569  ! 7      8      149    !   ! 149    15     6      ! 7      8      1459   ! 149    3      2      !   ! 14789  178    479    ! 2      149    3      ! 6      149    5      !   +----------------------+----------------------+----------------------+   ! 2      567    57     ! 1489   1469   1469   ! 1489   1479   3      !   ! 3      9      1      ! 48     2456   7      ! 248    45     46     !   ! 67     4      8      ! 139    123569 12569  ! 129    1579   169    !   +----------------------+----------------------+----------------------+   ! 5      1237   23479  ! 1349   123479 1249   ! 149    6      8      !   ! 14789  1378   3479   ! 6      13479  149    ! 5      2      149    !   ! 1469   126    249    ! 5      1249   8      ! 3      149    7      !   +----------------------+----------------------+----------------------+188 candidates.`

Some easy cleaning (eliminating useless "guardians")
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`hidden-pairs-in-a-row: r1{n2 n3}{c2 c3} ==> r1c3≠9, r1c3≠5, r1c3≠4, r1c2≠5, r1c2≠1singles ==> r2c2=5, r4c3=5hidden-pairs-in-a-block: b2{n5 n6}{r1c5 r1c6} ==> r1c6≠9, r1c6≠4, r1c6≠1, r1c5≠9, r1c5≠4, r1c5≠1   +----------------------+----------------------+----------------------+   ! 149    23     23     ! 149    56     56     ! 7      8      149    !   ! 149    5      6      ! 7      8      149    ! 149    3      2      !   ! 14789  178    479    ! 2      149    3      ! 6      149    5      !   +----------------------+----------------------+----------------------+   ! 2      67     5      ! 1489   1469   1469   ! 1489   1479   3      !   ! 3      9      1      ! 48     2456   7      ! 248    45     46     !   ! 67     4      8      ! 139    123569 12569  ! 129    1579   169    !   +----------------------+----------------------+----------------------+   ! 5      1237   23479  ! 1349   123479 1249   ! 149    6      8      !   ! 14789  1378   3479   ! 6      13479  149    ! 5      2      149    !   ! 1469   126    249    ! 5      1249   8      ! 3      149    7      !   +----------------------+----------------------+----------------------+`

That's where Tridagon Forcing Whips come into play
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`tridagon link (same block) for digits 1, 4 and 9 in blocks:        b8, with cells: r9c5 (link cell), r7c4 (link cell), r8c6        b9, with cells: r9c8, r7c7, r8c9        b2, with cells: r3c5, r1c4, r2c6        b3, with cells: r3c8, r1c9, r2c7 ==> tridagon-link[12](n2r9c5, n3r7c4)tridagon-forcing-whip-elim[13] based on tridagon-link(n2r9c5, n3r7c4)....for n2r9c5: -....for n3r7c4: partial-whip[1]: c5n3{r8 r6} - ==> r6c5≠2whip[3]: c6n6{r6 r1} - c6n5{r1 r6} - b5n2{r6c6 .} ==> r5c5≠6hidden-single-in-a-row ==> r5c9=6whip[5]: b5n2{r5c5 r6c6} - r6n5{c6 c8} - r6n7{c8 c1} - r6n6{c1 c5} - r1c5{n6 .} ==> r5c5≠5hidden-single-in-a-row ==> r5c8=5whip[4]: c6n2{r7 r6} - r6n5{c6 c5} - c5n3{r6 r8} - c5n7{r8 .} ==> r7c5≠2tridagon-forcing-whip-elim[15] based on tridagon-link(n3r7c4, n2r9c5)....for n3r7c4: partial-whip[1]: c5n3{r8 r6} -....for n2r9c5: partial-whip[2]: c6n2{r7 r6} - r6n5{c6 c5} - ==> r6c5≠9, r6c5≠6, r6c5≠1tridagon-forcing-whip-elim[15] based on tridagon-link(n2r9c5, n3r7c4)....for n2r9c5: partial-whip[1]: c6n2{r7 r6} -....for n3r7c4: partial-whip[2]: c5n3{r8 r6} - r6n5{c5 c6} - ==> r6c6≠9, r6c6≠6singles ==> r6c1=6, r4c2=7, r6c8=7, r9c2=6hidden-pairs-in-a-column: c1{n7 n8}{r3 r8} ==> r8c1≠9, r8c1≠4, r8c1≠1, r3c1≠9, r3c1≠4, r3c1≠1whip[6]: c5n7{r7 r8} - b8n3{r8c5 r7c4} - c5n3{r7 r6} - r6n5{c5 c6} - c6n2{r6 r7} - r7c2{n2 .} ==> r7c5≠1whip[6]: r9n2{c3 c5} - r5n2{c5 c7} - c7n8{r5 r4} - b6n4{r4c7 r4c8} - r3n4{c8 c5} - r5c5{n4 .} ==> r9c3≠4tridagon-forcing-whip-elim[15] based on tridagon-link(n2r9c5, n3r7c4)....for n2r9c5: partial-whip[1]: c6n2{r7 r6} -....for n3r7c4: partial-whip[2]: c5n3{r8 r6} - r6n5{c5 c6} - ==> r6c6≠1whip[4]: r5c5{n4 n2} - r6c6{n2 n5} - r1c6{n5 n6} - r4n6{c6 .} ==> r4c5≠4tridagon-forcing-whip-elim[20] based on tridagon-link(n3r7c4, n2r9c5)....for n3r7c4: partial-whip[4]: c5n3{r8 r6} - r6n5{c5 c6} - c6n2{r6 r7} - r7c2{n2 n1} -....for n2r9c5: partial-whip[4]: r5n2{c5 c7} - c7n8{r5 r4} - b6n4{r4c7 r4c8} - r9n4{c8 c1} - ==> r9c1≠1`

The end in W6 is easy (and could probably be simplified): Show
whip[1]: c1n1{r2 .} ==> r3c2≠1
singles ==> r3c2=8, r3c1=7, r8c1=8
x-wing-in-rows: n1{r3 r9}{c5 c8} ==> r8c5≠1, r4c8≠1, r4c5≠1
whip[5]: c4n3{r7 r6} - r6c5{n3 n5} - r1c5{n5 n6} - r4n6{c5 c6} - b5n1{r4c6 .} ==> r7c4≠1
whip[6]: r4n8{c4 c7} - r4n1{c7 c6} - r4n4{c6 c8} - r5c7{n4 n2} - c5n2{r5 r9} - b8n1{r9c5 .} ==> r4c4≠9
whip[6]: r3c3{n4 n9} - b7n9{r7c3 r9c1} - c8n9{r9 r4} - b5n9{r4c5 r6c4} - r1n9{c4 c9} - c9n4{r1 .} ==> r8c3≠4
whip[3]: b7n4{r7c3 r9c1} - b9n4{r9c8 r8c9} - r1n4{c9 .} ==> r7c4≠4
whip[4]: c6n2{r7 r6} - r6n5{c6 c5} - r6n3{c5 c4} - r7c4{n3 .} ==> r7c6≠9
whip[5]: r4c8{n4 n9} - r6c9{n9 n1} - r8c9{n1 n9} - b3n9{r1c9 r2c7} - c6n9{r2 .} ==> r9c8≠4
whip[2]: r9n4{c5 c1} - b1n4{r1c1 .} ==> r3c5≠4
whip[2]: b2n4{r2c6 r1c4} - c9n4{r1 .} ==> r8c6≠4
whip[3]: c6n2{r7 r6} - r5c5{n2 n4} - b8n4{r9c5 .} ==> r7c6≠1
whip[3]: r9n2{c3 c5} - r9n1{c5 c8} - r7n1{c7 .} ==> r7c2≠2
singles ==> r1c2=2, r1c3=3
whip[3]: r8n3{c2 c5} - r7c4{n3 n9} - r8c6{n9 .} ==> r8c2≠1
singles ==> r8c2=3, r7c2=1
whip[3]: r8n1{c6 c9} - c9n4{r8 r1} - b2n4{r1c4 .} ==> r2c6≠1
whip[3]: r7c7{n9 n4} - r7c6{n4 n2} - r6n2{c6 .} ==> r6c7≠9
whip[3]: r7c7{n9 n4} - r8n4{c9 c5} - c5n7{r8 .} ==> r7c5≠9
whip[4]: r4n6{c6 c5} - b5n9{r4c5 r6c4} - r6c9{n9 n1} - r8n1{c9 .} ==> r4c6≠1
singles ==> r8c6=1, r9c8=1, r3c5=1
hidden-pairs-in-a-row: r4{n1 n8}{c4 c7} ==> r4c7≠9, r4c7≠4, r4c4≠4
finned-x-wing-in-columns: n9{c6 c8}{r4 r2} ==> r2c7≠9
singles ==> r7c7=9, r7c4=3, r8c9=4, r6c5=3, r6c6=5, r1c6=6, r1c5=5, r4c5=6, r5c5=2, r6c7=2, r7c6=2, r9c3=2
x-wing-in-columns: n9{c4 c9}{r1 r6} ==> r1c1≠9
finned-x-wing-in-columns: n4{c6 c7}{r2 r4} ==> r4c8≠4
stte
denis_berthier
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### Re: The tridagon rule

.
THE ANTI-TRIDAGON PATTERN

While I use the expression "trivalue oddagon pattern" for (and only for) the precise contradictory pattern (a pattern that can appear in no well-formed puzzle), let me call anti-trivalue-oddagon pattern or in short anti-tridagon pattern the following very broad set of conditions (which, in and of itself, allows no assertion or elimination, but only a big OR conclusion). It is defined to cover all the cases of any number of additional candidates in any number of cells of the "trivalue oddagon pattern".
(As such, it covers the trivalue oddagon pattern as well as the pattern of the tridagon rule in the 2st post or the patterns of the Tridagon Forcing Whips. But it covers many more cases.)

The anti-tridagon pattern
Let there be four blocks forming a rectangle in two bands and two stacks:
b11 b12
b21 b22
Let there be three digits, say 1 2 3, such that:
in each of the four blocks, there are three cells in different rows and different columns such that:
-- the same additional conditions on the 4x3 cells as in the first post of this thread are satisfied;
-- each of these 4x3 cells contains the three digits
----------notice that this doesn't cover the case where digits might be missing in some cell - but I haven't seen any such case as yet;
----------notice also the only difference with the contradictory trivalue oddagon pattern: each cell may contain any number of additional candidates

First remark: in any such case, none of the three digits can be a given in the 2 bands and 2 stacks of the 4 blocks. As a result, at least two of the three digits must be given in the uniquely defined block not in the 2 stacks and not in the 2 bands. This justifies mith's additional criterion and may provide speed improvements in some implementations.

First result: each of the 63,137 min-expand puzzles in mith's database has at least one anti-tridagon pattern (and at most one possibility for the broader pattern of blocks and digits). (Thanks mith for pointing out an error in the first version of this statement.)

Second result: classification of the number of additional candidates when Subsets, Finned-Fish, whips[≤7], Tridagons and Tridagon-Forcing-Whips(≤22) are active:
Code: Select all
`1:  33,579 (including 29,356 solved in SFin+Trid+W7)2:  19,016 (including 9089 solved in SFin+Trid+W7+TFW22 but not in SFin+Trid+W7)3:  7,4434:  2,2015:  6476:  1917:  428:  79:  710: 211: 112: 1Total: 63,137`

[Edit:] I think the above results can best be appreciated if compared to the raw results after allowing only Subsets and Finned Fish:
Code: Select all
`1:  23,2792:  16,8303:  10,8514:  6,1655:  3,2546:  1,7187:  5628:  2339:  12110: 5911: 4012: 2013: 114: 015: 4Total: 63137`
Last edited by denis_berthier on Wed Jul 27, 2022 1:40 am, edited 2 times in total.
denis_berthier
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### Re: The tridagon rule

If you don't want to use UR it becomes very hard, but you can use the tri pattern and then its stte.
Code: Select all
`.-------------------.-----------------.--------------.| 149    23    23   | 149   56    56  | 7   8    149 || 149    5     6    | 7     8     149 | 19  3    2   || 14789  178   479  | 2     149   3   | 6   149  5   |:-------------------+-----------------+--------------:| 2      67    5    | 1489  1469  149 | 48  79   3   || 3      9     1    | 48    2     7   | 48  5    6   || 67     4     8    | 39    356   56  | 2   179  19  |:-------------------+-----------------+--------------:| 5      137   349  | 1349  37    2   | 19  6    8   || 1489   1378  3479 | 6     379   149 | 5   2    149 || 69     126   249  | 5     149   8   | 3   149  7   |'-------------------'-----------------'--------------'`

Now only a generalized definition is needed to find all patterns. Quad patterns are in larger puzzles. Or a 6 block pattern:
Hidden Text: Show
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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  123456789 | 123        123456789  123456789 || 123456789  123        123456789 | 123456789  123456789  123456789 | 123456789  123        123456789 || 123456789  123456789  123       | 123456789  123456789  123456789 | 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 |'---------------------------------'---------------------------------'---------------------------------'`

The exclusions are simple if you find a valid pattern. I don't even know how to hardcode this single pattern.
I just want a generic fast way of finding exclusions using the smallest pattern without coding all those patterns.
creint

Posts: 393
Joined: 20 January 2018

### Re: The tridagon rule

denis_berthier
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Location: Paris

### Re: The tridagon rule

First result: each of the 63,137 min-expand puzzles in mith's database has one and only one anti-tridagon pattern.

There are definitely puzzles where the pattern of *cells* is not unique after subsets (for example Tanngrisnir and Tanngnjóstr, ID 19252 which has two valid choices in box 4, each of which can be used to place a guardian digit as shown in marek's solution) - maybe the addition of finned fish and whips[≤7] is enough to disambiguate in all cases, I'm not sure. (Obviously the pattern of digits and boxes is unique to all of these puzzles.)

Also, I finally have a working desktop again, so I'll get my scripts merged together and get some more results out. Current count in the expanded db is over 469k.
mith

Posts: 988
Joined: 14 July 2020

### Re: The tridagon rule

mith wrote:
First result: each of the 63,137 min-expand puzzles in mith's database has one and only one anti-tridagon pattern.

There are definitely puzzles where the pattern of *cells* is not unique after subsets (for example Tanngrisnir and Tanngnjóstr, ID 19252 which has two valid choices in box 4, each of which can be used to place a guardian digit as shown in marek's solution) - maybe the addition of finned fish and whips[≤7] is enough to disambiguate in all cases, I'm not sure. (Obviously the pattern of digits and boxes is unique to all of these puzzles.).

OK. I corrected my first statement. (The number of occurrences was not properly counted.) Thanks.
This doesn't change the main point of my classification results: after applying rather simple rules (simple for a puzzle in T&E(3)), the number of remaining "guardians" is much less than at first sight.
denis_berthier
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Location: Paris

### Re: The tridagon rule

.
[Added for clarity of this thread]: This post marks a new turn in the use of the anti-tridagon pattern; see the post referenced below for the definition of new ORk-chains.

Using the OR-k-Forcing-Whips introduced here http://forum.enjoysudoku.com/or-k-forcing-whips-t40189.html,
here is a solution of min-expand #339, one of the puzzles that still has 4 guardians after applying W7.
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`     +-------+-------+-------+     ! . . . ! 4 . 6 ! 7 8 . !      ! . . . ! . . . ! 2 . . !      ! . 8 . ! 2 7 . ! . . . !      +-------+-------+-------+     ! 2 . 8 ! 3 4 . ! . . 7 !      ! 3 7 . ! . 6 . ! . . . !      ! . 4 6 ! 8 . 7 ! . . . !      +-------+-------+-------+     ! . 6 . ! . . . ! 1 9 4 !      ! . 3 4 ! . . . ! 5 2 . !      ! . . . ! . . 4 ! . 7 3 !      +-------+-------+-------+...4.678.......2...8.27....2.834...737..6.....468.7....6....194.34...52......4.73;111;44367`

Code: Select all
`Resolution state after Singles and whips[1]:   +----------------------+----------------------+----------------------+    ! 159    1259   12359  ! 4      1359   6      ! 7      8      159    !    ! 145679 159    13579  ! 159    13589  13589  ! 2      13456  1569   !    ! 14569  8      1359   ! 2      7      1359   ! 3469   13456  1569   !    +----------------------+----------------------+----------------------+    ! 2      159    8      ! 3      4      159    ! 69     156    7      !    ! 3      7      159    ! 159    6      1259   ! 489    145    12589  !    ! 159    4      6      ! 8      1259   7      ! 39     135    1259   !    +----------------------+----------------------+----------------------+    ! 578    6      257    ! 57     2358   2358   ! 1      9      4      !    ! 1789   3      4      ! 1679   189    189    ! 5      2      68     !    ! 1589   1259   1259   ! 1569   12589  4      ! 68     7      3      !    +----------------------+----------------------+----------------------+ 184 candidates`

Code: Select all
`hidden-pairs-in-a-column: c1{n4 n6}{r2 r3} ==> r3c1≠9, r3c1≠5, r3c1≠1, r2c1≠9, r2c1≠7, r2c1≠5, r2c1≠1hidden-single-in-a-block ==> r2c3=7biv-chain[3]: r4c7{n9 n6} - b9n6{r9c7 r8c9} - c9n8{r8 r5} ==> r5c9≠9z-chain[3]: c4n6{r9 r8} - r8n7{c4 c1} - r8n1{c1 .} ==> r9c4≠1z-chain[3]: c4n6{r9 r8} - r8n7{c4 c1} - r8n9{c1 .} ==> r9c4≠9biv-chain[4]: r4c7{n9 n6} - b9n6{r9c7 r8c9} - c9n8{r8 r5} - b6n2{r5c9 r6c9} ==> r6c9≠9whip[1]: c9n9{r3 .} ==> r3c7≠9biv-chain[4]: r8c9{n6 n8} - b6n8{r5c9 r5c7} - c7n4{r5 r3} - r3c1{n4 n6} ==> r3c9≠6z-chain[5]: c1n7{r8 r7} - c1n8{r7 r9} - b9n8{r9c7 r8c9} - r8n6{c9 c4} - r8n7{c4 .} ==> r8c1≠1, r8c1≠9whip[1]: r8n9{c6 .} ==> r9c5≠9whip[1]: r8n1{c6 .} ==> r9c5≠1   +-------------------+-------------------+-------------------+    ! 159   1259  12359 ! 4     1359  6     ! 7     8     159   !    ! 46    159   7     ! 159   13589 13589 ! 2     13456 1569  !    ! 46    8     1359  ! 2     7     1359  ! 346   13456 159   !    +-------------------+-------------------+-------------------+    ! 2     159   8     ! 3     4     159   ! 69    156   7     !    ! 3     7     159   ! 159   6     1259  ! 489   145   1258  !    ! 159   4     6     ! 8     1259  7     ! 39    135   125   !    +-------------------+-------------------+-------------------+    ! 578   6     25    ! 57    2358  2358  ! 1     9     4     !    ! 78    3     4     ! 1679  189   189   ! 5     2     68    !    ! 1589  1259  1259  ! 56    258   4     ! 68    7     3     !    +-------------------+-------------------+-------------------+ OR4-anti-tridagon[12] (type diag) for digits 1, 5 and 9 in blocks:         b1, with cells: r1c1, r2c2, r3c3        b2, with cells: r1c5, r2c4, r3c6        b4, with cells: r6c1, r4c2, r5c3        b5, with cells: r6c5, r4c6, r5c4with 4 guardians: n3r1c5 n3r3c3 n3r3c6 n2r6c5`

Based on this OR4 relation, several OR4-Forcing-Whips will allow an easy solution.
Code: Select all
`OR4-forcing-whip-elim[4] based on OR4-anti-tridagon[12] for n3r3c6, n3r1c5, n3r3c3 and  n2r6c5:   || n3r3c6 -   || n3r1c5 -   || n3r3c3 - partial-whip[1]: b3n3{r3c8 r2c8} -   || n2r6c5 - partial-whip[2]: c6n2{r5 r7} - r7n3{c6 c5} - ==> r2c5≠3t-whip[5]: c6n8{r8 r2} - r2n3{c6 c8} - r2n4{c8 c1} - r2n6{c1 c9} - r8c9{n6 .} ==> r8c5≠8OR4-forcing-whip-elim[5] based on OR4-anti-tridagon[12] for n3r3c6, n3r3c3, n3r1c5 and  n2r6c5:   || n3r3c6 -   || n3r3c3 -   || n3r1c5 - partial-whip[1]: c3n3{r1 r3} -   || n2r6c5 - partial-whip[3]: r5n2{c6 c9} - r5n8{c9 c7} - c7n4{r5 r3} - ==> r3c7≠3hidden-single-in-a-column ==> r6c7=3naked-pairs-in-a-row: r3{c1 c7}{n4 n6} ==> r3c8≠6, r3c8≠4z-chain[5]: c6n2{r5 r7} - r7c3{n2 n5} - r5c3{n5 n9} - r6n9{c1 c5} - b5n2{r6c5 .} ==> r5c6≠1t-whip[6]: c5n3{r7 r1} - r2n3{c6 c8} - r2n4{c8 c1} - r2n6{c1 c9} - r8n6{c9 c4} - r9c4{n6 .} ==> r7c5≠5OR4-forcing-whip-elim[7] based on OR4-anti-tridagon[12] for n3r3c6, n3r3c3, n3r1c5 and  n2r6c5:   || n3r3c6 - partial-whip[1]: r2n3{c6 c8} -   || n3r3c3 - partial-whip[1]: b3n3{r3c8 r2c8} -   || n3r1c5 - partial-whip[1]: r2n3{c6 c8} -   || n2r6c5 - partial-whip[3]: r5n2{c6 c9} - c9n8{r5 r8} - c9n6{r8 r2} - ==> r2c8≠6hidden-single-in-a-column ==> r4c8=6naked-single ==> r4c7=9z-chain[7]: c6n2{r5 r7} - r7c3{n2 n5} - r5c3{n5 n1} - r4n1{c2 c6} - r8c6{n1 n8} - c9n8{r8 r5} - r5n2{c9 .} ==> r5c6≠9whip[5]: r8c5{n1 n9} - b5n9{r6c5 r5c4} - c4n1{r5 r8} - r8c6{n1 n8} - r2n8{c6 .} ==> r2c5≠1whip[7]: b3n4{r2c8 r3c7} - c7n6{r3 r9} - c4n6{r9 r8} - c4n1{r8 r5} - b6n1{r5c8 r6c9} - r6n2{c9 c5} - b5n9{r6c5 .} ==> r2c8≠1whip[7]: b5n9{r6c5 r5c4} - r8n9{c4 c6} - r8c5{n9 n1} - c4n1{r8 r2} - r2c2{n1 n5} - r4n5{c2 c6} - c6n1{r4 .} ==> r2c5≠9OR4-forcing-whip-elim[7] based on OR4-anti-tridagon[12] for n3r3c6, n3r3c3, n3r1c5 and  n2r6c5:   || n3r3c6 - partial-whip[1]: r2n3{c6 c8} -   || n3r3c3 - partial-whip[1]: b3n3{r3c8 r2c8} -   || n3r1c5 - partial-whip[1]: r2n3{c6 c8} -   || n2r6c5 - partial-whip[3]: r5n2{c6 c9} - r5n8{c9 c7} - c7n4{r5 r3} - ==> r2c8≠4: better notation for the OR-k-Forcing-Whips (their content is unchanged)`

The end is easy:
Code: Select all
`singles ==> r3c7=4, r3c1=6, r2c1=4, r5c7=8, r9c7=6, r8c9=8, r8c1=7, r9c4=5, r7c4=7, r2c9=6, r8c4=6, r5c8=4finned-x-wing-in-rows: n1{r4 r2}{c2 c6} ==> r3c6≠1biv-chain[2]: r4n1{c2 c6} - c4n1{r5 r2} ==> r2c2≠1whip[1]: r2n1{c6 .} ==> r1c5≠1biv-chain[3]: r4n1{c2 c6} - r5c4{n1 n9} - b4n9{r5c3 r6c1} ==> r6c1≠1biv-chain[4]: c2n2{r1 r9} - r9c5{n2 n8} - r2c5{n8 n5} - r2c2{n5 n9} ==> r1c2≠9biv-chain[3]: c2n9{r9 r2} - c4n9{r2 r5} - b4n9{r5c3 r6c1} ==> r9c1≠9biv-chain[4]: b7n9{r9c3 r9c2} - r2c2{n9 n5} - r2c5{n5 n8} - r9c5{n8 n2} ==> r9c3≠2biv-chain[3]: c1n1{r1 r9} - r9c3{n1 n9} - b4n9{r5c3 r6c1} ==> r1c1≠9singles ==> r6c1=9, r5c4=9, r2c4=1finned-x-wing-in-rows: n1{r5 r3}{c3 c9} ==> r1c9≠1whip[1]: b3n1{r3c9 .} ==> r3c3≠1biv-chain[3]: r5c3{n1 n5} - r7n5{c3 c1} - r1c1{n5 n1} ==> r1c3≠1biv-chain[3]: c1n5{r1 r7} - r7c3{n5 n2} - b1n2{r1c3 r1c2} ==> r1c2≠5biv-chain[3]: c3n2{r7 r1} - r1n3{c3 c5} - b8n3{r7c5 r7c6} ==> r7c6≠2singles ==> r5c6=2, r6c9=2finned-x-wing-in-columns: n5{c2 c6}{r4 r2} ==> r2c5≠5singles ==> r2c5=8, r9c5=2, r7c5=3, r7c6=8, r7c1=5, r1c1=1, r1c2=2, r9c1=8, r7c3=2, r1c3=3finned-x-wing-in-columns: n5{c3 c9}{r5 r3} ==> r3c8≠5finned-x-wing-in-columns: n5{c8 c5}{r6 r2} ==> r2c6≠5finned-x-wing-in-columns: n5{c6 c3}{r3 r4} ==> r4c2≠5stte`
Last edited by denis_berthier on Fri Aug 11, 2023 7:30 am, edited 2 times in total.
denis_berthier
2010 Supporter

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Location: Paris

### Re: The tridagon rule

Hidden Text: Show
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`Hidden Pair: 46 in r2c1,r3c1 => r2c1<>1579,r3c1<>159Hidden Single: 7 in r2 => r2c3=7Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c1<>63r3c3 - 3r3c78 = 3r2c8 - (3=15896)r2c245693r1c5 - (3=15896)r2c245693r3c6 - (3=15896)r2c245692r6c5 - (2=1594)r5c3468 - 4r2c8 = 4r2c1Hidden Single: 6 in c1 => r3c1=6Hidden Single: 4 in c1 => r2c1=4Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r9c5<>1,r25c4,r7c56,r9c5<>5,r9c5<>93r3c3 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p145673r1c5 - (3=15896)r2c24569 - (6=8)r8c9 - (8=16795)b8p145673r3c6 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p145672r6c5 - (2=15698)r12368c9 - (8=16795)b8p14567Naked Pair: in r2c4,r5c4 => r8c4<>19,r9c4<>19,Locked Candidates 1 (Pointing): 1 in b8 => r8c1<>1Locked Candidates 1 (Pointing): 9 in b8 => r8c1<>9Naked Triple: in r7c5,r7c6,r9c5 => r8c5<>8,r8c6<>8,Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c5<>33r3c3 - 3r3c78 = 3r2c83r1c53r3c62r6c5 - (2=83)r79c5UR Forcing Chain: Each true guardian of UR 38{r27c56} will all lead to: r2c6<>31r2c5 - (1=5693)r2c24895r2c5 - (5=1693)r2c24899r2c5 - (9=1563)r2c24891r2c65r2c69r2c6(2-3)r7c5 = 3r7c62r7c6 - (2=1593)r3458c6Hidden Single: 3 in r2 => r2c8=3Hidden Single: 6 in r2 => r2c9=6Hidden Single: 3 in r6 => r6c7=3Hidden Single: 6 in r8 => r8c4=6Hidden Single: 7 in r8 => r8c1=7Hidden Single: 7 in r7 => r7c4=7Hidden Single: 8 in r8 => r8c9=8Full House: r9c7=6Hidden Single: 6 in r4 => r4c8=6Hidden Single: 8 in r5 => r5c7=8Hidden Single: 4 in r5 => r5c8=4Hidden Single: 4 in r3 => r3c7=4Full House: r4c7=9Hidden Single: 5 in c4 => r9c4=5Finned X-Wing:1r24\c26 fr2c45 => r3c6<>1Finned X-Wing:5r24\c26 fr2c5 => r3c6<>5Finned Swordfish:1r248\c256 fr2c4 => r1c5<>1Locked Candidates 1 (Pointing): 1 in b2 => r2c2<>1Whip[3]: Supposing 9r1c3 would causes 3 to disappear in Row 1 => r1c3<>99r1c3 - r2c2(9=5) - 5b2(p6=p2) - 3r1(c5=.)Whip[3]: Supposing 1r6c1 would causes 9 to disappear in Box 4 => r6c1<>11r6c1 - 1r4(c2=c6) - r5c4(1=9) - 9b4(p6=.)Whip[4]: Supposing 9r1c2 would causes 5 to disappear in Box 2 => r1c2<>99r1c2 - r2c2(9=5) - r1c1(5=1) - r1c9(1=5) - 5b2(p2=.)Whip[3]: Supposing 9r9c1 would causes 9 to disappear in Column 2 => r9c1<>99r9c1 - 9r6(c1=c5) - 9c4(r5=r2) - 9c2(r2=.)Whip[4]: Supposing 5r1c9 would causes 5 to disappear in Row 5 => r1c9<>55r1c9 - 5r3(c8=c3) - r7c3(5=2) - 2c6(r7=r5) - 5r5(c6=.)Locked Candidates 1 (Pointing): 5 in b3 => r3c3<>5Whip[4]: Supposing 1r1c3 would causes 1 to disappear in Box 4 => r1c3<>11r1c3 - r1c9(1=9) - r1c1(9=5) - 5c2(r2=r4) - 1b4(p2=.)Whip[4]: Supposing 1r5c6 would causes 2 to disappear in Column 6 => r5c6<>11r5c6 - r5c4(1=9) - r5c3(9=5) - r7c3(5=2) - 2c6(r7=.)Whip[4]: Supposing 9r1c5 will result in all candidates in cell r2c4 being impossible => r1c5<>99r1c5 - r1c9(9=1) - 1r3(c8=c3) - 1r5(c3=c4) - r2c4(1=.)Whip[3]: Supposing 5r2c6 would causes 8 to disappear in Column 6 => r2c6<>55r2c6 - r1c5(5=3) - 3r7(c5=c6) - 8c6(r7=.)Locked Candidates 1 (Pointing): 5 in b2 => r6c5<>5Whip[4]: Supposing 5r1c3 would causes 5 to disappear in Column 2 => r1c3<>55r1c3 - r7c3(5=2) - 2c6(r7=r5) - 5c6(r5=r4) - 5c2(r4=.)Whip[3]: Supposing 1r9c2 would causes 1 to disappear in Column 1 => r9c2<>11r9c2 - r4c2(1=5) - 5b1(p5=p1) - 1c1(r1=.)Whip[4]: Supposing 5r1c2 would causes 9 to disappear in Box 7 => r1c2<>55r1c2 - r2c2(5=9) - r1c1(9=1) - 1r9(c1=c3) - 9b7(p9=.)Whip[3]: Supposing 8r7c5 would causes 3 to disappear in Column 5 => r7c5<>88r7c5 - r7c1(8=5) - 5r1(c1=c5) - 3c5(r1=.)Whip[4]: Supposing 9r2c2 would causes 9 to disappear in Box 2 => r2c2<>99r2c2 - 5c2(r2=r4) - r4c6(5=1) - r8c6(1=9) - 9b2(p9=.)Hidden Single: 9 in c2 => r9c2=9Hidden Single: 2 in c2 => r1c2=2Hidden Single: 1 in c2 => r4c2=1Full House: r4c6=5Full House: r2c2=5Hidden Single: 5 in r1 => r1c5=5Hidden Single: 3 in r1 => r1c3=3Hidden Single: 3 in r3 => r3c6=3Hidden Single: 3 in r7 => r7c5=3Whip[3]: Supposing 9r2c5 will result in all candidates in cell r8c5 being impossible => r2c5<>99r2c5 - r2c4(9=1) - 1c6(r2=r8) - r8c5(1=.)Whip[3]: Supposing 9r5c3 would causes 5 to disappear in Column 3 => r5c3<>99r5c3 - r5c6(9=2) - 2r7(c6=c3) - 5c3(r7=.)stte`

If i disable UR-Foring chain, the path will be longer.
Hidden Text: Show
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`Hidden Pair: 46 in r2c1,r3c1 => r2c1<>1579,r3c1<>159Hidden Single: 7 in r2 => r2c3=7Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c1<>63r3c3 - 3r3c78 = 3r2c8 - (3=15896)r2c245693r1c5 - (3=15896)r2c245693r3c6 - (3=15896)r2c245692r6c5 - (2=1594)r5c3468 - 4r2c8 = 4r2c1Hidden Single: 6 in c1 => r3c1=6Hidden Single: 4 in c1 => r2c1=4Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r9c5<>1,r25c4,r7c56,r9c5<>5,r9c5<>93r3c3 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p145673r1c5 - (3=15896)r2c24569 - (6=8)r8c9 - (8=16795)b8p145673r3c6 - (3=14596)b3p36789 - (6=8)r8c9 - (8=16795)b8p145672r6c5 - (2=15698)r12368c9 - (8=16795)b8p14567Naked Pair: in r2c4,r5c4 => r8c4<>19,r9c4<>19,Locked Candidates 1 (Pointing): 1 in b8 => r8c1<>1Locked Candidates 1 (Pointing): 9 in b8 => r8c1<>9Naked Triple: in r7c5,r7c6,r9c5 => r8c5<>8,r8c6<>8,Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c5<>33r3c3 - 3r3c78 = 3r2c83r1c53r3c62r6c5 - (2=83)r79c5Whip[3]: Supposing 9r5c9 would causes 8 to disappear in Box 6 => r5c9<>99r5c9 - r4c7(9=6) - r9c7(6=8) - 8b6(p4=.)Whip[4]: Supposing 9r3c7 would causes 4 to disappear in Box 3 => r3c7<>99r3c7 - r4c7(9=6) - 6c8(r4=r2) - 3b3(p5=p8) - 4b3(p8=.)Locked Candidates 2 (Claiming): 9 in c7 => r6c9<>9Whip[4]: Supposing 1r5c6 would causes 2 to disappear in Column 6 => r5c6<>11r5c6 - r5c4(1=9) - r5c3(9=5) - r7c3(5=2) - 2c6(r7=.)Whip[4]: Supposing 9r5c6 would causes 2 to disappear in Column 6 => r5c6<>99r5c6 - r5c4(9=1) - r5c3(1=5) - r7c3(5=2) - 2c6(r7=.)Whip[6]: Supposing 1r2c8 would causes 1 to disappear in Column 4 => r2c8<>11r2c8 - 3r2(c8=c6) - 8r2(c6=c5) - r9c5(8=2) - 2r6(c5=c9) - 1b6(p9=p6) - 1c4(r5=.)Whip[7]: Supposing 1r2c5 will result in all candidates in cell r8c5 being impossible => r2c5<>11r2c5 - r2c4(1=9) - r2c2(9=5) - r2c9(5=6) - 6c8(r2=r4) - r4c7(6=9) - 9c6(r4=r8) - r8c5(9=.)Whip[7]: Supposing 9r2c5 will result in all candidates in cell r8c5 being impossible => r2c5<>99r2c5 - r2c4(9=1) - r2c2(1=5) - r2c9(5=6) - 6c8(r2=r4) - 5r4(c8=c6) - 1c6(r4=r8) - r8c5(1=.)Whip[8]: Supposing 1r1c2 will result in all candidates in cell r5c4 being impossible => r1c2<>11r1c2 - 2r1(c2=c3) - 3r1(c3=c5) - 3r2(c6=c8) - 6c8(r2=r4) - 1r4(c8=c6) - r8c6(1=9) - 9c5(r8=r6) - r5c4(9=.)Whip[8]: Supposing 9r1c2 will result in all candidates in cell r9c7 being impossible => r1c2<>99r1c2 - 2c2(r1=r9) - r9c5(2=8) - r2c5(8=5) - r2c2(5=1) - 1c4(r2=r5) - 1r4(c6=c8) - 6r4(c8=c7) - r9c7(6=.)Whip[8]: Supposing 2r9c2 would causes 1 to disappear in Column 2 => r9c2<>22r9c2 - r7c3(2=5) - 5r9(c1=c4) - 6r9(c4=c7) - r4c7(6=9) - 9c2(r4=r2) - 9c4(r2=r5) - r5c3(9=1) - 1c2(r4=.)Hidden Single: 2 in c2 => r1c2=2Whip[8]: Supposing 2r7c5 would causes 2 to disappear in Box 5 => r7c5<>22r7c5 - r7c3(2=5) - r7c4(5=7) - r7c1(7=8) - 8c6(r7=r2) - r2c5(8=5) - 5c2(r2=r4) - 5b5(p3=p6) - 2b5(p6=.)Whip[8]: Supposing 9r1c3 would causes 3 to disappear in Row 2 => r1c3<>99r1c3 - 3r1(c3=c5) - r7c5(3=8) - r2c5(8=5) - r2c2(5=1) - 1c4(r2=r5) - 1r4(c6=c8) - 6c8(r4=r2) - 3r2(c8=.)Whip[10]: Supposing 1r1c3 will result in all candidates in cell r1c1 being impossible => r1c3<>11r1c3 - 3r1(c3=c5) - r7c5(3=8) - r2c5(8=5) - r2c2(5=9) - r2c4(9=1) - r2c9(1=6) - r8c9(6=8) - r8c1(8=7) - r7c1(7=5) - r1c1(5=.)Whip[10]: Supposing 5r1c3 will result in all candidates in cell r5c4 being impossible => r1c3<>55r1c3 - 3r1(c3=c5) - 3r2(c6=c8) - 6c8(r2=r4) - 6c7(r4=r9) - r9c4(6=5) - 5c2(r9=r4) - 1r4(c2=c6) - r8c6(1=9) - 9c5(r8=r6) - r5c4(9=.)Naked Single: r1c3=3Hidden Single: 3 in c5 => r7c5=3Whip[10]: Supposing 5r2c8 would causes 5 to disappear in Row 6 => r2c8<>55r2c8 - r2c5(5=8) - r9c5(8=2) - 2r6(c5=c9) - 5b6(p9=p6) - 8r5(c9=c7) - 8r9(c7=c1) - r8c1(8=7) - r7c1(7=5) - 5r1(c1=c5) - 5r6(c5=.)Whip[9]: Supposing 5r5c9 will result in all candidates in cell r5c6 being impossible => r5c9<>55r5c9 - 5c8(r6=r3) - 4r3(c8=c7) - 3b3(p7=p5) - 6r2(c8=c9) - 6r8(c9=c4) - r9c4(6=5) - 5c3(r9=r7) - 2r7(c3=c6) - r5c6(2=.)Whip[6]: Supposing 5r2c6 would causes 6 to disappear in Column 8 => r2c6<>55r2c6 - 3r2(c6=c8) - r3c7(3=4) - 4r5(c7=c8) - 5r5(c8=c3) - 5r4(c2=c8) - 6c8(r4=.)Whip[8]: Supposing 9r2c2 would causes 9 to disappear in Column 3 => r2c2<>99r2c2 - 9c4(r2=r5) - 9r4(c6=c7) - 6r4(c7=c8) - 6r2(c8=c9) - 5r2(c9=c5) - 8c5(r2=r9) - 2r9(c5=c3) - 9c3(r9=.)Whip[5]: Supposing 5r2c5 would causes 8 to disappear in Column 5 => r2c5<>55r2c5 - r2c2(5=1) - 1c4(r2=r5) - 1c3(r5=r9) - 2r9(c3=c5) - 8c5(r9=.)Naked Single: r2c5=8Hidden Single: 8 in c6 => r7c6=8Hidden Single: 2 in r7 => r7c3=2Hidden Single: 2 in r9 => r9c5=2Hidden Single: 2 in r6 => r6c9=2Hidden Single: 2 in r5 => r5c6=2Locked Candidates 1 (Pointing): 5 in b6 => r3c8<>5Whip[5]: Supposing 1r2c6 would causes 1 to disappear in Column 4 => r2c6<>11r2c6 - 3r2(c6=c8) - r3c7(3=4) - r3c8(4=1) - 1c9(r1=r5) - 1c4(r5=.)Whip[5]: Supposing 5r3c3 would causes 5 to disappear in Column 6 => r3c3<>55r3c3 - r2c2(5=1) - 1c4(r2=r5) - r5c3(1=9) - r4c2(9=5) - 5c6(r4=.)Whip[4]: Supposing 5r4c2 would causes 1 to disappear in Column 4 => r4c2<>55r4c2 - r2c2(5=1) - r3c3(1=9) - r5c3(9=1) - 1c4(r5=.)Whip[5]: Supposing 5r1c1 would causes 5 to disappear in Box 2 => r1c1<>55r1c1 - r2c2(5=1) - 1c4(r2=r5) - 1r4(c6=c8) - 5r4(c8=c6) - 5b2(p9=.)Hidden Single: 5 in b1 => r2c2=5Whip[5]: Supposing 9r5c3 would causes 9 to disappear in Column 4 => r5c3<>99r5c3 - 5r5(c3=c8) - 5r4(c8=c6) - 5r3(c6=c9) - 9r3(c9=c6) - 9c4(r2=.)Whip[3]: Supposing 1r9c1 will result in all candidates in cell r1c1 being impossible => r9c1<>11r9c1 - r9c2(1=9) - 9c3(r9=r3) - r1c1(9=.)Whip[6]: Supposing 1r1c5 will result in all candidates in cell r2c4 being impossible => r1c5<>11r1c5 - r1c1(1=9) - r1c9(9=5) - 5c5(r1=r6) - 9r6(c5=c7) - 9r5(c7=c4) - r2c4(9=.)Whip[3]: Supposing 9r3c9 would causes 5 to disappear in Box 3 => r3c9<>99r3c9 - r3c3(9=1) - 1r1(c1=c9) - 5b3(p3=.)Whip[3]: Supposing 9r4c6 will result in all candidates in cell r5c4 being impossible => r4c6<>99r4c6 - r8c6(9=1) - 1c5(r8=r6) - r5c4(1=.)Whip[4]: Supposing 9r1c1 would causes 1 to disappear in Column 1 => r1c1<>99r1c1 - 9r3(c3=c6) - r8c6(9=1) - 1c5(r8=r6) - 1c1(r6=.)Hidden Single: 9 in b1 => r3c3=9Full House: r1c1=1Whip[3]: Supposing 1r4c6 would causes 5 to disappear in Box 5 => r4c6<>11r4c6 - r4c2(1=9) - r6c1(9=5) - 5b5(p8=.)Naked Single: r4c6=5Hidden Single: 5 in r3 => r3c9=5Hidden Single: 5 in r1 => r1c5=5Full House: r1c9=9X-Wing:1c49\r25  => r5c38<>1stte`
yzfwsf

Posts: 864
Joined: 16 April 2019

### Re: The tridagon rule

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Hi yzfwsf

I'm quite surprised that in the few solutions you have given based on your equivalent of my OR3 or OR4 Forcing chains, you have very long cumulated ALS-chains; e.g. for the first in your previous post:
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`Triplet Oddagon Forcing Chain: Each true guardian of Triplet Oddagon will all lead To: r2c1<>63r3c3 - 3r3c78 = 3r2c8 - (3=15896)r2c245693r1c5 - (3=15896)r2c245693r3c6 - (3=15896)r2c245692r6c5 - (2=1594)r5c3468 - 4r2c8 = 4r2c1`

Total length: 1 (for the OR4) + 6 + 5 + 5 + 5 = 22
The longest I need is 7.
denis_berthier
2010 Supporter

Posts: 4045
Joined: 19 June 2007
Location: Paris

### Re: The tridagon rule

Hi denis_berthier:
The algorithm of my solver is speed priority, not for rating the puzzle, so the Cell/Region Foring chain does not use the simplest strategy, but a by-product of searching for the Als chain, basically without any additional consumption.If desired, users can use the "Search all possible steps" feature to create their own solution paths.
yzfwsf

Posts: 864
Joined: 16 April 2019

### Re: The tridagon rule

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A puzzle with lots of guardians: #1182 in mith's list of 63,137 min-expands.
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`+-------+-------+-------+! . 2 . ! 4 . . ! . . . !! . . 7 ! . . . ! . . 6 !! 6 . 8 ! . . . ! . 1 5 !+-------+-------+-------+! . . . ! 5 . 4 ! . 6 1 !! . . . ! . 9 . ! . . 2 !! . 6 . ! . 1 2 ! . 9 . !+-------+-------+-------+! . . . ! . . 5 ! 1 . . !! 5 . . ! . . . ! . 2 4 !! 9 . 1 ! 2 4 . ! . . . !+-------+-------+-------+.2.4.......7.....66.8....15...5.4.61....9...2.6..12.9......51..5......249.124....;236;29694SER = 11.7`

If we try to find an anti-tridagon pattern just after Singles, we find two, one with 9 guardians and one with 12:
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`+----------------------+----------------------+----------------------+! 13     2      359    ! 4      35678  136789 ! 3789   378    3789   !! 134    13459  7      ! 1389   2358   1389   ! 23489  348    6      !! 6      349    8      ! 379    237    379    ! 23479  1      5      !+----------------------+----------------------+----------------------+! 2378   3789   239    ! 5      378    4      ! 378    6      1      !! 13478  134578 345    ! 3678   9      3678   ! 34578  34578  2      !! 3478   6      345    ! 378    1      2      ! 34578  9      378    !+----------------------+----------------------+----------------------+! 23478  3478   2346   ! 36789  3678   5      ! 1      378    3789   !! 5      378    36     ! 136789 3678   136789 ! 36789  2      4      !! 9      378    1      ! 2      4      3678   ! 35678  3578   378    !+----------------------+----------------------+----------------------+OR9-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:     b5, with cells: r4c5, r5c6, r6c4     b6, with cells: r4c7, r5c8, r6c9     b8, with cells: r8c5, r9c6, r7c4     b9, with cells: r8c7, r9c9, r7c8with 9 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c4 n9r7c4 n6r8c5 n6r8c7 n9r8c7 n6r9c6OR12-anti-tridagon[12] (type antidiag) for digits 7, 8 and 3 in blocks:     b5, with cells: r4c5, r5c6, r6c4     b6, with cells: r4c7, r5c8, r6c9     b8, with cells: r7c5, r9c6, r8c4     b9, with cells: r7c9, r9c8, r8c7with 12 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c5 n9r7c9 n1r8c4 n6r8c4 n9r8c4 n6r8c7 n9r8c7 n6r9c6 n5r9c8`

In instead of Singles, we allow Subsets + Finned Fish (IMO, the minimal reasonable thing to do before looking for any anti-tridagon), the situation is hardly better:
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`hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3   +----------------------+----------------------+----------------------+   ! 13     2      359    ! 4      35678  136789 ! 3789   378    3789   !   ! 134    15     7      ! 1389   2358   1389   ! 23489  348    6      !   ! 6      349    8      ! 379    237    379    ! 23479  1      5      !   +----------------------+----------------------+----------------------+   ! 2378   3789   239    ! 5      378    4      ! 378    6      1      !   ! 13478  15     345    ! 3678   9      3678   ! 34578  34578  2      !   ! 3478   6      345    ! 378    1      2      ! 34578  9      378    !   +----------------------+----------------------+----------------------+   ! 23478  3478   2346   ! 36789  3678   5      ! 1      378    3789   !   ! 5      378    36     ! 136789 3678   136789 ! 36789  2      4      !   ! 9      378    1      ! 2      4      3678   ! 35678  3578   378    !   +----------------------+----------------------+----------------------+OR9-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:        b5, with cells: r4c5, r5c6, r6c4        b6, with cells: r4c7, r5c8, r6c9        b8, with cells: r8c5, r9c6, r7c4        b9, with cells: r8c7, r9c9, r7c8with 9 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c4 n9r7c4 n6r8c5 n6r8c7 n9r8c7 n6r9c6OR12-anti-tridagon[12] (type antidiag) for digits 7, 8 and 3 in blocks:        b5, with cells: r4c5, r5c6, r6c4        b6, with cells: r4c7, r5c8, r6c9        b8, with cells: r7c5, r9c6, r8c4        b9, with cells: r7c9, r9c8, r8c7with 12 guardians: n6r5c6 n4r5c8 n5r5c8 n6r7c5 n9r7c9 n1r8c4 n6r8c4 n9r8c4 n6r8c7 n9r8c7 n6r9c6 n5r9c8`

The situation is quite better is one starts with Subsets + Finned Fish + Whips[≤4]: there remains only one anti-tridagon-pattern and it has only 5 guardians:
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`hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3whip[3]: r1c1{n3 n1} - r2c2{n1 n5} - c5n5{r2 .} ==> r1c5≠3whip[4]: c5n6{r8 r1} - r1n5{c5 c3} - r2c2{n5 n1} - c4n1{r2 .} ==> r8c4≠6whip[4]: r3n4{c7 c2} - b1n9{r3c2 r1c3} - b3n9{r1c9 r3c7} - c7n2{r3 .} ==> r2c7≠4whip[4]: r2c2{n1 n5} - r1n5{c3 c5} - r1n6{c5 c6} - r1n1{c6 .} ==> r2c1≠1whip[3]: c7n2{r2 r3} - r3n4{c7 c2} - r2c1{n4 .} ==> r2c7≠3whip[4]: r2n9{c6 c7} - r3n9{c7 c2} - r3n4{c2 c7} - c7n2{r3 .} ==> r1c6≠9whip[4]: r3n7{c6 c7} - r3n4{c7 c2} - b1n9{r3c2 r1c3} - r1n5{c3 .} ==> r1c5≠7whip[4]: r6n5{c7 c3} - r1n5{c3 c5} - r1n6{c5 c6} - r9n6{c6 .} ==> r9c7≠5hidden-single-in-a-block ==> r9c8=5whip[4]: r1c1{n3 n1} - r2c2{n1 n5} - r1n5{c3 c5} - r1n6{c5 .} ==> r1c6≠3whip[3]: r3n4{c7 c2} - r2c1{n4 n3} - b2n3{r2c6 .} ==> r3c7≠3whip[4]: r3n4{c7 c2} - b1n9{r3c2 r1c3} - b3n9{r1c9 r2c7} - c7n2{r2 .} ==> r3c7≠7whip[1]: r3n7{c6 .} ==> r1c6≠7whip[4]: r2n2{c7 c5} - c5n5{r2 r1} - r1n8{c5 c6} - r1n6{c6 .} ==> r2c7≠8whip[4]: r1n6{c6 c5} - r1n5{c5 c3} - r2c2{n5 n1} - b2n1{r2c4 .} ==> r1c6≠8whip[4]: r2c1{n3 n4} - r2c8{n4 n8} - r1n8{c9 c5} - c5n5{r1 .} ==> r2c5≠3whip[4]: r1n6{c5 c6} - r1n1{c6 c1} - r2c2{n1 n5} - c5n5{r2 .} ==> r1c5≠8whip[1]: r1n8{c9 .} ==> r2c8≠8naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c6≠3, r2c4≠3whip[1]: b2n3{r3c6 .} ==> r3c2≠3whip[3]: c3n2{r7 r4} - r4n9{c3 c2} - c2n3{r4 .} ==> r7c3≠3whip[3]: r4n2{c1 c3} - c3n9{r4 r1} - b1n3{r1c3 .} ==> r4c1≠3whip[4]: c3n6{r8 r7} - c3n2{r7 r4} - r4n9{c3 c2} - c2n3{r4 .} ==> r8c3≠3singles ==> r8c3=6, r9c7=6   +-------------------+-------------------+-------------------+   ! 13    2     359   ! 4     56    16    ! 3789  378   3789  !   ! 34    15    7     ! 189   258   189   ! 29    34    6     !   ! 6     49    8     ! 379   237   379   ! 249   1     5     !   +-------------------+-------------------+-------------------+   ! 278   3789  239   ! 5     378   4     ! 378   6     1     !   ! 13478 15    345   ! 3678  9     3678  ! 34578 3478  2     !   ! 3478  6     345   ! 378   1     2     ! 34578 9     378   !   +-------------------+-------------------+-------------------+   ! 23478 3478  24    ! 36789 3678  5     ! 1     378   3789  !   ! 5     378   6     ! 13789 378   13789 ! 3789  2     4     !   ! 9     378   1     ! 2     4     378   ! 6     5     378   !   +-------------------+-------------------+-------------------+OR5-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:        b5, with cells: r4c5, r5c6, r6c4        b6, with cells: r4c7, r5c8, r6c9        b8, with cells: r8c5, r9c6, r7c4        b9, with cells: r8c7, r9c9, r7c8with 5 guardians: n6r5c6 n4r5c8 n6r7c4 n9r7c4 n9r8c7`

Now, the first thing one may want to try is wether the puzzle can be solved in W4+OR5FW4. Unfortunately, the answer is negative.
From this point on, one can progressively increase the max sizes of allowed chains, independently the size of normal chains (bivalue, z-, whips...) and the size of ORk-Forcing-Whips.
For this puzzle, I found a good balance with all chains ≤ 6 and Forcing-Whips ≤ 9:
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`hidden-pairs-in-a-column: c2{n1 n5}{r2 r5} ==> r5c2≠8, r5c2≠7, r5c2≠4, r5c2≠3, r2c2≠9, r2c2≠4, r2c2≠3biv-chain[3]: r1c1{n3 n1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} ==> r1c5≠3biv-chain[4]: r1c1{n3 n1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} - b2n6{r1c5 r1c6} ==> r1c6≠3biv-chain[4]: r1n1{c1 c6} - b2n6{r1c6 r1c5} - b2n5{r1c5 r2c5} - r2c2{n5 n1} ==> r2c1≠1biv-chain[3]: r2c1{n3 n4} - r3n4{c2 c7} - b3n2{r3c7 r2c7} ==> r2c7≠3whip[3]: r3n4{c7 c2} - r2c1{n4 n3} - b2n3{r2c4 .} ==> r3c7≠3biv-chain[4]: r1n1{c6 c1} - r2c2{n1 n5} - b2n5{r2c5 r1c5} - b2n6{r1c5 r1c6} ==> r1c6≠7, r1c6≠8, r1c6≠9biv-chain[3]: r8n1{c4 c6} - r1c6{n1 n6} - b5n6{r5c6 r5c4} ==> r8c4≠6z-chain[3]: r1n8{c9 c5} - c5n5{r1 r2} - r2n2{c5 .} ==> r2c7≠8biv-chain[4]: r1n5{c5 c3} - r2c2{n5 n1} - r1n1{c1 c6} - b2n6{r1c6 r1c5} ==> r1c5≠7, r1c5≠8whip[1]: r1n8{c9 .} ==> r2c8≠8whip[1]: r1n7{c9 .} ==> r3c7≠7naked-pairs-in-a-row: r2{c1 c8}{n3 n4} ==> r2c7≠4, r2c6≠3, r2c5≠3, r2c4≠3whip[1]: b2n3{r3c6 .} ==> r3c2≠3z-chain[3]: c2n3{r9 r4} - r4n9{c2 c3} - c3n2{r4 .} ==> r7c3≠3z-chain[3]: b1n3{r2c1 r1c3} - c3n9{r1 r4} - r4n2{c3 .} ==> r4c1≠3biv-chain[4]: r6n5{c7 c3} - r1n5{c3 c5} - b2n6{r1c5 r1c6} - r9n6{c6 c7} ==> r9c7≠5hidden-single-in-a-block ==> r9c8=5z-chain[4]: c2n3{r9 r4} - r4n9{c2 c3} - c3n2{r4 r7} - c3n6{r7 .} ==> r8c3≠3naked-single ==> r8c3=6hidden-single-in-a-block ==> r9c7=6z-chain[5]: b6n5{r6c7 r5c7} - b6n4{r5c7 r5c8} - r2n4{c8 c1} - r6n4{c1 c3} - r6n5{c3 .} ==> r6c7≠3, r6c7≠8, r6c7≠7whip[6]: r5n6{c4 c6} - r1n6{c6 c5} - r1n5{c5 c3} - r5c3{n5 n4} - r6n4{c3 c7} - r6n5{c7 .} ==> r5c4≠3   +-------------------+-------------------+-------------------+   ! 13    2     359   ! 4     56    16    ! 3789  378   3789  !   ! 34    15    7     ! 189   258   189   ! 29    34    6     !   ! 6     49    8     ! 379   237   379   ! 249   1     5     !   +-------------------+-------------------+-------------------+   ! 278   3789  239   ! 5     378   4     ! 378   6     1     !   ! 13478 15    345   ! 678   9     3678  ! 34578 3478  2     !   ! 3478  6     345   ! 378   1     2     ! 45    9     378   !   +-------------------+-------------------+-------------------+   ! 23478 3478  24    ! 36789 3678  5     ! 1     378   3789  !   ! 5     378   6     ! 13789 378   13789 ! 3789  2     4     !   ! 9     378   1     ! 2     4     378   ! 6     5     378   !   +-------------------+-------------------+-------------------+OR5-anti-tridagon[12] (type diag) for digits 7, 8 and 3 in blocks:        b5, with cells: r4c5, r5c6, r6c4        b6, with cells: r4c7, r5c8, r6c9        b8, with cells: r8c5, r9c6, r7c4        b9, with cells: r8c7, r9c9, r7c8with 5 guardians: n6r5c6 n4r5c8 n6r7c4 n9r7c4 n9r8c7OR5-forcing-whip-elim[9] based on OR5-anti-tridagon[12] for n9r8c7, n4r5c8, n9r7c4, n6r5c6 and  n6r7c4:   || n9r8c7 -   || n4r5c8 - partial-whip[1]: c7n4{r6 r3} -   || n9r7c4 - partial-whip[1]: c9n9{r7 r1} -   || n6r5c6 - partial-whip[3]: r1n6{c6 c5} - r1n5{c5 c3} - b1n9{r1c3 r3c2} -   || n6r7c4 - partial-whip[3]: c5n6{r7 r1} - r1n5{c5 c3} - b1n9{r1c3 r3c2} - ==> r3c7≠9OR5-forcing-whip-elim[9] based on OR5-anti-tridagon[12] for n9r7c4, n9r8c7, n4r5c8, n6r5c6 and  n6r7c4:   || n9r7c4 - partial-whip[1]: c9n9{r7 r1} -   || n9r8c7 - partial-whip[1]: c9n9{r7 r1} -   || n4r5c8 - partial-whip[2]: r2n4{c8 c1} - r3c2{n4 n9} -   || n6r5c6 - partial-whip[2]: r1n6{c6 c5} - r1n5{c5 c3} -   || n6r7c4 - partial-whip[2]: c5n6{r7 r1} - r1n5{c5 c3} - ==> r1c3≠9singles ==> r3c2=9, r2c1=4, r2c8=3, r3c7=4, r6c7=5, r2c7=2, r3c5=2, r5c8=4, r6c3=4, r7c3=2, r4c1=2, r7c2=4, r4c3=9z-chain[5]: c5n6{r7 r1} - r1c6{n6 n1} - r1c1{n1 n3} - r7c1{n3 n8} - r7c8{n8 .} ==> r7c5≠7z-chain[5]: c5n6{r7 r1} - r1c6{n6 n1} - r1c1{n1 n3} - r7c1{n3 n7} - r7c8{n7 .} ==> r7c5≠8z-chain[4]: r7c5{n3 n6} - r1n6{c5 c6} - c6n1{r1 r2} - c6n9{r2 .} ==> r8c6≠3z-chain[5]: r3c4{n7 n3} - r6c4{n3 n8} - r4c5{n8 n3} - r7c5{n3 n6} - c4n6{r7 .} ==> r5c4≠7biv-chain[4]: r5c4{n8 n6} - r7n6{c4 c5} - r1c5{n6 n5} - r2c5{n5 n8} ==> r4c5≠8, r2c4≠8biv-chain[3]: r8n1{c6 c4} - r2c4{n1 n9} - c6n9{r2 r8} ==> r8c6≠7, r8c6≠8whip[3]: r4n8{c7 c2} - b7n8{r8c2 r7c1} - c8n8{r7 .} ==> r1c7≠8biv-chain[3]: r7c8{n7 n8} - r1n8{c8 c9} - c9n9{r1 r7} ==> r7c9≠7whip[6]: r4n8{c7 c2} - r9n8{c2 c6} - r5n8{c6 c4} - r5n6{c4 c6} - c6n3{r5 r3} - c6n7{r3 .} ==> r8c7≠8whip[1]: c7n8{r5 .} ==> r6c9≠8whip[4]: r4n8{c7 c2} - r6n8{c1 c4} - r8n8{c4 c5} - c5n7{r8 .} ==> r4c7≠7z-chain[3]: b6n7{r5c7 r6c9} - r9n7{c9 c2} - r4n7{c2 .} ==> r5c6≠7z-chain[3]: c5n3{r8 r4} - b5n7{r4c5 r6c4} - r3c4{n7 .} ==> r8c4≠3, r7c4≠3whip[4]: r5n7{c1 c7} - r6c9{n7 n3} - b5n3{r6c4 r4c5} - r7n3{c5 .} ==> r5c1≠3whip[4]: b5n7{r4c5 r6c4} - r6c9{n7 n3} - r7n3{c9 c1} - c2n3{r8 .} ==> r4c5≠3naked-single ==> r4c5=7whip[1]: c2n7{r9 .} ==> r7c1≠7whip[1]: c5n3{r8 .} ==> r9c6≠3finned-x-wing-in-rows: n3{r9 r4}{c2 c9} ==> r6c9≠3singles ==> r6c9=7, r5c1=7, r5c2=1, r2c2=5, r1c3=3, r1c1=1, r1c6=6, r1c5=5, r5c3=5, r2c5=8, r8c5=3, r7c5=6, r5c4=6finned-x-wing-in-rows: n8{r8 r6}{c4 c2} ==> r4c2≠8stte`
denis_berthier
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