The New Sudoku Players' Forum

Website · by **denis_berthier** » Mon Aug 01, 2022 8:27 am

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ORk-Forcing-Whips

I introduced ordinary Forcing-Whips (based on a bivalue relation) long ago in PBCS1
I introduced Tridagon-Forcing-Whips (based on a tridagon link) more recently here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859-31.html

The more general idea of ORk-Forcing-Whips is trivial, but there are small subtleties about complexity.

When an OR relation between k candidates (Z1 Z2 ... Zk, k > 1) has been proven, if there are k partial whips (of respective lengths p1, p2, ... pk), with respective targets Z1, Z2 ... Zk, then:
- if the last (right-linking) candidate of all the k partial whips is the same, then one can assert it as True;
- if a candidate is linked to the ends (i.e. the last right-linking candidates) of all the k partial whips, then it can be eliminated.
This is a trivial case of "reasoning by cases".
(For simplicity of formulation, a direct contradiction link between two candidates is considered as a partial-whip[0])

As k streams of reasoning must be followed in parallel and complexity is more or less exponential in the length of chains, the global length of the ORk-Forcing-Whip can be defined consistently in two and only two ways:
1) either consider that the OK-k relation has been proven independently and doesn't have to be counted in the complexity of the Forcing-Whip, in which case the total length is 1 + p1 + p2 ...+ pk (the +1 for consistency with the basic forcing-whips.
2) or consider that the OK-k relation doesn't lead to any elimination and must therefore be counted in the complexity of the Forcing-Whip, in which case the total length must be (length of the ORk relation) + p1 + p2 ...+ pk.

In the definition of Tridagon-Forcing-Whips, I have adopted the second view.

In the definition of the general ORk-Forcing-Whips, I'll adopt the first view. This corresponds to an extended resolution model, where OR relations can be asserted as intermediate results (in addition to only values and candidates). This is mainly intended for use with exotic patterns that lead to ORk relations. Typically in such cases, one wants to give some priority to the exotic pattern and to what can be deduced from it (while maintaining some consistency with the complexities of the ingredients, i.e. partial-whips)
(For Tridagon-Forcing-Whips, one will be able to choose any of the two views.)

[Edit]:added ORk-Contrad-Whips in the title.
[Edit2]:added ORk-Whips in the title.

Website · by **denis_berthier** » Thu Aug 11, 2022 5:09 am

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OR-k-Forcing-Whips based on anti-tridagon OR-k relations

Examples can be found in the following links:
tridagon + anti-tridagon OR3- and OR4- Forcing-Whips: http://forum.enjoysudoku.com/tanngrisnir-and-tanngnjostr-ser-11-7-te3-id-19252-t40126-3.html
anti-tridagon OR4-Forcing-Whips: http://forum.enjoysudoku.com/njor-r-ser-10-4-te3-id-289988-t40129-3.html
anti-tridagon OR4-Forcing-Whips: http://forum.enjoysudoku.com/the-tridagon-rule-t39859-100.html
anti-tridagon OR5-Forcing-Whips: http://forum.enjoysudoku.com/vor-r-ser-10-7-te3-id-155557-t40135-2.html
anti-tridagon OR5-Forcing-Whips: http://forum.enjoysudoku.com/mith-s-te3-min-expand-62886-63137-t40199-2.html
anti-tridagon OR5-Forcing-Whips in a puzzle with lots of guardians: http://forum.enjoysudoku.com/the-tridagon-rule-t39859-104.html

Website · by **denis_berthier** » Thu Aug 11, 2022 5:23 am

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ORk-Forcing-Whips based on bivalue, trivalue... relations

An obvious special case of ORk-Forcing-Whips appears when their base relation is a mere bivalue, trivalue... relation.
Note that, in such a case, the conventional length (1 + p1 + p2 +...) of the ORk-Forcing-Whip coincides with its normal length in CSP-Rules.
Considering the importance of these special cases, I've given them their own names:
Forcing2-Whips (equivalent to the old Forcing-whips) when k=2, with associated rating FW or F2W
Forcing3-Whips when k=3, with associated rating F3W
Forcing4-Whips when k=3, with associated rating F4W....

They can be selected in the SudoRules configuration file by setting global variable ?*Forcing2-Whips * (resp. ?*Forcing3-Whips *, ?*Forcing4-Whips *,...) to TRUE.
Note the implications (necessary for consistency of the ratings): Forcing4-Whips[n] => Forcing3-Whips[n] => Forcing2-Whips[n] =>Whips[n]

A first result of these new generic rules is an addition to the comparisons of ratings in the cbg-000 collection.
See here: http://forum.enjoysudoku.com/pattern-based-constraint-satisfaction-2nd-3rd-eds-t32567-15.html
and here for all the details: https://github.com/denis-berthier/CSP-Rules-Examples/tree/main/Sudoku/cbg-000

Website · by **denis_berthier** » Tue Sep 06, 2022 5:49 am

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ORk-whips[1], k≥2
I should have mentioned a special (but not rare) case of ORk-Forcing-Whips that appears when each of the k partial whips has length 0.
Though still an ORk-Forcing-Whip[1], the pattern can also be considered as a mere ORk-Whip[1] (by reversing the direction we look at it), because it has the main property of whips[1]: the target is linked to all the candidates in the ORk relation.

Example (with k=2) where this is enough to solve a puzzle with SER 11.7: puzzle #6530 in mith's list of 63,137 min-expand puzzles in T&E(3).

Code: Select all: +-------+-------+-------+ ! . 2 3 ! . . . ! 7 . 9 ! ! 4 5 . ! . . . ! . . . ! ! . . 9 ! 2 . 3 ! . 4 5 ! +-------+-------+-------+ ! 2 9 . ! . . . ! 5 7 . ! ! 3 . . ! . . . ! 4 . 2 ! ! . 4 . ! . 2 . ! . 9 3 ! +-------+-------+-------+ ! 5 3 . ! 9 . 8 ! . . . ! ! . 7 . ! 6 . . ! . . . ! ! . . . ! . 4 2 ! . . . ! +-------+-------+-------+ .23...7.945.........92.3.4529....57.3.....4.2.4..2..9353.9.8....7.6.........42...;1400;22497 SER = 11.7

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 168 2 3 ! 1458 1568 1456 ! 7 168 9 ! ! 4 5 1678 ! 178 16789 1679 ! 12368 12368 168 ! ! 1678 168 9 ! 2 1678 3 ! 168 4 5 ! +----------------------+----------------------+----------------------+ ! 2 9 168 ! 1348 1368 146 ! 5 7 168 ! ! 3 168 15678 ! 1578 156789 15679 ! 4 168 2 ! ! 1678 4 15678 ! 1578 2 1567 ! 168 9 3 ! +----------------------+----------------------+----------------------+ ! 5 3 1246 ! 9 17 8 ! 126 126 1467 ! ! 189 7 1248 ! 6 135 15 ! 12389 12358 148 ! ! 1689 168 168 ! 1357 4 2 ! 13689 13568 1678 ! +----------------------+----------------------+----------------------+ 193 candidates.

Code: Select all: hidden-pairs-in-a-column: c3{n2 n4}{r7 r8} ==> r8c3≠8, r8c3≠1, r7c3≠6, r7c3≠1 whip[1]: r7n6{c9 .} ==> r9c7≠6, r9c8≠6, r9c9≠6 hidden-pairs-in-a-row: r2{n2 n3}{c7 c8} ==> r2c8≠8, r2c8≠6, r2c8≠1, r2c7≠8, r2c7≠6, r2c7≠1 z-chain[3]: c9n4{r8 r7} - r7n7{c9 c5} - r7n1{c5 .} ==> r8c9≠1 z-chain[3]: c8n5{r9 r8} - r8c6{n5 n1} - b7n1{r8c1 .} ==> r9c8≠1 t-whip[4]: r7n1{c9 c5} - b8n7{r7c5 r9c4} - r9n5{c4 c8} - r9n3{c8 .} ==> r9c7≠1 t-whip[5]: r7n1{c9 c5} - b8n7{r7c5 r9c4} - r9n5{c4 c8} - r9n3{c8 c7} - c7n9{r9 .} ==> r8c7≠1 +----------------------+----------------------+----------------------+ ! 168 2 3 ! 1458 1568 1456 ! 7 168 9 ! ! 4 5 1678 ! 178 16789 1679 ! 23 23 168 ! ! 1678 168 9 ! 2 1678 3 ! 168 4 5 ! +----------------------+----------------------+----------------------+ ! 2 9 168 ! 1348 1368 146 ! 5 7 168 ! ! 3 168 15678 ! 1578 156789 15679 ! 4 168 2 ! ! 1678 4 15678 ! 1578 2 1567 ! 168 9 3 ! +----------------------+----------------------+----------------------+ ! 5 3 24 ! 9 17 8 ! 126 126 1467 ! ! 189 7 24 ! 6 135 15 ! 2389 12358 48 ! ! 1689 168 168 ! 1357 4 2 ! 389 358 178 ! +----------------------+----------------------+----------------------+

Code: Select all: OR2-anti-tridagon[12] (type antidiag) for digits 6, 8 and 1 in blocks: b1, with cells: r1c1, r2c3, r3c2 b3, with cells: r1c8, r2c9, r3c7 b4, with cells: r6c1, r4c3, r5c2 b6, with cells: r6c7, r4c9, r5c8 with 2 guardians: n7r2c3 n7r6c1 OR2-whip[1] based on OR2-anti-tridagon[12] for n7r2c3 and n7r6c1: || n7r2c3 - || n7r6c1 - ==> r6c3≠7, r5c3≠7

The end is easy:

Code: Select all: singles ==> r6c1=7, r2c3=7, r3c5=7, r7c5=1, r8c6=5, r8c5=3, r9c4=7, r5c6=7, r5c5=9, r2c6=9, r1c5=5, r7c9=7, r8c9=4, r8c3=2, r7c3=4, r4c4=3, r4c6=4, r1c4=4, r9c8=5, r9c7=3, r2c7=2, r2c8=3, r7c7=6, r7c8=2, r8c7=9, r9c1=9 whip[1]: c1n6{r3 .} ==> r3c2≠6 hidden-single-in-a-row ==> r3c1=6 whip[1]: b2n8{r2c5 .} ==> r2c9≠8 x-wing-in-rows: n8{r1 r8}{c1 c8} ==> r5c8≠8 x-wing-in-columns: n6{c5 c9}{r2 r4} ==> r4c3≠6 finned-x-wing-in-columns: n1{c6 c7}{r6 r1} ==> r1c8≠1 biv-chain[2]: c7n1{r6 r3} - r2n1{c9 c4} ==> r6c4≠1 finned-swordfish-in-columns: n1{c8 c1 c4}{r5 r8 r1} ==> r1c6≠1 stte

Website · by **denis_berthier** » Tue Sep 06, 2022 10:43 am

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ORk-Contrad-Whips

When an application-specific ORk relation between k candidates has been proven, the most natural way to take advantage of it is via the generic ORk-Forcing-Whips defined in the first post of this thread.
In this case, the ORk relation serves as a natural starting point for the forcing whip.

However, there is also another way of using it to eliminate candidates. AFAIK, the first time it has been used this way is by DEFISE, in this puzzle: http://forum.enjoysudoku.com/23427-more-anti-tridagon-guardians-t40295.html

The formal definition is almost obvious:
given an ORk relation between candidates C1, C2, ... Ck, an ORk-Contradiction-Whip[n] (or ORk-Contrad-Whip[n] for short) based on the ORk relation, with target Z, is an ordinary partial-whip[n-1] W with target Z, such that each of the Ck's is linked to (at least) a right-linking candidate of W.
In such a situation, the target Z can be eliminated.

Proof: obvious. If Z was True, all the right-linking candidates would be True and there would be no possibility for any of the Ci's to be True.
Here, the ORk relation plays the same role as an n-th CSP-Variable.

Note: obviously, ORk-Forcing-Whips and ORk-Contrad-Whips could be extended to g-Whips, Braids and g-Braids.

Here is a simple example, with 2 OR2CW3 eliminations: #950 in mith's list of 63137 min-expands

Code: Select all: +-------+-------+-------+ ! . 2 . ! . . . ! . . 9 ! ! . . . ! . 8 9 ! 1 3 2 ! ! . . . ! . . . ! 5 6 . ! +-------+-------+-------+ ! 2 3 . ! 8 9 . ! 4 7 . ! ! . 8 7 ! . 2 4 ! 3 9 . ! ! . . . ! 3 . 7 ! . . . ! +-------+-------+-------+ ! 3 . 8 ! . 7 . ! . . . ! ! . 9 2 ! . 4 . ! . . . ! ! 7 4 . ! . . . ! . . . ! +-------+-------+-------+ .2......9....89132......56.23.89.47..87.2439....3.7...3.8.7.....92.4....74.......;218;256011

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 14568 2 13456 ! 14567 1356 1356 ! 78 48 9 ! ! 456 567 456 ! 4567 8 9 ! 1 3 2 ! ! 1489 17 1349 ! 1247 13 123 ! 5 6 478 ! +----------------------+----------------------+----------------------+ ! 2 3 156 ! 8 9 156 ! 4 7 156 ! ! 156 8 7 ! 156 2 4 ! 3 9 156 ! ! 14569 156 14569 ! 3 156 7 ! 268 1258 1568 ! +----------------------+----------------------+----------------------+ ! 3 156 8 ! 12569 7 1256 ! 269 1245 1456 ! ! 156 9 2 ! 156 4 13568 ! 678 158 135678 ! ! 7 4 156 ! 12569 1356 123568 ! 2689 1258 13568 ! +----------------------+----------------------+----------------------+ 184 candidates.

Code: Select all: hidden-pairs-in-a-row: r6{n4 n9}{c1 c3} ==> r6c3≠6, r6c3≠5, r6c3≠1, r6c1≠6, r6c1≠5, r6c1≠1 +----------------------+----------------------+----------------------+ ! 14568 2 13456 ! 14567 1356 1356 ! 78 48 9 ! ! 456 567 456 ! 4567 8 9 ! 1 3 2 ! ! 1489 17 1349 ! 1247 13 123 ! 5 6 478 ! +----------------------+----------------------+----------------------+ ! 2 3 156 ! 8 9 156 ! 4 7 156 ! ! 156 8 7 ! 156 2 4 ! 3 9 156 ! ! 49 156 49 ! 3 156 7 ! 268 1258 1568 ! +----------------------+----------------------+----------------------+ ! 3 156 8 ! 12569 7 1256 ! 269 1245 1456 ! ! 156 9 2 ! 156 4 13568 ! 678 158 135678 ! ! 7 4 156 ! 12569 1356 123568 ! 2689 1258 13568 ! +----------------------+----------------------+----------------------+ OR2-anti-tridagon[12] (type antidiag) for digits 1, 5 and 6 in blocks: b4, with cells: r4c3, r5c1, r6c2 b5, with cells: r4c6, r5c4, r6c5 b7, with cells: r9c3, r8c1, r7c2 b8, with cells: r9c5, r8c4, r7c6 with 2 guardians: n2r7c6 n3r9c5 OR2-contrad-whip[3] based on OR2-anti-tridagon[12] for n3r9c5 and n2r7c6: partial-whip[2]: r3c5{n1 n3} - r3c6{n3 n2} - ==> r3c1≠1 OR2-contrad-whip[3] based on OR2-anti-tridagon[12] for n3r9c5 and n2r7c6: partial-whip[2]: r3c5{n1 n3} - r3c6{n3 n2} - ==> r3c2≠1

Code: Select all: singles ==> r3c2=7, r2c4=7, r1c7=7, r8c9=7, r9c9=3, r8c6=3, r9c6=8 whip[1]: r2n4{c3 .} ==> r1c1≠4, r1c3≠4, r3c1≠4, r3c3≠4 whip[1]: r2n5{c3 .} ==> r1c1≠5, r1c3≠5 whip[1]: r2n6{c3 .} ==> r1c1≠6, r1c3≠6 finned-x-wing-in-columns: n1{c2 c9}{r7 r6} ==> r6c8≠1 whip[1]: c8n1{r9 .} ==> r7c9≠1 +-------------------+-------------------+-------------------+ ! 18 2 13 ! 1456 1356 156 ! 7 48 9 ! ! 456 56 456 ! 7 8 9 ! 1 3 2 ! ! 89 7 139 ! 124 13 12 ! 5 6 48 ! +-------------------+-------------------+-------------------+ ! 2 3 156 ! 8 9 156 ! 4 7 156 ! ! 156 8 7 ! 156 2 4 ! 3 9 156 ! ! 49 156 49 ! 3 156 7 ! 268 258 1568 ! +-------------------+-------------------+-------------------+ ! 3 156 8 ! 12569 7 1256 ! 269 1245 456 ! ! 156 9 2 ! 156 4 3 ! 68 158 7 ! ! 7 4 156 ! 12569 156 8 ! 269 125 3 ! +-------------------+-------------------+-------------------+ tridagon for digits 1, 5 and 6 in blocks: b8, with cells: r7c6 (target cell), r8c4, r9c5 b7, with cells: r7c2, r8c1, r9c3 b5, with cells: r4c6, r5c4, r6c5 b4, with cells: r4c3, r5c1, r6c2 ==> r7c6≠1,5,6

Code: Select all: singles ==> r7c6=2, r3c6=1, r3c5=3, r3c3=9, r3c1=8, r1c1=1, r1c3=3, r3c9=4, r1c8=8, r3c4=2, r6c3=4, r6c1=9, r2c1=4, r8c7=8, r6c9=8, r7c8=4, r1c4=4 finned-x-wing-in-rows: n1{r7 r6}{c2 c4} ==> r5c4≠1 singles ==> r6c5=1, r4c3=1, r5c9=1, r7c2=1 naked-pairs-in-a-row: r9{c3 c5}{n5 n6} ==> r9c8≠5, r9c7≠6, r9c4≠6, r9c4≠5 whip[1]: b9n6{r7c9 .} ==> r7c4≠6 finned-x-wing-in-columns: n5{c8 c1}{r8 r6} ==> r6c2≠5 stte

Here is a harder example, with 7 OR3/4-CW eliminations: #25911 in the same list

Code: Select all: +-------+-------+-------+ ! 1 . 3 ! . . . ! . . . ! ! . 5 . ! 1 . 9 ! . . 6 ! ! 6 9 . ! 2 3 . ! . . . ! +-------+-------+-------+ ! 2 . . ! . . . ! 4 . 8 ! ! 3 6 . ! . . . ! 1 2 7 ! ! . . . ! . . 2 ! . 9 5 ! +-------+-------+-------+ ! 5 . 6 ! . 2 3 ! . . . ! ! . . 1 ! 6 9 . ! . 5 . ! ! 9 . . ! 5 . 1 ! . . . ! +-------+-------+-------+ 1.3.......5.1.9..669.23....2.....4.836....127.....2.955.6.23.....169..5.9..5.1...;5250;73296 SER = 11.0

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 1 2478 3 ! 478 45678 45678 ! 25789 478 249 ! ! 478 5 2478 ! 1 478 9 ! 2378 3478 6 ! ! 6 9 478 ! 2 3 4578 ! 578 1478 14 ! +-------------------+-------------------+-------------------+ ! 2 17 579 ! 379 1567 567 ! 4 36 8 ! ! 3 6 4589 ! 489 458 458 ! 1 2 7 ! ! 478 1478 478 ! 3478 14678 2 ! 36 9 5 ! +-------------------+-------------------+-------------------+ ! 5 478 6 ! 478 2 3 ! 789 1478 149 ! ! 478 23478 1 ! 6 9 478 ! 278 5 234 ! ! 9 23478 2478 ! 5 478 1 ! 2678 4678 234 ! +-------------------+-------------------+-------------------+ 169 candidates.

Code: Select all: hidden-pairs-in-a-column: c3{n5 n9}{r4 r5} ==> r5c3≠8, r5c3≠4, r4c3≠7 whip[1]: r5n4{c6 .} ==> r6c4≠4, r6c5≠4 whip[1]: r5n8{c6 .} ==> r6c4≠8, r6c5≠8 z-chain[5]: c2n3{r8 r9} - b7n2{r9c2 r9c3} - r2n2{c3 c7} - r8n2{c7 c9} - r8n3{c9 .} ==> r8c2≠4, r8c2≠8, r8c2≠7 whip[5]: c8n6{r9 r4} - r4n3{c8 c4} - r6c4{n3 n7} - r7n7{c4 c2} - r4n7{c2 .} ==> r9c8≠7 +-------------------+-------------------+-------------------+ ! 1 2478 3 ! 478 45678 45678 ! 25789 478 249 ! ! 478 5 2478 ! 1 478 9 ! 2378 3478 6 ! ! 6 9 478 ! 2 3 4578 ! 578 1478 14 ! +-------------------+-------------------+-------------------+ ! 2 17 59 ! 379 1567 567 ! 4 36 8 ! ! 3 6 59 ! 489 458 458 ! 1 2 7 ! ! 478 1478 478 ! 37 167 2 ! 36 9 5 ! +-------------------+-------------------+-------------------+ ! 5 478 6 ! 478 2 3 ! 789 1478 149 ! ! 478 23 1 ! 6 9 478 ! 278 5 234 ! ! 9 23478 2478 ! 5 478 1 ! 2678 468 234 ! +-------------------+-------------------+-------------------+

Code: Select all: OR3-anti-tridagon[12] (type diag) for digits 4, 7 and 8 in blocks: b1, with cells: r1c2, r2c1, r3c3 b2, with cells: r1c4, r2c5, r3c6 b7, with cells: r7c2, r8c1, r9c3 b8, with cells: r7c4, r8c6, r9c5 with 3 guardians: n2r1c2 n5r3c6 n2r9c3 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[3]: c3n2{r9 r2} - r1n2{c2 c7} - c7n5{r1 r3} - ==> r9c9≠2 OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[4]: r8c2{n3 n2} - c9n2{r8 r1} - r1n9{c9 c7} - c7n5{r1 r3} - ==> r9c2≠3 singles ==> r8c2=3, r9c9=3 whip[1]: r8n2{c9 .} ==> r9c7≠2 t-whip[5]: c2n2{r9 r1} - r2n2{c3 c7} - c7n3{r2 r6} - r6c4{n3 n7} - r4n7{c6 .} ==> r9c2≠7 OR3-contrad-whip[5] based on OR3-anti-tridagon[12] for n2r9c3, n5r3c6 and n2r1c2: partial-whip[4]: r2n2{c7 c3} - r1n2{c2 c9} - r1n9{c9 c7} - c7n5{r1 r3} - ==> r8c7≠2 hidden-single-in-a-block ==> r8c9=2 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠4 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠7 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: r2n2{c7 c3} - r1c2{n2 n8} - b2n8{r1c4 r3c6} - ==> r2c7≠8 OR3-contrad-whip[4] based on OR3-anti-tridagon[12] for n2r1c2, n5r3c6 and n2r9c3: partial-whip[3]: b1n2{r1c2 r2c3} - b3n2{r2c7 r1c7} - c7n5{r1 r3} - ==> r1c2≠8 stte

[Edit 1]:
Note 1: Both ORk-Forcing-Whips and ORk-Contrad-Whips are available in CSP-Rules / SudoRules, for k=1, 2, ...6 and n = 1...36
Note 2: ORk-Forcing-Whips and ORk-Contrad-Whips can be activated at the same time; they are built on the same partial-whips
Note 3: a first, quick evaluation, to be made more precise, shows ORk-Forcing-Whips are more powerful than ORk-Contrad-Whips (with same k and n)

[Edit2]: change the proof. This version is more in the spirit of whips (impossibility occurring in the final CSP / ORk).

Website · by **denis_berthier** » Fri Sep 09, 2022 10:40 am

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a reader of [PBCS] and user of CSP-Rules that prefers to remain anonymous wrote:You've always said that you don't like forcing-whips but now you are promoting ORk-Forcing-Whips. Isn't that contradictory?

Good question. And very easy to answer.

Forcing-chains in general are very inelegant, because they start with a big OR and they suppose one follows two (or more) streams of reasoning in parallel. Mathematically, they amount to reasoning by cases. With respect to T&E, they amount to having some T&E step right at the start (any OR-branching step in a chain introduces one level of T&E).

Ordinary forcing whips have no argument to make forgive this inelegance: they are not really more powerful than whips (see section 6.5.1 of [PBCS3]).

Tridagon-based ORk-Forcing-Whips are radically different:
1) in terms of resolution power (as I will soon show in the tridagon thread (http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html)
2) and in terms of focus: their OR starting point is concentrated on very specific candidates (defined by the ORk relation - i.e. in the current context, the guardians of a precise and generally unique anti-tridagon pattern).
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Website · by **denis_berthier** » Mon Sep 19, 2022 5:41 am

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ORk-whips

An ORk-whip is a pattern more general than my previously defined ORk-contrad-whips: the ORk part may appear at another place than the final (contradiction) one.
Informally speaking, an ORk-whip (based on some ORk-relation, e.g. an anti-tridagon one) is like a whip, except that some of its CSP-Variables is replaced by an ORk relation, where k-1 candidates are linked to the target or to previous right-linking candidates and the remaining one is used as a right linking-candidate for the next step.

The notation for an ORk-whip based on an ORk-relation is as follows:
- the full name is ORname-ORk-whip[n], with "n" the total length as usual, "k" the size or the OR relation and "OR name" the short name of this relation.
- I use double curly braces for the ORk part (to recall that it's not a CSP-Variable).

Example based on an anti-tridagon OR3 relation:
Trid-OR3-whip[5]: r7c1{n1 n6} - r9n6{c1 c4} - r7c6{n6 n8} - OR3{{n8r5c6 n6r3c4 | n4r1c5}} - r7c5{n4 .} ==> r7c3≠1

For a first fully developed example, see here: http://forum.enjoysudoku.com/1359-in-t-e-3-min-expands-t40363-5.html

Website · by **denis_berthier** » Mon Sep 19, 2022 5:50 am

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Here is an easier example, #1167 in mith's list of 63137 min-expand puzzle in T&E(3):

Code: Select all: +-------+-------+-------+ ! . . . ! 4 5 . ! . 8 . ! ! 4 5 . ! . . . ! 2 . 3 ! ! 6 8 . ! 3 . 2 ! 4 5 . ! +-------+-------+-------+ ! . . . ! . 3 4 ! 5 . 8 ! ! . . . ! 6 . 5 ! . . 2 ! ! . . 5 ! 8 2 . ! 6 . . ! +-------+-------+-------+ ! . 7 . ! . . . ! . . 5 ! ! . 9 . ! . . . ! 8 . 4 ! ! . . 2 ! . . 3 ! . . . ! +-------+-------+-------+ ...45..8.45....2.368.3.245.....345.8...6.5..2..582.6...7......5.9....8.4..2..3...;234;17556 SER = 11.7

Code: Select all: Resolution state after Singles (and whips[1]): +----------------------+----------------------+----------------------+ ! 12379 123 1379 ! 4 5 1679 ! 179 8 1679 ! ! 4 5 179 ! 179 16789 16789 ! 2 1679 3 ! ! 6 8 179 ! 3 179 2 ! 4 5 179 ! +----------------------+----------------------+----------------------+ ! 1279 126 1679 ! 179 3 4 ! 5 179 8 ! ! 13789 134 134789 ! 6 179 5 ! 1379 13479 2 ! ! 1379 134 5 ! 8 2 179 ! 6 13479 179 ! +----------------------+----------------------+----------------------+ ! 138 7 13468 ! 129 14689 1689 ! 139 12369 5 ! ! 135 9 136 ! 1257 167 167 ! 8 12367 4 ! ! 158 146 2 ! 1579 146789 3 ! 179 1679 1679 ! +----------------------+----------------------+----------------------+ 189 candidates.

As usual, using the simplest-first strategy, let's first do some easy cleaning:

Code: Select all: hidden-pairs-in-a-column: c3{n4 n8}{r5 r7} ==> r7c3≠6, r7c3≠3, r7c3≠1, r5c3≠9, r5c3≠7, r5c3≠3, r5c3≠1 biv-chain[3]: r9n8{c1 c5} - b8n4{r9c5 r7c5} - r7c3{n4 n8} ==> r7c1≠8 biv-chain[3]: r9n8{c5 c1} - r7c3{n8 n4} - b8n4{r7c5 r9c5} ==> r9c5≠1, r9c5≠6, r9c5≠7, r9c5≠9 z-chain[3]: r9n7{c9 c4} - c4n5{r9 r8} - r8n2{c4 .} ==> r8c8≠7 whip[1]: b9n7{r9c9 .} ==> r9c4≠7 z-chain[3]: r7c1{n1 n3} - b9n3{r7c7 r8c8} - c8n2{r8 .} ==> r7c8≠1 z-chain[4]: r7c1{n1 n3} - r8n3{c3 c8} - r8n2{c8 c4} - r8n5{c4 .} ==> r8c1≠1 +-------------------+-------------------+-------------------+ ! 12379 123 1379 ! 4 5 1679 ! 179 8 1679 ! ! 4 5 179 ! 179 16789 16789 ! 2 1679 3 ! ! 6 8 179 ! 3 179 2 ! 4 5 179 ! +-------------------+-------------------+-------------------+ ! 1279 126 1679 ! 179 3 4 ! 5 179 8 ! ! 13789 134 48 ! 6 179 5 ! 1379 13479 2 ! ! 1379 134 5 ! 8 2 179 ! 6 13479 179 ! +-------------------+-------------------+-------------------+ ! 13 7 48 ! 129 14689 1689 ! 139 2369 5 ! ! 35 9 136 ! 1257 167 167 ! 8 1236 4 ! ! 158 146 2 ! 159 48 3 ! 179 1679 1679 ! +-------------------+-------------------+-------------------+ OR3-anti-tridagon[12] for digits 7, 9 and 1 in blocks: b2, with cells: r1c6, r2c4, r3c5 b3, with cells: r1c7, r2c8, r3c9 b5, with cells: r6c6, r4c4, r5c5 b6, with cells: r6c9, r4c8, r5c7 with 3 guardians: n6r1c6 n6r2c8 n3r5c7

The part with the ORk-whips;
Trid-OR3-whip[4]: c8n2{r7 r8} - b9n3{r8c8 r7c7} - OR3{{n3r5c7 n6r2c8 | n6r1c6}} - c9n6{r1 .} ==> r7c8≠6
whip[1]: r7n6{c6 .} ==> r8c5≠6, r8c6≠6
naked-pairs-in-a-block: b8{r8c5 r8c6}{n1 n7} ==> r9c4≠1, r8c4≠7, r8c4≠1, r7c6≠1, r7c5≠1, r7c4≠1
whip[1]: b8n1{r8c6 .} ==> r8c3≠1, r8c8≠1
hidden-pairs-in-a-column: c4{n1 n7}{r2 r4} ==> r4c4≠9, r2c4≠9
whip[1]: c4n9{r9 .} ==> r7c5≠9, r7c6≠9
naked-triplets-in-a-column: c5{r3 r5 r8}{n1 n9 n7} ==> r2c5≠9, r2c5≠7, r2c5≠1
z-chain[3]: b7n1{r9c1 r9c2} - c2n6{r9 r4} - r4n2{c2 .} ==> r4c1≠1
z-chain[3]: c3n1{r3 r4} - r4n6{c3 c2} - c2n2{r4 .} ==> r1c2≠1
Trid-OR3-whip[4]: b7n3{r8c1 r8c3} - r8n6{c3 c8} - OR3{{n6r2c8 n3r5c7 | n6r1c6}} - c9n6{r1 .} ==> r5c1≠3
Trid-OR3-whip[4]: c8n2{r8 r7} - b9n3{r7c8 r7c7} - OR3{{n3r5c7 n6r2c8 | n6r1c6}} - c9n6{r1 .} ==> r8c8≠6

The end is easy:

Code: Select all: singles ==> r8c3=6,r4c2=6, r4c1=2, r1c2=2, r1c3=3, r6c1≠3 hidden-pairs-in-a-row: r6{n3 n4}{c2 c8} ==> r6c8≠9, r6c8≠7, r6c8≠1, r6c2≠1 biv-chain[2]: r7n1{c7 c1} - c2n1{r9 r5} ==> r5c7≠1 biv-chain[3]: c2n1{r5 r9} - r7c1{n1 n3} - c7n3{r7 r5} ==> r5c2≠3 singles ==> r6c2=3, r6c8=4 z-chain[3]: r4n9{c8 c3} - r2n9{c3 c6} - b5n9{r6c6 .} ==> r5c8≠9 z-chain[3]: r4n9{c8 c3} - r3n9{c3 c5} - b5n9{r5c5 .} ==> r6c9≠9 t-whip[3]: r4n9{c8 c3} - r6n9{c1 c6} - r2n9{c6 .} ==> r9c8≠9, r7c8≠9 naked-pairs-in-a-block: b9{r7c8 r8c8}{n2 n3} ==> r7c7≠3 singles ==> r5c7=3, r4c8=9 whip[1]: b4n9{r6c1 .} ==> r1c1≠9 biv-chain[2]: b6n1{r6c9 r5c8} - c2n1{r5 r9} ==> r9c9≠1 finned-swordfish-in-columns: n1{c8 c2 c4}{r2 r9 r5} ==> r5c5≠1 biv-chain[3]: r4n7{c3 c4} - r5c5{n7 n9} - b4n9{r5c1 r6c1} ==> r6c1≠7 biv-chain[2]: b6n7{r6c9 r5c8} - c1n7{r5 r1} ==> r1c9≠7 biv-chain[3]: c7n7{r9 r1} - c1n7{r1 r5} - b6n7{r5c8 r6c9} ==> r9c9≠7 biv-chain[3]: r9n7{c7 c8} - r5c8{n7 n1} - c2n1{r5 r9} ==> r9c7≠1 biv-chain[3]: r2c4{n7 n1} - c5n1{r3 r8} - b8n7{r8c5 r8c6} ==> r1c6≠7, r2c6≠7 biv-chain[2]: b2n7{r3c5 r2c4} - r4n7{c4 c3} ==> r3c3≠7 x-wing-in-columns: n7{c3 c4}{r2 r4} ==> r2c8≠7 finned-x-wing-in-rows: n7{r3 r6}{c9 c5} ==> r5c5≠7 singles ==> r5c5=9,> r6c1=9 naked-pairs-in-a-block: b2{r2c4 r3c5}{n1 n7} ==> r2c6≠1, r1c6≠1 biv-chain[2]: b2n1{r3c5 r2c4} - r4n1{c4 c3} ==> r3c3≠1 stte

Website · by **denis_berthier** » Sun Jan 08, 2023 9:11 am

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Previously, I have spoken of the ultra-persitency of ORk-relations (see the Augmented User Manual own GitHub or on ResearchGate https://www.researchgate.net/publication/365186265_Augmented_User_Manual_for_CSP-Rules-V21).

This post is about splitting ORk-relations. The idea of doing such things was more or less inspired by solutions proposed by other people in the Puzzles section of this forum. What I've done is find a generic way to express some of the ad hoc arguments made there.

First define a c-chain[n] as a sequence of n+1 different candidates C1... Cn+1 such that for each 1 ≤ i ≤ n, Ci and Ci+1 are related by a (3D) bivalue relation.
It is obvious that, if n is even then C1 and Cn+1 have the same truth value.
This is a generic definition and a generic property.

Theorem (generic): given an ORk relation between candidates Z1, ... Zk, if there is any even c-chain[n] between two of the Zi's, say Za and Zb, then the original ORk-relation can be split into two ORk-1 relations, between k-1 candidates, respectively {Z1, ... Zk}-Za and {Z1, ... Zk}-Zb.
[Edit, added]:Corollary: if k=2, both Z1 and Z2 can be asserted.

Proof: obvious

Note: "between two of the Zi's, say Za and Zb" means that C1 = Za and Cn+1 = Zb

For an example of using the theorem, see second post here: http://forum.enjoysudoku.com/51007-in-63137-t-e-3-min-expands-t40721-3.html and next post.
For an example of using the corollary, see here: http://forum.enjoysudoku.com/other-hard-one-from-mith-s-trivalue-oddagon-list-t40727-4.html.
For an example of using the theorem repeatedly in conjunction with ORk-reduction (ultra-persistency, see here: http://forum.enjoysudoku.com/1182-in-63137-list-of-t-e-3-min-expands-t40739-3.html

Note: the way such splitting is coded in CSP-Rules is still experimental.

Website · by **denis_berthier** » Fri Jan 13, 2023 12:21 pm

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a reader of this thread wrote:How do you know that c-chains of even lengths are the only way ORk-relations can be reduced?

1) ORk relations are reduced when a guardian is eliminated (this is part of their ultra-persistency in CSP-Rules ).
But your question (about my previous post) is about a totally different topic.

2) Indeed, I don't know what you claim. On the contrary, there are probably many kinds of chain patterns that could be used as a basis for reducing ORk-relations.

What I've been looking for is rules that:
- allowed to reduce the number of guardians; it appeared that trying to split ORk-relations (instead of trying to reduce them, i.e. to eliminate guardians from them) was the most natural thing to do;
- involved only generic conditions (with no ad-hoc reasoning);
- were simple enough to be integrated in the hierarchy of resolution rules (though they are not resolution rules).
The last two conditions are trivially satisfied by c-chains (a c-chain[n] being posited at level Ln, just before the ORk-chains[n] that might rely on their results).

Now, can these splitting rules take into account the different ad hoc reductions that have been proposed for anti-tridagon puzzles in the Puzzles section? I haven't checked and it's not that important here for the following reason:
my approach has always been to find universal rules/techniques that can be applied in a systematic way. They may not be the smartest ones for a particular puzzle, but they aim at being the smartest ones in the mean.
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Website · by **denis_berthier** » Fri Jan 10, 2025 1:13 pm

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ORk-chains are generic and can be used in combination with ORk-relations generated by any type of patterns.
This applies in particular to Deadly Patterns (DPs) if one accepts the assumption of uniqueness.

Out of sheer curiosity about their effectiveness in this con text, I've now coded the 17 Deadly Patterns on 9 or fewer cells defined here: https://www.sudopedia.org/wiki/Deadly_Pattern
The rules are similar to those for eleven 630 impossible 3-digit patterns and they produce similar ORk-relations.
Because Deadly patterns are unstable, I've set their priorities very high, just after whips[1].
The following example illustrates how they can be used. Having very high priorities, they are expected to have many guardians and to be subjected to many ORk-ultra persitency and ORk-splitting rules before becoming applicable.

Here's an example of a puzzle in SFin2+W3 but in SFin2+W2+DP-OR6W2, puzzle #5733 in the cbg-000 controlled-bias collection. The gain (from L3 to l2) is not much in this case, but this is just an illustration of the idea.

Code: Select all: Puzzle cbg-000 #37 +-------+-------+-------+ ! . . 3 ! 4 . . ! . 8 . ! ! . . 6 ! . 8 . ! . . . ! ! 7 . . ! . . . ! . . 5 ! +-------+-------+-------+ ! 2 3 . ! . 9 . ! 6 . 7 ! ! . . . ! . 7 4 ! . . 8 ! ! . . . ! . . 1 ! . . . ! +-------+-------+-------+ ! . . . ! . 1 . ! . . . ! ! . . 8 ! 3 . . ! . 9 . ! ! . 7 . ! . 6 . ! 5 3 . ! +-------+-------+-------+ ..34...8...6.8....7.......523..9.6.7....74..8.....1.......1......83...9..7..6.53. 136942

Code: Select all: Resolution state after Singles and whips[1]: +----------------+----------------+----------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +----------------+----------------+----------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169 169 159 ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +----------------+----------------+----------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +----------------+----------------+----------------+ 98 candidates.

At this point, there are 8 different deadly patterns with 8 or fewer guardians. Only one of them will be useful - and it isn't the one with the minimum number of guardians. In what follows, I keep the 8 DPs, but I keep track only of the useful one. It's interesting to see how the associated ORk-relations is progressively split into ORk-relations with smaller values of k.

The 8 DPs:

Code: Select all: DP4-2-1s-OR6-relation in cells (marked #): (r6c8 r6c7 r2c8 r2c7) with 6 guardians (in cells marked @) : n4r6c8 n5r6c8 n4r6c7 n9r6c7 n4r2c8 n4r2c7 +-------------------+-------------------+-------------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124#@ 24#@ 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +-------------------+-------------------+-------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169 169 159 ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249#@ 245#@ 29 ! +-------------------+-------------------+-------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +-------------------+-------------------+-------------------+ ;;; the only one that will be used: DP4-2-1s-OR8-relation in cells (marked #): (r8c2 r8c1 r2c2 r2c1) with 8 guardians (in cells marked @) : n5r8c2 n6r8c2 n5r8c1 n6r8c1 n4r2c2 n5r2c2 n4r2c1 n5r2c1 +----------------------+----------------------+----------------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145#@ 1245#@ 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +----------------------+----------------------+----------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169 169 159 ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +----------------------+----------------------+----------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156#@ 156#@ 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +----------------------+----------------------+----------------------+ DP4-2-1s-OR8-relation in cells (marked #): (r8c2 r8c1 r5c2 r5c1) with 8 guardians (in cells marked @) : n5r8c2 n6r8c2 n5r8c1 n6r8c1 n6r5c2 n9r5c2 n6r5c1 n9r5c1 +-------------------+-------------------+-------------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +-------------------+-------------------+-------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169#@ 169#@ 159 ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +-------------------+-------------------+-------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156#@ 156#@ 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +-------------------+-------------------+-------------------+ DP4-2-1s-OR8-relation in cells (marked #): (r9c3 r9c1 r5c3 r5c1) with 8 guardians (in cells marked @) : n4r9c3 n9r9c3 n4r9c1 n9r9c1 n5r5c3 n9r5c3 n6r5c1 n9r5c1 +----------------------+----------------------+----------------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +----------------------+----------------------+----------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169#@ 169 159#@ ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +----------------------+----------------------+----------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12 9 126 ! ! 149#@ 7 1249#@ ! 89 6 28 ! 5 3 14 ! +----------------------+----------------------+----------------------+ DP4-2-1s-OR6-relation in cells (marked #): (r5c2 r5c1 r1c2 r1c1) with 6 guardians (in cells marked @) : n6r5c2 n9r5c2 n6r5c1 n9r5c1 n9r1c2 n9r1c1 +-------------------+-------------------+-------------------+ ! 19#@ 129#@ 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +-------------------+-------------------+-------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169#@ 169#@ 159 ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +-------------------+-------------------+-------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +-------------------+-------------------+-------------------+ DP4-2-1-OR7-relation in cells (marked #): (r6c8 r6c4 r5c8 r5c4) with 7 guardians (in cells marked @) : n4r6c8 n5r6c8 n5r6c4 n6r6c4 n5r5c8 n5r5c4 n6r5c4 +-------------------+-------------------+-------------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +-------------------+-------------------+-------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169 169 159 ! 256#@ 7 4 ! 3 125#@ 8 ! ! 8 46 7 ! 256#@ 3 1 ! 249 245#@ 29 ! +-------------------+-------------------+-------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +-------------------+-------------------+-------------------+ DP4-2-1-OR7-relation in cells (marked #): (r5c8 r5c3 r4c8 r4c3) with 7 guardians (in cells marked @) : n5r5c8 n5r5c3 n9r5c3 n4r4c8 n5r4c8 n4r4c3 n5r4c3 +-------------------+-------------------+-------------------+ ! 19 129 3 ! 4 5 6 ! 7 8 129 ! ! 145 1245 6 ! 7 8 9 ! 124 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +-------------------+-------------------+-------------------+ ! 2 3 145#@ ! 58 9 58 ! 6 145#@ 7 ! ! 169 169 159#@ ! 256 7 4 ! 3 125#@ 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +-------------------+-------------------+-------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +-------------------+-------------------+-------------------+ DP6-2-2s-OR6-relation in cells (marked #): (r2c2 r2c7 r1c2 r1c9 r8c7 r8c9) with 6 guardians (in cells marked @) : n4r2c2 n5r2c2 n4r2c7 n9r1c2 n9r1c9 n6r8c9 +----------------------+----------------------+----------------------+ ! 19 129#@ 3 ! 4 5 6 ! 7 8 129#@ ! ! 145 1245#@ 6 ! 7 8 9 ! 124#@ 24 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +----------------------+----------------------+----------------------+ ! 2 3 145 ! 58 9 58 ! 6 145 7 ! ! 169 169 159 ! 256 7 4 ! 3 125 8 ! ! 8 46 7 ! 256 3 1 ! 249 245 29 ! +----------------------+----------------------+----------------------+ ! 3 469 249 ! 59 1 25 ! 8 7 46 ! ! 156 156 8 ! 3 4 7 ! 12# 9 126#@ ! ! 149 7 1249 ! 89 6 28 ! 5 3 14 ! +----------------------+----------------------+----------------------+

naked-pairs-in-a-row: r4{c4 c6}{n5 n8} ==> r4c8≠5, r4c3≠5
singles ==> r5c3=5, r6c8=5
finned-x-wing-in-rows: n4{r6 r3}{c7 c2} ==> r2c2≠4

At least one candidate of a previous DP4-2-1s-OR8-relation between candidates n5r8c2 n6r8c2 n5r8c1 n6r8c1 n4r2c2 n5r2c2 n4r2c1 n5r2c1 has just been eliminated.
There remains a DP4-2-1s-OR7-relation between candidates: n5r8c2 n6r8c2 n5r8c1 n6r8c1 n5r2c2 n4r2c1 n5r2c1

finned-x-wing-in-rows: n4{r4 r3}{c3 c8} ==> r2c8≠4
singles ==> r2c8=2, r5c8=1, r4c8=4, r4c3=1, r6c2=4, r6c4=6, r5c4=2, r1c2=2
naked-pairs-in-a-column: c2{r5 r7}{n6 n9} ==> r8c2≠6

Code: Select all: +-------------+-------------+-------------+ ! 19 2 3 ! 4 5 6 ! 7 8 19 ! ! 145 15 6 ! 7 8 9 ! 14 2 3 ! ! 7 8 49 ! 1 2 3 ! 49 6 5 ! +-------------+-------------+-------------+ ! 2 3 1 ! 58 9 58 ! 6 4 7 ! ! 69 69 5 ! 2 7 4 ! 3 1 8 ! ! 8 4 7 ! 6 3 1 ! 29 5 29 ! +-------------+-------------+-------------+ ! 3 69 249 ! 59 1 25 ! 8 7 46 ! ! 156 15 8 ! 3 4 7 ! 12 9 126 ! ! 149 7 249 ! 89 6 28 ! 5 3 14 ! +-------------+-------------+-------------+

At least one candidate of a previous DP4-2-1s-OR7-relation between candidates n5r8c2 n6r8c2 n5r8c1 n6r8c1 n5r2c2 n4r2c1 n5r2c1 has just been eliminated.
There remains a DP4-2-1s-OR6-relation between candidates: n5r8c2 n5r8c1 n6r8c1 n5r2c2 n4r2c1 n5r2c1

x-wing-in-columns: n1{c2 c7}{r2 r8} ==> r8c9≠1, r8c1≠1, r2c1≠1

DP4-2-1s-OR6-relation between candidates n5r8c2, n5r8c1, n6r8c1, n5r2c2, n4r2c1 and n5r2c1
+ same valence for candidates n6r8c1 and n5r2c1 via c-chain[2]: n6r8c1,n5r8c1,n5r2c1
==> DP4-2-1s-OR6-relation can be split into two DP4-2-1s-OR5-relations with respective lists of guardians:
n5r8c2 n5r8c1 n5r2c2 n4r2c1 n5r2c1 and n5r8c2 n5r8c1 n6r8c1 n5r2c2 n4r2c1 .

DP4-2-1s-OR4-relation between candidates n5r8c2, n5r8c1, n5r2c2 and n4r2c1
+ same valence for candidates n5r8c1 and n5r2c2 via c-chain[2]: n5r8c1,n5r8c2,n5r2c2
==> DP4-2-1s-OR4-relation can be split into two DP4-2-1s-OR3-relations with respective lists of guardians:
n5r8c2 n5r2c2 n4r2c1 and n5r8c2 n5r8c1 n4r2c1 .

DP4-2-1s-OR3-relation between candidates n5r8c2, n5r8c1 and n4r2c1
+ same valence for candidates n5r8c1 and n4r2c1 via c-chain[2]: n5r8c1,n5r2c1,n4r2c1
==> DP4-2-1s-OR3-relation can be split into two DP4-2-1s-OR2-relations with respective lists of guardians:
n5r8c2 n4r2c1 and n5r8c2 n5r8c1 .

DP4-2-1s-OR2-whip[2]: OR2{{n4r2c1 | n5r8c2}} - b7n1{r8c2 .} ==> r9c1≠4
stte

Note: DP4-2-1 is the usual UR2, but with more guardians than usual.
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The New Sudoku Players' Forum

ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips

ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips

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Re: OR-k-Forcing-Whips

Re: OR-k-Forcing-Whips

Re: ORk-Forcing-Whips and ORk-Contrad-Whips

Re: ORk-Forcing-Whips and ORk-Contrad-Whips

Re: ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips

Re: ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips

Re: ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips

Re: ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips

Re: ORk-Forcing-Whips, ORk-Contrad-Whips and ORk-whips