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Website · by **denis_berthier** » Fri Dec 23, 2022 6:43 am

.
That's a hard one again.
I continue writing the SER, though it is now clear it doesn't mean much when anti-tridagon rules can be applied.

As I said in a previous post, I've now coded the full ORk-g-whips (soon to be published on GitHub). What I observe is, they don't add much resolution power to ORk-whips. This puzzle is one more of the few cases where they do.

Code: Select all: +-------+-------+-------+ ! . . . ! . . . ! 7 8 . ! ! . . . ! 7 8 9 ! 1 . 2 ! ! . . . ! . . . ! . 6 4 ! +-------+-------+-------+ ! 2 . 5 ! 9 1 . ! 8 . . ! ! . . . ! . 2 8 ! . . . ! ! 8 . . ! 5 . 7 ! 2 . . ! +-------+-------+-------+ ! . 7 1 ! 8 9 . ! . . 5 ! ! 5 . . ! . . . ! . . . ! ! 9 . 2 ! . 7 5 ! . . . ! +-------+-------+-------+ ......78....7891.2.......642.591.8......28...8..5.72...7189...55........9.2.75...;4024;280382 SER = 10.4

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 13 12359 39 ! 12346 3456 12346 ! 7 8 39 ! ! 346 3456 346 ! 7 8 9 ! 1 35 2 ! ! 137 123589 3789 ! 123 35 123 ! 359 6 4 ! +----------------------+----------------------+----------------------+ ! 2 346 5 ! 9 1 346 ! 8 347 367 ! ! 13467 13469 34679 ! 346 2 8 ! 34569 13459 1369 ! ! 8 13469 3469 ! 5 346 7 ! 2 1349 1369 ! +----------------------+----------------------+----------------------+ ! 346 7 1 ! 8 9 2346 ! 346 234 5 ! ! 5 3468 3468 ! 12346 346 12346 ! 3469 123479 136789 ! ! 9 3468 2 ! 1346 7 5 ! 346 134 1368 ! +----------------------+----------------------+----------------------+ 193 candidates.

by **totuan** » Fri Dec 23, 2022 1:58 pm

denis_berthier wrote:.
That's a hard one again.
Code: Select all
+-------+-------+-------+ ! . . . ! . . . ! 7 8 . ! ! . . . ! 7 8 9 ! 1 . 2 ! ! . . . ! . . . ! . 6 4 ! +-------+-------+-------+ ! 2 . 5 ! 9 1 . ! 8 . . ! ! . . . ! . 2 8 ! . . . ! ! 8 . . ! 5 . 7 ! 2 . . ! +-------+-------+-------+ ! . 7 1 ! 8 9 . ! . . 5 ! ! 5 . . ! . . . ! . . . ! ! 9 . 2 ! . 7 5 ! . . . ! +-------+-------+-------+ ......78....7891.2.......642.591.8......28...8..5.72...7189...55........9.2.75...;4024;280382 SER = 10.4

Code: Select all
Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 13 12359 39 ! 12346 3456 12346 ! 7 8 39 ! ! 346 3456 346 ! 7 8 9 ! 1 35 2 ! ! 137 123589 3789 ! 123 35 123 ! 359 6 4 ! +----------------------+----------------------+----------------------+ ! 2 346 5 ! 9 1 346 ! 8 347 367 ! ! 13467 13469 34679 ! 346 2 8 ! 34569 13459 1369 ! ! 8 13469 3469 ! 5 346 7 ! 2 1349 1369 ! +----------------------+----------------------+----------------------+ ! 346 7 1 ! 8 9 2346 ! 346 234 5 ! ! 5 3468 3468 ! 12346 346 12346 ! 3469 123479 136789 ! ! 9 3468 2 ! 1346 7 5 ! 346 134 1368 ! +----------------------+----------------------+----------------------+ 193 candidates.

Hmmm...!
Don't know how this one is hard??? :roll:

All guardians of Tridagon (346) lead to r5c8=5 then... nothing :lol:

Thanks for the puzzle!
totuan

by **Cenoman** » Fri Dec 23, 2022 10:28 pm

Code: Select all: +-----------------------+------------------------+-----------------------------+ | 1 25 b39 | 246 456 246 | 7 8 c39 | | 346 3456 346 | 7 8 9 | 1 35 2 | | 7 2589 B89 | 123 35 123 | 5-9 6 4 | +-----------------------+------------------------+-----------------------------+ | 2 346* 5 | 9 1 346* | 8 347 367 | | 346* 13469 7 | 346* 2 8 | 34569 13459 1369 | | 8 13469 a3469* | 5 346* 7 | 2 1349 1369 | +-----------------------+------------------------+-----------------------------+ | 346* 7 1 | 8 9 F2346* |zGg346 G234 5 | | 5 3468 A3468* | 12346 346* 12346 |zGg3469 123479 136789 | | 9 f3468* 2 | y1346* 7 5 |zGg346 zg134 g1368 | +-----------------------+------------------------+-----------------------------+

1. TH(346)b4578 having five guardians: 9r6c3, 8r8c3, 8r9c2, 2r7c6, 1r9c4
(9)r6c3 - r1c3 = (9)r1c9
(8)r8c3 - (8=9)r3c3
(8)r9c2 - (8=13469)b9p14789
(2)r7c6 - (2=3469)b9p1247
(1)r9c4 - (1=3469)b9p1478
=> -9 r3c7; 9 placements

Code: Select all: +----------------------+-----------------------+-------------------------+ | 1 2 3 | 46 5 46 | 7 8 9 | | 46 5 46 | 7 8 9 | 1 3 2 | | 7 89 89 | 12* 3 12* | 5 6 4 | +----------------------+-----------------------+-------------------------+ | 2 346 5 | 9 1 346 | 8 b47 367 | | 346 13469 7 | 346 2 8 | 3469 5 136 | | 8 13469 469 | 5 46 7 | 2 149 136 | +----------------------+-----------------------+-------------------------+ | 346 7 1 | 8 9 2346 | 346 24 5 | | 5 3468 468 | 12346* 46 12346* | 3469 c12-479 d3678-1 | | 9 3468 2 | 1346 7 5 | 346 a14 d368-1 | +----------------------+-----------------------+-------------------------+

2. (1=4)r9c8 - (4=7)r4c8 - r8c8 = (78)r89c9 => -1 r89c9
3. UR(12)r48c46 using externals => +12 r8c8

Code: Select all: +------------------+-----------------------+-----------------+ | 1 2 3 | 46 5 46 | 7 8 9 | | 46 5 46 | 7 8 9 | 1 3 2 | | 7 8 9 | 12 3 12 | 5 6 4 | +------------------+-----------------------+-----------------+ | 2 d34* 5 | 9 1 d34* | 8 7 6 | | c36 9 7 | b36 2 8 | 4 5 1 | | 8 1 46 | 5 a46 7 | 2 9 3 | +------------------+-----------------------+-----------------+ | d34* 7 1 | 8 9 fd2346* | e36# 24 5 | | 5 346 8 | 12346 4-6 12346 | 9 12 7 | | 9 346 2 | 1346 7 5 | 36 14 8 | +------------------+-----------------------+-----------------+

Almost kite:
4. (6)r6c5 = r5c4 - (6=3)r5c1 - [r4c2 = r4c6 - r7c6 = r7c1] = (3-6)r7c7 = (6)r7c6 => -6 r8c5; ste

by **eleven** » Fri Dec 23, 2022 10:46 pm

Very similar. 5 guardians => 5r3c7
9r6c3 - (9=3)r1c3 - r1c9 = (3-5)r2c8 = 5r3c7
2r7c6 - r7c8 = (2-7)r8c8 = 79r8c97 - (9=5)r3c7
8r8c3 - (8=9)r3c3 - (9=5)r3c7
8r9c2 - r9c9 = 879r8c7 - (9=5)r3c7
1r9c4 - r9c89 = 179r8c897 - (9=5)r3c7

Then there is a UR type 3 (again!). Without it it would be harder!

Code: Select all: +----------------------+-----------------------+----------------------+ | 1 2 3 | 46 5 46 | 7 8 9 | | 46 5 46 | 7 8 9 | 1 3 2 | | 7 89 89 |#12 3 #12 | 5 6 4 | +----------------------+------- ----------------+----------------------+ | 2 346 5 | 9 1 346 | 8 47 367 | | 346 13469 7 | 346 2 8 | 3469 5 136 | | 8 13469 469 | 5 46 7 | 2 149 136 | +----------------------+-----------------------+----------------------+ | 346 7 1 | 8 9 2346 | 346 24 5 | | 5 *3468 *468 |#12+346 *46 #12+346 | 9-346 1279 178-36 | | 9 3468 2 | 1346 7 5 | 346 14 1368 | +----------------------+-----------------------+----------------------+

UR 12 r38c46: quad 3468r8c23456 => -346r7c7,-36r7c9

singles and locked 6c2b7.

Code: Select all: +----------------------+----------------------+----------------------+ | 1 2 3 | 46 5 46 | 7 8 9 | | 46 5 46 | 7 8 9 | 1 3 2 | | 7 8 9 | 12 3 12 | 5 6 4 | +----------------------+----------------------+----------------------+ | 2 34 5 | 9 1 c34 | 8 7 6 | |e36 9 7 | 36 2 8 | 4 5 1 | | 8 1 a46 | 5 b46 7 | 2 9 3 | +----------------------+----------------------+----------------------+ |e34 7 1 | 8 9 d2346 | 36 d24 5 | | 5 346 8 | 12346 c46 12346 | 9 12 7 | | 9 346 2 | 1346 7 5 | 36 14 8 | +----------------------+----------------------+----------------------+

4r6c3 = r6c5 - (4=36)r4c6,r8c5 - (3|6=24)r7c68 - (4=36)r5c1 => -6r6c3, stte

by **marek stefanik** » Fri Dec 23, 2022 11:53 pm

After the same start, avoiding the UR:

Code: Select all: .-----------------.------------------.--------------------. | 1 2 3 | 46 5 46 | 7 8 9 | | 46 5 46 | 7 8 9 | 1 3 2 | | 7 89 *89 | 12 3 12 | 5 6 4 | :-----------------+------------------+--------------------: | 2 #346 5 | 9 1 #346 | 8 47 367 | |#346 13469 7 |#346 2 8 | 3469 5 136 | | 8 13469 #46+9| 5 #46 7 | 2 149 136 | :-----------------+------------------+--------------------: |#346 7 1 | 8 9 α#346+2 | 346 β24 5 | | 5 3468 #46+8| 12346 #46 12346 | 3469 *127–49*178–36| | 9 a#346+8 2 |A#346+1 7 5 | 346 B14 bB1368 | '-----------------'------------------'--------------------'

TH 346#, 3 has been eliminated from the rectangle => no RTs
Rectangle guardians (9r6c3 and 8r8c3) are linked via r3c3 => mutually exclusive
Hence another guardian is needed (8r9c2 or 1r9c4 or 2r7c6).

Code: Select all: |8r9c2 – 8r9c9 = 78r8c89 |1r9c4 – 1r9c89 = 17r8c89 |2r7c6 – 2r7c8 = 27r8c89 => –49r8c8, -36r8c9

Reaching the same state as with the UR.

Marek

by **eleven** » Sat Dec 24, 2022 1:17 am

marek stefanik wrote:Hence another guardian is needed (8r9c2 or 1r9c4 or 2r7c6).

So this pattern is impossible

Code: Select all: *-----------------------------------* | . 123 . | . . 123 | | 123 . . | 123 . . | | . . 12 | . 12 . | |------------------+----------------| | 123 . . | . . 123 | | . . . | . 12 . | | . 123 . | 123 . . | *-----------------------------------*

I need 2 cases to prove it (r1c2=12 or 3). is that a "hard" pattern ? [Added: sure now, no]

by **marek stefanik** » Sat Dec 24, 2022 2:06 am

Code: Select all: *-----------------------------------* | . A123 . | . . A123 | |B123 . . |B123 . . | | . . C12 | . C12 . | |------------------+----------------| | 123 . . | . . A123 | | . . . | . C12 . | | . 123 . |B123 . . | *-----------------------------------*

Yeah, it's not that hard. I like the parity-based proof a bit more:
The permutations in boxes (in order ABC) all have the same parity and C-marked cells in b15 contain the same digit.
Hence the permutations in b15 are identical and the digits in the A- and B-marked cells are both eliminated from b4p18, i.e. contra.

Marek

Website · by **denis_berthier** » Sat Dec 24, 2022 4:37 am

eleven wrote:So this pattern is impossible
Code: Select all
*-----------------------------------* | . 123 . | . . 123 | | 123 . . | 123 . . | | . . 12 | . 12 . | |------------------+----------------| | 123 . . | . . 123 | | . . . | . 12 . | | . 123 . | 123 . . | *-----------------------------------*

is that a "hard" pattern ? [Added: sure now, no]

If you mean hard in the sense of what's needed to prove it contradictory:
- it doesn't require T&E(3)
- it doesn't even require T&E(2), not even some T&E(Bp, 1), p>0
- it requires only T&E(1)

Indeed, it can be proven contradictory using only whips[≤ 10]. So, not hard compared to the T&E(3) patterns, but not easy when considered as a T&E(1) pattern.

In CSP-Rules, this proof can be done by using function "solve-k-digit-pattern-string" with the string ".1...1...1..1.......1.1....1....1.......1.....1.1....." corresponding to the above pattern. This function supposes that all the cells in the pattern have exactly k digits 1, 2 ... k; but I've already shown how to do for the "incomplete" cells:

Code: Select all: (bind ?*simulated-eliminations* (create$ (nrc-to-label 3 3 3) (nrc-to-label 3 3 5 ) (nrc-to-label 3 5 5) )) (solve-k-digit-pattern-string 3 ".1...1...1..1.......1.1....1....1.......1.....1.1.....")

(Some day, I may write a more general function solve-kl-digit-patterns.)

Code: Select all: Resolution state after Singles and whips[1]: +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123 123456789 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 12 ! 123456789 12 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123 123456789 123456789 ! 123456789 123456789 123 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 12 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123 123456789 ! 123 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! +-------------------------------+-------------------------------+-------------------------------+ 660 candidates. biv-chain[2]: r3c3{n1 n2} - r3c5{n2 n1} ==> r3c1≠1, r3c2≠1, r3c7≠1, r3c8≠1, r3c9≠1, r3c4≠1, r3c6≠1 biv-chain[2]: r3c3{n2 n1} - r3c5{n1 n2} ==> r3c1≠2, r3c2≠2, r3c7≠2, r3c8≠2, r3c9≠2, r3c4≠2, r3c6≠2 biv-chain[2]: r3c5{n1 n2} - r5c5{n2 n1} ==> r1c5≠1, r2c5≠1, r7c5≠1, r8c5≠1, r9c5≠1, r4c5≠1, r6c5≠1 biv-chain[2]: r3c5{n2 n1} - r5c5{n1 n2} ==> r1c5≠2, r2c5≠2, r7c5≠2, r8c5≠2, r9c5≠2, r4c5≠2, r6c5≠2 whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r1c1≠1 whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r1c1≠2 whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r1c3≠1 whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r1c3≠2 whip[3]: r3c5{n1 n2} - r1c6{n2 n3} - r2c4{n3 .} ==> r1c4≠1 whip[3]: r3c5{n2 n1} - r1c6{n1 n3} - r2c4{n3 .} ==> r1c4≠2 whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r2c2≠1 whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r2c2≠2 whip[3]: r3c3{n1 n2} - r1c2{n2 n3} - r2c1{n3 .} ==> r2c3≠1 whip[3]: r3c3{n2 n1} - r1c2{n1 n3} - r2c1{n3 .} ==> r2c3≠2 whip[3]: r3c5{n1 n2} - r1c6{n2 n3} - r2c4{n3 .} ==> r2c6≠1 whip[3]: r3c5{n2 n1} - r1c6{n1 n3} - r2c4{n3 .} ==> r2c6≠2 whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r4c4≠1 whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r4c4≠2 whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r5c4≠1 whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r5c4≠2 whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r5c6≠1 whip[3]: r5c5{n1 n2} - r4c6{n2 n3} - r6c4{n3 .} ==> r6c6≠1 whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r5c6≠2 whip[3]: r5c5{n2 n1} - r4c6{n1 n3} - r6c4{n3 .} ==> r6c6≠2 whip[10]: r5c5{n1 n2} - r3n2{c5 c3} - b1n1{r3c3 r1c2} - r2c1{n1 n3} - r4c1{n3 n2} - r6c2{n2 n3} - r6c4{n3 n1} - r2c4{n1 n2} - r1c6{n2 n3} - r4c6{n3 .} ==> r5c1≠1 whip[10]: r5c5{n2 n1} - r3n1{c5 c3} - b1n2{r3c3 r1c2} - r2c1{n2 n3} - r4c1{n3 n1} - r6c2{n1 n3} - r6c4{n3 n2} - r2c4{n2 n1} - r1c6{n1 n3} - r4c6{n3 .} ==> r5c1≠2 whip[10]: r5c5{n1 n2} - r3n2{c5 c3} - b1n1{r3c3 r2c1} - r1c2{n1 n3} - r6c2{n3 n2} - r4c1{n2 n3} - r4c6{n3 n1} - r1c6{n1 n2} - r2c4{n2 n3} - r6c4{n3 .} ==> r5c2≠1 whip[10]: r5c5{n2 n1} - r3n1{c5 c3} - b1n2{r3c3 r2c1} - r1c2{n2 n3} - r6c2{n3 n1} - r4c1{n1 n3} - r4c6{n3 n2} - r1c6{n2 n1} - r2c4{n1 n3} - r6c4{n3 .} ==> r5c2≠2 whip[10]: r3c3{n1 n2} - c5n2{r3 r5} - b5n1{r5c5 r6c4} - r4c6{n1 n3} - r4c1{n3 n2} - r6c2{n2 n3} - r1c2{n3 n1} - r1c6{n1 n2} - r2c4{n2 n3} - r2c1{n3 .} ==> r4c3≠1 whip[10]: r3c3{n1 n2} - c5n2{r3 r5} - b5n1{r5c5 r4c6} - r6c4{n1 n3} - r6c2{n3 n2} - r4c1{n2 n3} - r2c1{n3 n1} - r2c4{n1 n2} - r1c6{n2 n3} - r1c2{n3 .} ==> r6c3≠1 whip[10]: r3c3{n2 n1} - c5n1{r3 r5} - b5n2{r5c5 r6c4} - r4c6{n2 n3} - r4c1{n3 n1} - r6c2{n1 n3} - r1c2{n3 n2} - r1c6{n2 n1} - r2c4{n1 n3} - r2c1{n3 .} ==> r4c3≠2 whip[10]: r3c3{n2 n1} - c5n1{r3 r5} - b5n2{r5c5 r4c6} - r6c4{n2 n3} - r6c2{n3 n1} - r4c1{n1 n3} - r2c1{n3 n2} - r2c4{n2 n1} - r1c6{n1 n3} - r1c2{n3 .} ==> r6c3≠2 whip[10]: r3n1{c3 c5} - r5c5{n1 n2} - r3n2{c5 c3} - r2c1{n2 n3} - r2c4{n3 n2} - r1c6{n2 n3} - r4c6{n3 n1} - r4c1{n1 n2} - r6c2{n2 n3} - r6c4{n3 .} ==> r1c2≠1 t-whip[3]: r1c2{n3 n2} - r3c3{n2 n1} - r2c1{n1 .} ==> r3c2≠3, r3c1≠3, r2c3≠3, r2c2≠3, r1c3≠3, r1c1≠3 z-chain[4]: r3c5{n2 n1} - r1c6{n1 n3} - r1c2{n3 n2} - r3n2{c3 .} ==> r2c4≠2 z-chain[3]: r2c4{n3 n1} - r1c6{n1 n2} - r1c2{n2 .} ==> r1c5≠3, r1c4≠3 t-whip[3]: r2c4{n3 n1} - r3c5{n1 n2} - r1c6{n2 .} ==> r3c6≠3, r3c4≠3, r2c6≠3, r2c5≠3 whip[1]: r3n3{c9 .} ==> r1c7≠3, r1c8≠3, r1c9≠3, r2c7≠3, r2c8≠3, r2c9≠3 z-chain[4]: r5c5{n1 n2} - r6c4{n2 n3} - r2c4{n3 n1} - c5n1{r3 .} ==> r4c6≠1 z-chain[3]: r4c6{n3 n2} - r6c4{n2 n1} - r2c4{n1 .} ==> r5c4≠3, r4c4≠3 t-whip[3]: r4c6{n3 n2} - r5c5{n2 n1} - r6c4{n1 .} ==> r6c6≠3, r6c5≠3, r5c6≠3, r4c5≠3 whip[1]: c5n3{r9 .} ==> r7c4≠3, r7c6≠3, r8c4≠3, r8c6≠3, r9c4≠3, r9c6≠3 biv-chain[2]: b5n3{r6c4 r4c6} - r1n3{c6 c2} ==> r6c2≠3 biv-chain[2]: r2n3{c1 c4} - b5n3{r6c4 r4c6} ==> r4c1≠3 biv-chain[2]: r4c1{n2 n1} - r6c2{n1 n2} ==> r4c2≠2, r5c3≠2, r6c1≠2 biv-chain[2]: r6c2{n1 n2} - r4c1{n2 n1} ==> r4c2≠1, r5c3≠1, r6c1≠1 biv-chain[3]: c6n3{r4 r1} - r1c2{n3 n2} - b4n2{r6c2 r4c1} ==> r4c6≠2 singles ==> r4c6=3, r2c4=3, r1c2=3 biv-chain[2]: r6c4{n2 n1} - r6c2{n1 n2} ==> r6c7≠2, r6c8≠2, r6c9≠2 biv-chain[2]: r6c2{n1 n2} - r6c4{n2 n1} ==> r6c7≠1, r6c8≠1, r6c9≠1 biv-chain[2]: r2c1{n2 n1} - r4c1{n1 n2} ==> r7c1≠2, r8c1≠2, r9c1≠2 biv-chain[2]: r4c1{n1 n2} - r2c1{n2 n1} ==> r7c1≠1, r8c1≠1, r9c1≠1 biv-chain[3]: b4n2{r6c2 r4c1} - b1n2{r2c1 r3c3} - c5n2{r3 r5} ==> r6c4≠2 singles ==> r6c4=1, r5c5=2, r3c5=1, r1c6=2, r3c3=2, r2c1=1 PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR BN-CELL b4n1

[Edit]: There's a simpler proof if you allow Subsets, but it is NOT in T&E(1):

Code: Select all: naked-pairs-in-a-column: c5{r3 r5}{n1 n2} ==> r9c5≠2, r9c5≠1, r8c5≠2, r8c5≠1, r7c5≠2, r7c5≠1, r6c5≠2, r6c5≠1, r4c5≠2, r4c5≠1, r2c5≠2, r2c5≠1, r1c5≠2, r1c5≠1 naked-pairs-in-a-row: r3{c3 c5}{n1 n2} ==> r3c9≠2, r3c9≠1, r3c8≠2, r3c8≠1, r3c7≠2, r3c7≠1, r3c6≠2, r3c6≠1, r3c4≠2, r3c4≠1, r3c2≠2, r3c2≠1, r3c1≠2, r3c1≠1 naked-triplets-in-a-block: b5{r4c6 r5c5 r6c4}{n3 n2 n1} ==> r6c6≠3, r6c6≠2, r6c6≠1, r6c5≠3, r5c6≠3, r5c6≠2, r5c6≠1, r5c4≠3, r5c4≠2, r5c4≠1, r4c5≠3, r4c4≠3, r4c4≠2, r4c4≠1 naked-triplets-in-a-block: b2{r1c6 r2c4 r3c5}{n2 n3 n1} ==> r3c6≠3, r3c4≠3, r2c6≠3, r2c6≠2, r2c6≠1, r2c5≠3, r1c5≠3, r1c4≠3, r1c4≠2, r1c4≠1 whip[1]: c5n3{r9 .} ==> r7c4≠3, r7c6≠3, r8c4≠3, r8c6≠3, r9c4≠3, r9c6≠3 naked-triplets-in-a-block: b1{r1c2 r2c1 r3c3}{n2 n3 n1} ==> r3c2≠3, r3c1≠3, r2c3≠3, r2c3≠2, r2c3≠1, r2c2≠3, r2c2≠2, r2c2≠1, r1c3≠3, r1c3≠2, r1c3≠1, r1c1≠3, r1c1≠2, r1c1≠1 whip[1]: r3n3{c9 .} ==> r1c7≠3, r1c8≠3, r1c9≠3, r2c7≠3, r2c8≠3, r2c9≠3 biv-chain[2]: b1n3{r2c1 r1c2} - c6n3{r1 r4} ==> r4c1≠3 biv-chain[2]: c4n3{r6 r2} - b1n3{r2c1 r1c2} ==> r6c2≠3 naked-pairs-in-a-block: b4{r4c1 r6c2}{n1 n2} ==> r6c3≠2, r6c3≠1, r6c1≠2, r6c1≠1, r5c3≠2, r5c3≠1, r5c2≠2, r5c2≠1, r5c1≠2, r5c1≠1, r4c3≠2, r4c3≠1, r4c2≠2, r4c2≠1 whip[6]: r1n3{c2 c6} - r2n3{c4 c1} - b1n1{r2c1 r3c3} - c5n1{r3 r5} - r4c6{n1 n2} - b4n2{r4c1 .} ==> r1c2≠2 biv-chain[3]: b4n2{r6c2 r4c1} - b1n2{r2c1 r3c3} - c5n2{r3 r5} ==> r6c4≠2 biv-chain[3]: b4n1{r4c1 r6c2} - r6c4{n1 n3} - r2n3{c4 c1} ==> r2c1≠1 biv-chain[3]: b4n2{r6c2 r4c1} - r2c1{n2 n3} - r1c2{n3 n1} ==> r6c2≠1 singles ==> r6c2=2, r4c1=1 biv-chain[3]: c5n2{r3 r5} - r4c6{n2 n3} - b2n3{r1c6 r2c4} ==> r2c4≠2 whip[1]: c4n2{r9 .} ==> r7c6≠2, r8c6≠2, r9c6≠2 naked-pairs-in-a-column: c4{r2 r6}{n1 n3} ==> r9c4≠1, r8c4≠1, r7c4≠1 whip[1]: b8n1{r9c6 .} ==> r1c6≠1 biv-chain[3]: r3c5{n2 n1} - r2c4{n1 n3} - r2c1{n3 n2} ==> r3c3≠2 singles ==> r3c3=1, r1c2=3, r1c6=2 PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RN-CELL r3n2

Website · by **denis_berthier** » Sat Dec 24, 2022 5:26 am

denis_berthier wrote:(Some day, I may write a more general function solve-kl-digit-patterns.)

Done.
The above proof can now be called by:

Code: Select all: (solve-kl-digit-pattern-string 2 3 ".3...3...3..3.......2.2....3....3.......2.....3.3.....")

(Dots can also be 0s).

Note that k and l in the pattern string stand respectively for candidates 1...k and 1...l in the corresponding cell.
As was the case for function solve-k-digit-pattern-string, "incomplete" strings (i.e. shorter than 81) are automatically complete with 0s.

Website · by **denis_berthier** » Fri Dec 30, 2022 7:24 am

.
I realise I've forgotten to give my solution to this puzzle. I'll give two, one based on ORk-g-whips (next post) and one based on ORk-forcing-g-whips (second next post).

Starting from the RS after whips[1], both have the same start:

Code: Select all: naked-pairs-in-a-row: r1{c3 c9}{n3 n9} ==> r1c6≠3, r1c5≠3, r1c4≠3, r1c2≠9, r1c2≠3, r1c1≠3 naked-single ==> r1c1=1 whip[1]: b2n3{r3c6 .} ==> r3c1≠3, r3c2≠3, r3c3≠3, r3c7≠3 singles ==> r3c1=7, r5c3=7 147 g-candidates, 693 csp-glinks and 384 non-csp glinks +----------------------+----------------------+----------------------+ ! 1 25 39 ! 246 456 246 ! 7 8 39 ! ! 346 3456 346 ! 7 8 9 ! 1 35 2 ! ! 7 2589 89 ! 123 35 123 ! 59 6 4 ! +----------------------+----------------------+----------------------+ ! 2 346 5 ! 9 1 346 ! 8 347 367 ! ! 346 13469 7 ! 346 2 8 ! 34569 13459 1369 ! ! 8 13469 3469 ! 5 346 7 ! 2 1349 1369 ! +----------------------+----------------------+----------------------+ ! 346 7 1 ! 8 9 2346 ! 346 234 5 ! ! 5 3468 3468 ! 12346 346 12346 ! 3469 123479 136789 ! ! 9 3468 2 ! 1346 7 5 ! 346 134 1368 ! +----------------------+----------------------+----------------------+ OR5-anti-tridagon[12] for digits 4, 6 and 3 in blocks: b4, with cells: r4c2, r5c1, r6c3 b5, with cells: r4c6, r5c4, r6c5 b7, with cells: r9c2, r7c1, r8c3 b8, with cells: r9c4, r7c6, r8c5 with 5 guardians: n9r6c3 n2r7c6 n8r8c3 n8r9c2 n1r9c4 z-chain[3]: r2c8{n3 n5} - c7n5{r3 r5} - c7n3{r5 .} ==> r7c8≠3, r9c8≠3, r8c8≠3 t-whip[3]: r9c8{n1 n4} - c7n4{r9 r5} - r5n5{c7 .} ==> r5c8≠1 t-whip[6]: r9c8{n1 n4} - c7n4{r9 r5} - r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - b6n9{r6c9 .} ==> r6c8≠1 whip[1]: c8n1{r9 .} ==> r8c9≠1, r9c9≠1 whip[5]: c2n1{r6 r5} - b4n9{r5c2 r6c3} - r6c8{n9 n3} - b3n3{r2c8 r1c9} - r1n9{c9 .} ==> r6c2≠4 t-whip[8]: c7n6{r9 r5} - r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - b6n9{r6c9 r6c8} - c3n9{r6 r3} - r3n8{c3 c2} - r9n8{c2 .} ==> r9c9≠6 t-whip[8]: r9c9{n8 n3} - c7n3{r9 r5} - r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - b6n9{r6c9 r6c8} - c3n9{r6 r3} - c3n8{r3 .} ==> r8c9≠8, r9c2≠8 At least one candidate of a previous Trid-OR5-relation has just been eliminated. There remains a Trid-OR4-relation between candidates: n9r6c3 n2r7c6 n8r8c3 n1r9c4 +-------------------+-------------------+-------------------+ ! 1 25 39 ! 246 456 246 ! 7 8 39 ! ! 346 3456 346 ! 7 8 9 ! 1 35 2 ! ! 7 2589 89 ! 123 35 123 ! 59 6 4 ! +-------------------+-------------------+-------------------+ ! 2 346 5 ! 9 1 346 ! 8 347 367 ! ! 346 13469 7 ! 346 2 8 ! 34569 3459 1369 ! ! 8 1369 3469 ! 5 346 7 ! 2 349 1369 ! +-------------------+-------------------+-------------------+ ! 346 7 1 ! 8 9 2346 ! 346 24 5 ! ! 5 3468 3468 ! 12346 346 12346 ! 3469 12479 3679 ! ! 9 346 2 ! 1346 7 5 ! 346 14 38 ! +-------------------+-------------------+-------------------+ hidden-single-in-a-row ==> r9c9=8

RESOLUTION STATE RS2, common starting point for the two solutions:

Code: Select all: +-------------------+-------------------+-------------------+ ! 1 25 39 ! 246 456 246 ! 7 8 39 ! ! 346 3456 346 ! 7 8 9 ! 1 35 2 ! ! 7 2589 89 ! 123 35 123 ! 59 6 4 ! +-------------------+-------------------+-------------------+ ! 2 346 5 ! 9 1 346 ! 8 347 367 ! ! 346 13469 7 ! 346 2 8 ! 34569 3459 1369 ! ! 8 1369 3469 ! 5 346 7 ! 2 349 1369 ! +-------------------+-------------------+-------------------+ ! 346 7 1 ! 8 9 2346 ! 346 24 5 ! ! 5 3468 3468 ! 12346 346 12346 ! 3469 12479 3679 ! ! 9 346 2 ! 1346 7 5 ! 346 14 8 ! +-------------------+-------------------+-------------------+

Website · by **denis_berthier** » Fri Dec 30, 2022 7:30 am

.
Solution with ORk-g-whips:

The ORk-part in gWW8+OR4gW8:
whip[8]: c2n1{r6 r5} - b4n9{r5c2 r6c3} - r1n9{c3 c9} - b3n3{r1c9 r2c8} - b3n5{r2c8 r3c7} - r3c5{n5 n3} - r6n3{c5 c9} - r6n1{c9 .} ==> r6c2≠6
Trid-OR4-whip[8]: r3n2{c6 c2} - r3n8{c2 c3} - b1n9{r3c3 r1c3} - OR4{{n9r6c3 n2r7c6 n8r8c3 | n1r9c4}} - r9c8{n1 n4} - c7n4{r9 r5} - c7n5{r5 r3} - r3n9{c7 .} ==> r1c6≠2
whip[5]: c6n1{r8 r3} - c6n2{r3 r7} - r7c8{n2 n4} - r4n4{c8 c2} - b7n4{r8c2 .} ==> r8c6≠4
whip[6]: c6n1{r3 r8} - c6n2{r8 r7} - r7c8{n2 n4} - b6n4{r4c8 r5c7} - c7n5{r5 r3} - r3c5{n5 .} ==> r3c6≠3
Trid-OR4-gwhip[8]: r3n8{c2 c3} - b1n9{r3c3 r1c3} - r3n9{c2 c7} - c7n5{r3 r5} - b6n4{r5c7 r456c8} - r9c8{n4 n1} - OR4{{n1r9c4 n8r8c3 n9r6c3 | n2r7c6}} - r7c8{n2 .} ==> r3c2≠2
singles ==> r1c2=2, r1c5=5, r3c5=3
whip[7]: r5n5{c7 c8} - r2c8{n5 n3} - r1c9{n3 n9} - r5n9{c9 c2} - c2n1{r5 r6} - r6n3{c2 c3} - r1c3{n3 .} ==> r5c7≠3
whip[1]: c7n3{r9 .} ==> r8c9≠3
whip[7]: r1n6{c6 c4} - r1n4{c4 c6} - r4c6{n4 n3} - r7c6{n3 n2} - r7c8{n2 n4} - b6n4{r4c8 r5c7} - r5c4{n4 .} ==> r8c6≠6
whip[8]: r8c5{n4 n6} - r6c5{n6 n4} - r4n4{c6 c8} - c8n7{r4 r8} - r8c9{n7 n9} - r1n9{c9 c3} - r3c3{n9 n8} - r8n8{c3 .} ==> r8c2≠4
Trid-OR4-whip[8]: b3n9{r1c9 r3c7} - r3c3{n9 n8} - r3c2{n8 n5} - c2n9{r3 r6} - c8n9{r6 r8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r5c9≠9
Trid-OR4-whip[8]: b3n9{r1c9 r3c7} - r3c3{n9 n8} - r3c2{n8 n5} - c2n9{r3 r5} - c8n9{r5 r8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r6c9≠9
Trid-OR4-whip[6]: r1c3{n3 n9} - c9n9{r1 r8} - r8n7{c9 c8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r8c3≠3
Trid-OR4-whip[7]: r3n8{c2 c3} - b1n9{r3c3 r1c3} - c9n9{r1 r8} - r8n7{c9 c8} - c8n2{r8 r7} - OR4{{n2r7c6 n8r8c3 n9r6c3 | n1r9c4}} - c8n1{r9 .} ==> r3c2≠5

The end is easy:

Code: Select all: singles ==> r2c2=5, r2c8=3, r1c9=9, r1c3=3, r3c7=5, r5c8=5 hidden-pairs-in-a-row: r6{n1 n3}{c2 c9} ==> r6c9≠6, r6c2≠9 finned-x-wing-in-columns: n3{c1 c4}{r5 r7} ==> r7c6≠3 biv-chain[3]: c5n4{r8 r6} - r6c8{n4 n9} - b9n9{r8c8 r8c7} ==> r8c7≠4 biv-chain[3]: r2c3{n4 n6} - r6n6{c3 c5} - c5n4{r6 r8} ==> r8c3≠4 z-chain[3]: b7n4{r7c1 r9c2} - r4n4{c2 c8} - r8n4{c8 .} ==> r7c6≠4 biv-chain[3]: r7c8{n4 n2} - r7c6{n2 n6} - r8c5{n6 n4} ==> r8c8≠4 whip[1]: r8n4{c5 .} ==> r9c4≠4 biv-chain[3]: r1c4{n6 n4} - b8n4{r8c4 r8c5} - c5n6{r8 r6} ==> r5c4≠6 biv-chain[3]: r5c4{n4 n3} - c1n3{r5 r7} - b7n4{r7c1 r9c2} ==> r5c2≠4 biv-chain[4]: r8c3{n6 n8} - r3c3{n8 n9} - c2n9{r3 r5} - c7n9{r5 r8} ==> r8c7≠6 z-chain[4]: r8c3{n6 n8} - r8c2{n8 n3} - c6n3{r8 r4} - b5n6{r4c6 .} ==> r8c5≠6 singles ==> r8c5=4, r6c5=6 biv-chain[2]: r6n4{c8 c3} - c2n4{r4 r9} ==> r9c8≠4 naked-single ==> r9c8=1 At least one candidate of a previous Trid-OR4-relation has just been eliminated. There remains a Trid-OR3-relation between candidates: n9r6c3 n2r7c6 n8r8c3 +----------------+----------------+----------------+ ! 1 2 3 ! 46 5 46 ! 7 8 9 ! ! 46 5 46 ! 7 8 9 ! 1 3 2 ! ! 7 89 89 ! 12 3 12 ! 5 6 4 ! +----------------+----------------+----------------+ ! 2 346 5 ! 9 1 34 ! 8 47 367 ! ! 346 1369 7 ! 34 2 8 ! 469 5 136 ! ! 8 13 49 ! 5 6 7 ! 2 49 13 ! +----------------+----------------+----------------+ ! 346 7 1 ! 8 9 26 ! 346 24 5 ! ! 5 368 68 ! 1236 4 123 ! 39 1279 67 ! ! 9 346 2 ! 36 7 5 ! 346 1 8 ! +----------------+----------------+----------------+ hidden-pairs-in-a-column: c4{n1 n2}{r3 r8} ==> r8c4≠6, r8c4≠3 finned-x-wing-in-columns: n3{c4 c1}{r5 r9} ==> r9c2≠3 finned-x-wing-in-rows: n6{r4 r8}{c9 c2} ==> r9c2≠6 naked-single ==> r9c2=4 biv-chain[3]: c1n3{r7 r5} - c4n3{r5 r9} - b8n6{r9c4 r7c6} ==> r7c1≠6 naked-single ==> r7c1=3 whip[1]: b7n6{r8c3 .} ==> r8c9≠6 stte

Website · by **denis_berthier** » Fri Dec 30, 2022 7:35 am

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Solution with ORk-forcing-g-whips:

The ORk part:

Code: Select all: OR4-forcing-gwhip-elim[7]: || n9r6c3 - || n1r9c4 - partial-gwhip[2]: b9n1{r9c8 r8c8} - c8n9{r8 r456} - || n2r7c6 - partial-gwhip[2]: r8n2{c6 c8} - c8n9{r8 r456} - || n8r8c3 - partial-whip[2]: r3c3{n8 n9} - r1n9{c3 c9} - ==> r6c9≠9 whip[8]: c2n1{r6 r5} - b4n9{r5c2 r6c3} - r1n9{c3 c9} - b3n3{r1c9 r2c8} - b3n5{r2c8 r3c7} - r3c5{n5 n3} - r6n3{c5 c9} - r6n1{c9 .} ==> r6c2≠6 Trid-OR4-forcing-whip-elim[8]: || n9r6c3 - partial-whip[1]: r1n9{c3 c9} - || n1r9c4 - partial-whip[2]: b9n1{r9c8 r8c8} - r8n7{c8 c9} - || n2r7c6 - partial-whip[2]: r8n2{c6 c8} - r8n7{c8 c9} - || n8r8c3 - partial-whip[2]: r3c3{n8 n9} - r1n9{c3 c9} - ==> r8c9≠9 biv-chain[3]: r8n9{c8 c7} - r3c7{n9 n5} - b6n5{r5c7 r5c8} ==> r5c8≠9 biv-chain[4]: r2c8{n3 n5} - r3c7{n5 n9} - r8n9{c7 c8} - c8n7{r8 r4} ==> r4c8≠3 Trid-OR4-forcing-whip-elim[7]: || n8r8c3 - partial-whip[1]: r3c3{n8 n9} - || n9r6c3 - partial-whip[1]: c2n9{r6 r3} - || n1r9c4 - partial-whip[2]: b9n1{r9c8 r8c8} - r8n9{c8 c7} - || n2r7c6 - partial-whip[2]: r8n2{c6 c8} - r8n9{c8 c7} - ==> r3c7≠9

The end is easy:

Code: Select all: singles ==> r3c7=5, r2c8=3, r1c9=9, r1c3=3, r3c5=3, r5c8=5, r2c2=5, r1c2=2, r1c5=5 hidden-pairs-in-a-row: r6{n1 n3}{c2 c9} ==> r6c9≠6, r6c2≠9 finned-x-wing-in-columns: n3{c1 c4}{r5 r7} ==> r7c6≠3 biv-chain[3]: r6n3{c9 c2} - c2n1{r6 r5} - r5n9{c2 c7} ==> r5c7≠3 whip[1]: c7n3{r9 .} ==> r8c9≠3 biv-chain[3]: c5n4{r8 r6} - r6c8{n4 n9} - b9n9{r8c8 r8c7} ==> r8c7≠4 biv-chain[3]: r2c3{n4 n6} - r6n6{c3 c5} - c5n4{r6 r8} ==> r8c3≠4 biv-chain[4]: r8c3{n6 n8} - r3c3{n8 n9} - c2n9{r3 r5} - c7n9{r5 r8} ==> r8c7≠6 z-chain[4]: b7n4{r9c2 r7c1} - r7n3{c1 c7} - r8c7{n3 n9} - r5n9{c7 .} ==> r5c2≠4 biv-chain[5]: r6c9{n3 n1} - b4n1{r6c2 r5c2} - r5n9{c2 c7} - r8c7{n9 n3} - c6n3{r8 r4} ==> r4c9≠3 naked-pairs-in-a-column: c9{r4 r8}{n6 n7} ==> r5c9≠6 finned-x-wing-in-columns: n6{c9 c5}{r8 r4} ==> r4c6≠6 biv-chain[3]: r1c4{n4 n6} - b5n6{r5c4 r6c5} - c5n4{r6 r8} ==> r8c4≠4, r9c4≠4 z-chain[3]: b8n4{r8c6 r7c6} - r4n4{c6 c2} - r9n4{c2 .} ==> r8c8≠4 whip[4]: c5n4{r8 r6} - r4c6{n4 n3} - r4c2{n3 n6} - r6n6{c3 .} ==> r8c2≠4 whip[1]: r8n4{c6 .} ==> r7c6≠4 z-chain[4]: b7n4{r9c2 r7c1} - r7n3{c1 c7} - b9n6{r7c7 r8c9} - r4n6{c9 .} ==> r9c2≠6 finned-x-wing-in-columns: n6{c5 c2}{r8 r6} ==> r6c3≠6 singles ==> r6c5=6, r8c5=4 biv-chain[2]: r6n4{c8 c3} - c2n4{r4 r9} ==> r9c8≠4 naked-single ==> r9c8=1 At least one candidate of a previous Trid-OR4-relation has just been eliminated. There remains a Trid-OR3-relation between candidates: n9r6c3 n2r7c6 n8r8c3 +----------------+----------------+----------------+ ! 1 2 3 ! 46 5 46 ! 7 8 9 ! ! 46 5 46 ! 7 8 9 ! 1 3 2 ! ! 7 89 89 ! 12 3 12 ! 5 6 4 ! +----------------+----------------+----------------+ ! 2 346 5 ! 9 1 34 ! 8 47 67 ! ! 346 1369 7 ! 34 2 8 ! 469 5 13 ! ! 8 13 49 ! 5 6 7 ! 2 49 13 ! +----------------+----------------+----------------+ ! 346 7 1 ! 8 9 26 ! 346 24 5 ! ! 5 368 68 ! 1236 4 1236 ! 39 1279 67 ! ! 9 34 2 ! 36 7 5 ! 346 1 8 ! +----------------+----------------+----------------+ hidden-pairs-in-a-column: c4{n1 n2}{r3 r8} ==> r8c4≠6, r8c4≠3 finned-x-wing-in-columns: n3{c4 c1}{r5 r9} ==> r9c2≠3 naked-single ==> r9c2=4 biv-chain[3]: c1n3{r7 r5} - r5c4{n3 n4} - c7n4{r5 r7} ==> r7c7≠3 hidden-single-in-a-row ==> r7c1=3 whip[1]: b7n6{r8c3 .} ==> r8c6≠6, r8c9≠6 stte

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#18913 in mith's 63137 T&E(3) min-expands

#18913 in mith's 63137 T&E(3) min-expands

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Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands

Re: #18913 in mith's 63137 T&E(3) min-expands