The New Sudoku Players' Forum

by **mith** » Wed Mar 03, 2021 9:29 pm

Code: Select all: +-------+-------+-------+ | 9 8 . | . . . | . . . | | 7 . . | . . 6 | 5 . . | | . 5 . | . 7 . | 9 6 . | +-------+-------+-------+ | . . 4 | . 8 . | . 3 . | | . . . | . . 7 | 8 . . | | . 7 . | 3 . . | . . 9 | +-------+-------+-------+ | . . . | . 5 . | . . 6 | | . . . | 6 . 9 | 3 8 . | | . . 2 | . . 3 | . 7 . | +-------+-------+-------+ 98.......7....65...5..7.96...4.8..3......78...7.3....9....5...6...6.938...2..3.7.

Code: Select all: +-------+-------+-------+ | . . . | . . . | . . . | | . . 9 | 8 . . | 7 6 . | | . 5 . | . 4 . | 9 8 . | +-------+-------+-------+ | . 9 . | . 3 . | . . . | | . . 7 | 6 . . | . . . | | 8 . . | . . 7 | 5 . . | +-------+-------+-------+ | 9 . 8 | 7 . . | 6 5 . | | 6 . . | 9 2 . | . . . | | . . . | . . 5 | . . . | +-------+-------+-------+ ...........98..76..5..4.98..9..3......76.....8....75..9.87..65.6..92.........5...

(Not an original idea, but I had fun playing around with it...)

Website · by **denis_berthier** » Thu Mar 04, 2021 7:48 am

mith wrote:
Code: Select all
+-------+-------+-------+ | . . . | . . . | . . . | | . . 9 | 8 . . | 7 6 . | | . 5 . | . 4 . | 9 8 . | +-------+-------+-------+ | . 9 . | . 3 . | . . . | | . . 7 | 6 . . | . . . | | 8 . . | . . 7 | 5 . . | +-------+-------+-------+ | 9 . 8 | 7 . . | 6 5 . | | 6 . . | 9 2 . | . . . | | . . . | . . 5 | . . . | +-------+-------+-------+ ...........98..76..5..4.98..9..3......76.....8....75..9.87..65.6..92.........5...

Second puzzle first.

Normal simplest-first SudoRules solution in W6: Show

After that, I tried to find a 1-step solution, but the lists of backdoors are not very promising:

Code: Select all: 1 BRT-ANTI-BACKDOOR FOUND: n1r2c4 1 W1-ANTI-BACKDOOR FOUND: n1r2c4 1 S-ANTI-BACKDOOR FOUND: n1r2c4

and indeed there is no whip, ... or g-braid 1-step solution.

I then tried the simplest anti-backdoor pairs:

Code: Select all: 1 BRT-ANTI-BACKDOOR-PAIR FOUND: n1r8c2, n3r6c3

but this doesn't allow a 2-whip solution.

Next step was to try the W1 anti-backdoor pairs:

Code: Select all: 17 W1-ANTI-BACKDOOR-PAIRS FOUND: n1r9c7, n3r6c3 n1r8c2, n3r6c3 n2r7c9, n3r6c3 n1r6c4, n3r9c4 n1r6c4, n4r7c6 n1r6c4, n4r6c4 n1r6c4, n3r6c3 n1r6c4, n3r2c6 n3r6c3, n4r7c2 n2r4c6, n2r5c6 n1r2c6, n3r9c4 n1r2c6, n4r7c6 n1r2c6, n4r6c4 n1r2c6, n3r6c3 n1r2c6, n3r2c6 n2r2c2, n3r6c3 n2r1c4, n2r3c4

Starting from the same resolution state as in the normal solution above (obtained after Singles only):

Code: Select all: 1234 8 6 123 7 9 1234 1234 5 1234 1234 9 8 5 123 7 6 1234 7 5 123 123 4 6 9 8 123 124 9 124 5 3 124 8 7 6 5 1234 7 6 8 124 1234 1234 9 8 6 1234 124 9 7 5 1234 1234 9 234 8 7 1 34 6 5 234 6 134 5 9 2 8 134 134 7 1234 7 1234 34 6 5 1234 9 8 134 candidates, 549 csp-links and 549 links. Density = 6.16%

I get eight 2-step solutions (not counting Singles or whips[1]):

Code: Select all: whip[6]: r7c6{n3 n4} - c4n4{r9 r6} - c9n4{r6 r2} - r1n4{c8 c1} - c1n3{r1 r2} - c6n3{r2 .} ==> r9c4 ≠ 3 singles ==> r9c4 = 4, r7c6 = 3 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c6n1{r5 r2} - c9n1{r2 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r6c4 ≠ 1 singles ==> r6c4 = 2, r2c6 = 2 whip[1]: r4n2{c3 .} ==> r5c2 ≠ 2 stte

Code: Select all: whip[6]: r7c6{n3 n4} - c4n4{r9 r6} - c9n4{r6 r2} - r1n4{c8 c1} - c1n3{r1 r2} - c6n3{r2 .} ==> r9c4 ≠ 3 singles ==> r9c4 = 4, r7c6 = 3 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c4n1{r3 r6} - c9n1{r6 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r2c6 ≠ 1 singles ==> r2c6 = 2, r6c4 = 2 whip[1]: b6n2{r5c8 .} ==> r5c2 ≠ 2 stte

Code: Select all: whip[6]: c6n3{r7 r2} - c4n3{r3 r9} - c4n4{r9 r6} - c9n4{r6 r2} - r1n4{c8 c1} - c1n3{r1 .} ==> r7c6 ≠ 4 singles ==> r7c6 = 3, r9c4 = 4 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c6n1{r5 r2} - c9n1{r2 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r6c4 ≠ 1 singles ==> r6c4 = 2, r2c6 = 2 whip[1]: r4n2{c3 .} ==> r5c2 ≠ 2 stte

Code: Select all: whip[6]: c6n3{r7 r2} - c4n3{r3 r9} - c4n4{r9 r6} - c9n4{r6 r2} - r1n4{c8 c1} - c1n3{r1 .} ==> r7c6 ≠ 4 singles ==> r7c6 = 3, r9c4 = 4 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c4n1{r3 r6} - c9n1{r6 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r2c6 ≠ 1 singles ==> r2c6 = 2, r6c4 = 2 whip[1]: b6n2{r5c8 .} ==> r5c2 ≠ 2 stte

Code: Select all: whip[6]: r9c4{n4 n3} - r7c6{n3 n4} - c9n4{r7 r2} - r1n4{c8 c1} - c1n3{r1 r2} - c6n3{r2 .} ==> r6c4 ≠ 4 singles ==> r9c4 = 4, r7c6 = 3 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c6n1{r5 r2} - c9n1{r2 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r6c4 ≠ 1 singles ==> r6c4 = 2, r2c6 = 2 whip[1]: r4n2{c3 .} ==> r5c2 ≠ 2

Code: Select all: whip[6]: r9c4{n4 n3} - r7c6{n3 n4} - c9n4{r7 r2} - r1n4{c8 c1} - c1n3{r1 r2} - c6n3{r2 .} ==> r6c4 ≠ 4 singles ==> r9c4 = 4, r7c6 = 3 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c4n1{r3 r6} - c9n1{r6 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r2c6 ≠ 1 singles ==> r2c6 = 2, r6c4 = 2 whip[1]: b6n2{r5c8 .} ==> r5c2 ≠ 2 stte

Code: Select all: whip[6]: c4n3{r3 r9} - r7c6{n3 n4} - c4n4{r9 r6} - c9n4{r6 r2} - r1n4{c8 c1} - c1n3{r1 .} ==> r2c6 ≠ 3 singles ==> r7c6 = 3, r9c4 = 4 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c6n1{r5 r2} - c9n1{r2 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r6c4 ≠ 1 singles ==> r6c4 = 2, r2c6 = 2 whip[1]: r4n2{c3 .} ==> r5c2 ≠ 2 stte

Code: Select all: whip[6]: c4n3{r3 r9} - r7c6{n3 n4} - c4n4{r9 r6} - c9n4{r6 r2} - r1n4{c8 c1} - c1n3{r1 .} ==> r2c6 ≠ 3 singles ==> r7c6 = 3, r9c4 = 4 whip[1]: b7n4{r8c2 .} ==> r2c2 ≠ 4, r5c2 ≠ 4 whip[1]: b1n4{r2c1 .} ==> r4c1 ≠ 4 whip[10]: c4n1{r3 r6} - c9n1{r6 r3} - r1n1{c8 c1} - r4n1{c1 c3} - b4n4{r4c3 r6c3} - b4n3{r6c3 r5c2} - r2c2{n3 n2} - r7c2{n2 n4} - c9n4{r7 r2} - c1n4{r2 .} ==> r2c6 ≠ 1 singles ==> r2c6 = 2, r6c4 = 2 whip[1]: b6n2{r5c8 .} ==> r5c2 ≠ 2 stte

The similarity of the solutions suggest there is some symmetry, but I don't have time now for checking it.

[Edit: re-ordered and grouped the 2-step solutions by pairs with the same first elimination.

Website · by **denis_berthier** » Thu Mar 04, 2021 10:04 am

[Edit]: First (not Second) puzzle

mith wrote:
Code: Select all
+-------+-------+-------+ | 9 8 . | . . . | . . . | | 7 . . | . . 6 | 5 . . | | . 5 . | . 7 . | 9 6 . | +-------+-------+-------+ | . . 4 | . 8 . | . 3 . | | . . . | . . 7 | 8 . . | | . 7 . | 3 . . | . . 9 | +-------+-------+-------+ | . . . | . 5 . | . . 6 | | . . . | 6 . 9 | 3 8 . | | . . 2 | . . 3 | . 7 . | +-------+-------+-------+ 98.......7....65...5..7.96...4.8..3......78...7.3....9....5...6...6.938...2..3.7.

Normal simplest-first SudoRules solution in W4: Show

As in the second puzzle case, single-elimination solutions are unlikely. The anti-backdoors are:

Code: Select all: 1 BRT-ANTI-BACKDOOR FOUND: n1r2c4 1 W1-ANTI-BACKDOOR FOUND: n1r2c4 1 S-ANTI-BACKDOOR FOUND: n1r2c4

and it doesn't give rise to a 1-step whip,... g-braid solution

So, I tried W1-anti-backdoor pairs:

Code: Select all: 126 W1-ANTI-BACKDOOR-PAIRS FOUND: n1r9c7, n2r8c9 n1r9c7, n4r8c5 n1r9c7, n2r7c6 n1r9c7, n4r5c9 n2r8c9, n4r9c5 n2r8c9, n4r8c9 n4r8c5, n4r9c5 n1r8c2, n1r9c7 n1r8c2, n4r9c5 n1r8c2, n2r8c9 n1r8c2, n1r8c5 n1r8c2, n4r7c7 n1r8c2, n2r7c6 n1r8c2, n4r5c9 n1r8c2, n4r5c4 n1r8c2, n4r2c8 n1r8c2, n4r1c6 n1r7c7, n1r9c7 n1r7c7, n4r9c5 n2r7c6, n4r9c5 n2r7c6, n4r8c9 n1r7c6, n2r8c9 n1r7c6, n2r7c6 n1r6c8, n2r6c8 n1r6c6, n1r9c7 n1r6c6, n4r9c5 n1r6c6, n1r8c2 n1r6c6, n1r7c6 n4r5c9, n4r9c5 n4r5c9, n4r8c9 n4r5c9, n4r6c7 n4r5c4, n4r7c1 n4r5c4, n4r6c6 n4r5c4, n4r5c9 n2r5c4, n4r5c4 n2r5c4, n4r2c8 n2r5c4, n4r1c6 n1r4c1, n4r5c4 n1r4c1, n4r2c8 n1r4c1, n4r1c6 n4r3c4, n4r5c4 n2r3c1, n4r5c4 n2r3c1, n4r2c8 n2r3c1, n4r1c6 n4r2c8, n4r7c1 n4r2c8, n4r6c6 n4r2c8, n4r5c9 n4r2c8, n4r3c4 n2r2c8, n4r5c4 n2r2c8, n4r2c8 n2r2c8, n4r1c6 n1r2c4, n1r9c7 n1r2c4, n4r9c5 n1r2c4, n4r8c9 n1r2c4, n2r8c9 n1r2c4, n4r8c5 n1r2c4, n1r8c5 n1r2c4, n1r8c2 n1r2c4, n4r7c7 n1r2c4, n1r7c7 n1r2c4, n2r7c6 n1r2c4, n1r7c6 n1r2c4, n4r7c2 n1r2c4, n1r7c2 n1r2c4, n4r7c1 n1r2c4, n3r7c1 n1r2c4, n2r6c8 n1r2c4, n1r6c8 n1r2c4, n6r6c7 n1r2c4, n4r6c7 n1r2c4, n2r6c7 n1r2c4, n4r6c6 n1r2c4, n1r6c6 n1r2c4, n4r6c5 n1r2c4, n2r6c5 n1r2c4, n1r6c5 n1r2c4, n4r5c9 n1r2c4, n1r5c9 n1r2c4, n6r5c5 n1r2c4, n2r5c5 n1r2c4, n1r5c5 n1r2c4, n4r5c4 n1r2c4, n2r5c4 n1r2c4, n3r5c2 n1r2c4, n2r5c2 n1r2c4, n1r5c2 n1r2c4, n2r5c1 n1r2c4, n1r5c1 n1r2c4, n2r4c9 n1r2c4, n1r4c9 n1r2c4, n7r4c7 n1r2c4, n2r4c7 n1r2c4, n1r4c7 n1r2c4, n6r4c2 n1r2c4, n2r4c2 n1r2c4, n1r4c1 n1r2c4, n4r3c4 n1r2c4, n2r3c1 n1r2c4, n4r2c8 n1r2c4, n2r2c8 n1r2c4, n2r2c4 n1r2c4, n4r2c2 n1r2c4, n7r1c9 n1r2c4, n2r1c9 n1r2c4, n4r1c8 n1r2c4, n4r1c7 n1r2c4, n2r1c7 n1r2c4, n4r1c6 n1r2c4, n2r1c6 n4r2c2, n4r5c4 n4r2c2, n4r2c8 n1r1c9, n1r2c4 n4r1c8, n4r5c4 n4r1c8, n4r2c8 n1r1c8, n1r6c8 n1r1c8, n1r2c4 n1r1c7, n1r2c4 n4r1c6, n4r7c1 n4r1c6, n4r6c6 n4r1c6, n4r5c9 n4r1c6, n4r3c4 n4r1c6, n4r2c2 n4r1c6, n4r1c8 n2r1c6, n4r5c4 n2r1c6, n4r2c8 n2r1c6, n4r1c6

31 of them give rise to a 2-step solution (not counting Singles and whips[1], starting from the same resolution state as in the normal solution.

Code: Select all: Starting non trivial part of solution with the following RESOLUTION STATE: 9 8 6 5 3 124 1247 124 1247 7 24 3 124 9 6 5 124 8 24 5 1 24 7 8 9 6 3 12 126 4 9 8 5 1267 3 127 123 1236 9 124 1246 7 8 5 124 8 7 5 3 1246 124 1246 124 9 134 134 8 7 5 124 124 9 6 5 14 7 6 124 9 3 8 124 6 9 2 8 14 3 14 7 5

I'll give only three of the simplest solutions:

Code: Select all: whip[4]: c5n2{r6 r8} - c6n2{r7 r1} - c9n2{r1 r4} - b4n2{r4c1 .} ==> r5c4 ≠ 2 whip[1]: c4n2{r3 .} ==> r1c6 ≠ 2 whip[1]: r1n2{c9 .} ==> r2c8 ≠ 2 whip[8]: c4n1{r5 r2} - r2n2{c4 c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r5c4 ≠ 4 stte

Code: Select all: whip[4]: r1n2{c9 c6} - c4n2{r3 r5} - r6n2{c6 c7} - r7n2{c7 .} ==> r2c8 ≠ 2 whip[1]: b3n2{r1c9 .} ==> r1c6 ≠ 2 whip[1]: b2n2{r3c4 .} ==> r5c4 ≠ 2 whip[8]: c4n1{r5 r2} - r2n2{c4 c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r5c4 ≠ 4 stte

Code: Select all: whip[4]: c4n2{r3 r5} - c5n2{r6 r8} - c9n2{r8 r4} - b4n2{r4c1 .} ==> r1c6 ≠ 2 whip[1]: r1n2{c9 .} ==> r2c8 ≠ 2 whip[1]: b2n2{r3c4 .} ==> r5c4 ≠ 2 whip[8]: c4n1{r5 r2} - r2n2{c4 c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r5c4 ≠ 4 stte

by **DEFISE** » Thu Mar 04, 2021 1:01 pm

Hi Denis,
I get like you for the two puzzles with my "few steps" algorithm (even without several random choices of targets).
Maybe it's luck.

Website · by **denis_berthier** » Thu Mar 04, 2021 1:56 pm

DEFISE wrote:Hi Denis,
I get like you for the two puzzles with my "few steps" algorithm (even without several random choices of targets).
Maybe it's luck.

Hi François,
For the hard one, I've listed all the possible 2-step solutions with whips, so you can only get one of them.

For the easier one, I've listed only part of the solutions involving the smaller whips. Whether or not you restrict a priori the whips length to 8, there's a part of luck.

Website · by **denis_berthier** » Fri Mar 05, 2021 7:04 am

denis_berthier wrote:Second puzzle
[...]I'll give only three of the simplest solutions

One should read First puzzle.
And, actually, it is interesting to give all the solutions (there are 12) involving a whip[4] and a whip[8 or 9], grouping them according to the first whip[4].

First group:

Code: Select all: whip[4]: c5n2{r6 r8} - c6n2{r7 r1} - c9n2{r1 r4} - b4n2{r4c1 .} ==> r5c4 ≠ 2 whip[1]: c4n2{r3 .} ==> r1c6 ≠ 2 whip[1]: r1n2{c9 .} ==> r2c8 ≠ 2

At this point there are three different whips[8] and one whip[9], each leading to a stte solution:

Code: Select all: whip[8]: c4n1{r5 r2} - r2n2{c4 c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r5c4 ≠ 4 OR whip[8]: r2n1{c8 c4} - r5c4{n1 n4} - c9n4{r5 r8} - c7n4{r9 r6} - r6n6{c7 c5} - r5n6{c5 c2} - c2n3{r5 r7} - c2n4{r7 .} ==> r2c8 ≠ 4 OR whip[8]: r3n4{c4 c1} - r2n4{c2 c8} - c4n4{r2 r5} - r6n4{c6 c7} - r7n4{c7 c2} - c2n3{r7 r5} - r5n6{c2 c5} - r6n6{c5 .} ==> r1c6 ≠ 4 OR whip[9]: r2c8{n1 n4} - r1n4{c7 c6} - c4n4{r2 r5} - r6n4{c6 c7} - c9n4{r5 r8} - c2n4{r8 r7} - c2n3{r7 r5} - r5n6{c2 c5} - r6n6{c5 .} ==> r2c4 ≠ 1

Second group:

Code: Select all: whip[4]: r1n2{c9 c6} - c4n2{r3 r5} - r6n2{c6 c7} - r7n2{c7 .} ==> r2c8 ≠ 2 whip[1]: b3n2{r1c9 .} ==> r1c6 ≠ 2 whip[1]: b2n2{r3c4 .} ==> r5c4 ≠ 2

At this point there are three different whips[8] and one whip[9], each leading to a stte solution:

Code: Select all: whip[8]: c4n1{r5 r2} - r2n2{c4 c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r5c4 ≠ 4 OR whip[8]: r1n4{c9 c6} - c4n4{r3 r5} - r6n4{c6 c7} - c9n4{r5 r8} - c2n4{r8 r7} - c2n3{r7 r5} - r5n6{c2 c5} - r6n6{c5 .} ==> r2c8 ≠ 4 OR whip[8]: r3n4{c4 c1} - r2n4{c2 c8} - c4n4{r2 r5} - r6n4{c6 c7} - r7n4{c7 c2} - c2n3{r7 r5} - r5n6{c2 c5} - r6n6{c5 .} ==> r1c6 ≠ 4 OR whip[9]: r5c4{n1 n4} - r3n4{c4 c1} - b1n2{r3c1 r2c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r2c4 ≠ 1

Third group:

Code: Select all: whip[4]: c4n2{r3 r5} - c5n2{r6 r8} - c9n2{r8 r4} - b4n2{r4c1 .} ==> r1c6 ≠ 2 whip[1]: r1n2{c9 .} ==> r2c8 ≠ 2 whip[1]: b2n2{r3c4 .} ==> r5c4 ≠ 2

At this point there are three different whips[8] and one whip[9], each leading to a stte solution:

Code: Select all: whip[8]: c4n1{r5 r2} - r2n2{c4 c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r5c4 ≠ 4 OR whip[8]: r1n4{c9 c6} - c4n4{r3 r5} - r6n4{c6 c7} - c9n4{r5 r8} - c2n4{r8 r7} - c2n3{r7 r5} - r5n6{c2 c5} - r6n6{c5 .} ==> r2c8 ≠ 4 OR whip[8]: r3n4{c4 c1} - r2n4{c2 c8} - c4n4{r2 r5} - r6n4{c6 c7} - r7n4{c7 c2} - c2n3{r7 r5} - r5n6{c2 c5} - r6n6{c5 .} ==> r1c6 ≠ 4 OR whip[9]: r5c4{n1 n4} - r3n4{c4 c1} - b1n2{r3c1 r2c2} - r2n4{c2 c8} - c9n4{r1 r8} - c7n4{r9 r6} - c7n6{r6 r4} - r4c2{n6 n1} - r8c2{n1 .} ==> r2c4 ≠ 1

As you can see, there are only three possibilities for the whip[4] (and they were all in my previous post), but each of them can be followed by one of three different whips[8] or a whip[9].
Notice also that the 3 groups of 4 whips involve very similar whips (same eliminations, with one of the whips being a variant of its homologue in another group).

François: finding one of the first whips[4] was unavoidable (admitting some remaining form of simplest-first), but then finding exactly the same among one the 3 (or 4) possible whips[8 or 9] could only be pure luck (1 chance in 3 or 4).

Finally, as in the case of the second puzzle, all this evokes some symmetry, but none obvious.

by **DEFISE** » Fri Mar 05, 2021 12:36 pm

Hi Denis,
Let's say that for these 2 puzzles the principle of my "few steps" algorithm was effective.
This is not always the case: once my "simplest first" gave less steps than "few steps" !
There we can speak of bad luck.

by **mith** » Fri Mar 05, 2021 3:24 pm

I'll leave this for another day or two, but I will say that the first puzzle does have a one step solution (of sorts). (The second does not, as far as I'm aware...)

by **Hajime** » Fri Mar 05, 2021 5:26 pm

mith wrote:I'll leave this for another day or two, but I will say that the first puzzle does have a one step solution (of sorts). (The second does not, as far as I'm aware...)

If you can prove r1c6=1 then stte

Website · by **denis_berthier** » Sat Mar 06, 2021 4:14 am

mith wrote:I'll leave this for another day or two, but I will say that the first puzzle does have a one step solution (of sorts). (The second does not, as far as I'm aware...)

Of sorts:

First puzzle, starting from the same resolution state as before:
FORCING[3]-T&E(S) applied to trivalue candidates n1r6c8, n2r6c8 and n4r6c8 :
===> 1 values decided in the three cases: n6r6c5
===> 29 candidates eliminated in the three cases: n2r1c6 n1r1c7 n2r1c7 n4r1c7 n1r1c8 n1r1c9 n2r2c8 n1r4c7 n2r4c7 n1r4c9 n2r4c9 n1r5c1 n2r5c1 n1r5c2 n2r5c2 n2r5c4 n6r5c5 n1r5c9 n4r5c9 n1r6c5 n2r6c5 n4r6c5 n1r6c6 n2r6c7 n6r6c7 n1r7c2 n4r7c7 n4r8c5 n2r8c9

Code: Select all: CURRENT RESOLUTION STATE: 9 8 6 5 3 14 7 24 247 7 24 3 124 9 6 5 14 8 24 5 1 24 7 8 9 6 3 12 126 4 9 8 5 67 3 7 3 36 9 14 124 7 8 5 2 8 7 5 3 6 24 14 124 9 134 34 8 7 5 124 12 9 6 5 14 7 6 12 9 3 8 14 6 9 2 8 14 3 14 7 5

stte

But then, the same technique works for the second puzzle:
FORCING[3]-T&E(S) applied to trivalue candidates n1r2c6, n2r2c6 and n3r2c6 :
===> 0 values decided in the three cases:
===> 36 candidates eliminated in the three cases: n4r1c1 n1r1c4 n3r1c7 n4r1c7 n2r1c8 n1r2c1 n2r2c1 n3r2c1 n3r2c2 n4r2c2 n1r2c9 n2r2c9 n2r3c3 n3r3c4 n4r4c1 n2r4c6 n2r5c2 n4r5c2 n4r5c6 n1r5c7 n2r5c7 n3r5c8 n1r6c3 n2r6c3 n1r6c8 n4r6c8 n2r6c9 n4r6c9 n3r7c9 n3r8c2 n4r8c7 n2r9c1 n4r9c1 n3r9c3 n4r9c3 n3r9c7

Code: Select all: CURRENT RESOLUTION STATE: 123 8 6 23 7 9 12 134 5 4 12 9 8 5 123 7 6 34 7 5 13 12 4 6 9 8 123 12 9 124 5 3 14 8 7 6 5 13 7 6 8 12 34 124 9 8 6 34 124 9 7 5 23 13 9 234 8 7 1 34 6 5 24 6 14 5 9 2 8 13 134 7 13 7 12 34 6 5 124 9 8

stte

by **Cenoman** » Sat Mar 06, 2021 6:12 pm

First puzzle, "reasonable" solution (five steps):

Code: Select all: +--------------------+---------------------+-------------------------+ | 9 8 6 | 5 3 14-2 |ZC1247 VzB124 C1247 | | 7 24 3 | c124 9 6 | 5 B124 8 | | 24 5 1 | c24 7 8 | 9 6 3 | +--------------------+---------------------+-------------------------+ | 12 126 4 | 9 8 5 | Z1267 3 127 | | 123 1236 9 |Xb14(2) 1246 7 | 8 5 124 | | 8 7 5 | 3 Xa1246 Xx124* | Y1246* WyA124 9 | +--------------------+---------------------+-------------------------+ | 134 134 8 | 7 5 w124* | 124* 9 6 | | 5 14 7 | 6 v124 9 | 3 8 u124 | | 6 9 2 | 8 14 3 | 14 7 5 | +--------------------+---------------------+-------------------------+

1. Kraken row (2)r6c5678
(2)r6c5 - r5c4 = r23c4
XW(2)r67\c67
(2)r6c8 - r12c8 = r1c79
=> -2 r1c6; (-2 r5c4, r2c8)

2. (2)r8c9 = r8c5 - r7c6 = r6c6 - r6c8 = r1c8 => -2 r1c9
3. (2)r1c8 = r6c8 - (2=146)b5p489 - r6c7 = (67)r14c7 => -2 r1c7

Hidden Text: Show

4. (1)r2c4 = (r5c4 & r2c8) - r6c568 = r6c7 - r79c7 = r8c9 - (1=42)r28c2 => -2 r2c4

Hidden Text: Show

5. Remote Pair (14)r5c4, r2c4, r2c8, r6c8 => -14 r5c9, r6c56; ste

One step solution (crazy) with krakens:

Code: Select all: +--------------------+---------------------+----------------------+ | 9 8 6 | 5 3 124 | 1247 124 1247 | | 7 24 3 | 24-1 9 6 | 5 124 8 | | 24 5 1 | 24 7 8 | 9 6 3 | +--------------------+---------------------+----------------------+ | 12 126 4 | 9 8 5 | 1267 3 127 | | 123 1236 9 | 124 1246 7 | 8 5 124 | | 8 7 5 | 3 1246 124 | 1246 124 9 | +--------------------+---------------------+----------------------+ | 134 134 8 | 7 5 124 | 124 9 6 | | 5 14 7 | 6 124 9 | 3 8 124 | | 6 9 2 | 8 14 3 | 14 7 5 | +--------------------+---------------------+----------------------+

Hidden Text: Show

Second puzzle, one step solution:

Code: Select all: +-----------------------+-------------------+-----------------------+ | 1234 8 6 | 123 7 9 | 1234 1234 5 | | 1234 1234 9 | 8 5 13+2 | 7 6 1234 | | 7 5 123 | 123 4 6 | 9 8 123 | +-----------------------+-------------------+-----------------------+ | 124 9 124 | 5 3 124 | 8 7 6 | | 5 1234 7 | 6 8 124 | 1234 1234 9 | | 8 6 1234 | 124 9 7 | 5 1234 1234 | +-----------------------+-------------------+-----------------------+ | 9 234 8 | 7 1 34 | 6 5 234 | | 6 134 5 | 9 2 8 | 134 134 7 | | 1234 7 1234 | 34 6 5 | 1234 9 8 | +-----------------------+-------------------+-----------------------+

Hidden Text: Show

by **mith** » Sat Mar 06, 2021 9:53 pm

So, as I mentioned in the OP, this is not a new idea, and in fact has been discussed on these forums a couple years ago (if not earlier).

Code: Select all: .--------------.----------------.-----------------. | 9 8 6 | 5 3 124 | 1247 124 1247 | | 7 24 3 | 124 9 6 | 5 124 8 | | 24 5 1 | 24 7 8 | 9 6 3 | :--------------+----------------+-----------------: | 12 126 4 | 9 8 5 | 1267 3 127 | | 123 1236 9 | 124 1246 7 | 8 5 124 | | 8 7 5 | 3 1246 124 | 1246 124 9 | :--------------+----------------+-----------------: | 134 134 8 | 7 5 124 | 124 9 6 | | 5 14 7 | 6 124 9 | 3 8 124 | | 6 9 2 | 8 14 3 | 14 7 5 | '--------------'----------------'-----------------'

The key is to note that, after singles, the digits 124 only appear once each, while almost all of the other digits are placed. Whatever the solution, the digits 124 will appear in some order in other houses. Let's pick column 4, for example, and we'll relabel r235c4 as ABC. We don't know how the digits 124 map to ABC, but let's look at a grid with ABC in place of all the 124s:

Code: Select all: .-----------------.-------------.------------------. | 9 8 6 | 5 3 C | ABC7 ABC ABC7 | | 7 BC 3 | A 9 6 | 5 BC 8 | | AC 5 AC | B 7 8 | 9 6 3 | :-----------------+-------------+------------------: | ABC ABC6 ABC | 9 8 5 | ABC67 3 ABC7 | | AB3 AB36 9 | C AB6 7 | 8 5 AB | | 8 7 5 | 3 AB6 AB | ABC6 ABC 9 | :-----------------+-------------+------------------: | AB3C AB3C 8 | 7 5 ABC | ABC 9 6 | | 5 ABC 7 | 6 ABC 9 | 3 8 ABC | | 6 9 ABC | 8 ABC 3 | ABC 7 5 | '-----------------'-------------'------------------'

But this grid just solves with singles:

Code: Select all: .---------.---------.---------. | 9 8 6 | 5 3 C | 7 B A | | 7 B 3 | A 9 6 | 5 C 8 | | A 5 C | B 7 8 | 9 6 3 | :---------+---------+---------: | B C A | 9 8 5 | 6 3 7 | | 3 6 9 | C A 7 | 8 5 B | | 8 7 5 | 3 6 B | C A 9 | :---------+---------+---------: | C 3 8 | 7 5 A | B 9 6 | | 5 A 7 | 6 B 9 | 3 8 C | | 6 9 B | 8 C 3 | A 7 5 | '---------'---------'---------'

And now we can match this to the original puzzle to find C=1, A=4, B=2. The second puzzle simplifies with the same trick, though not to singles. (The title is due to both puzzles having available complex fish with endo fins after singles, if not using the relabeling trick.)

I was reminded of this recently and have been investigating grids which allow this sort of relabeling trick; the four-digit version is quite rare, whereas the three-digit is quite common. I found only 13 four-digit examples out of the solution grids for known 17c puzzles (I'll have to run this on one of Denis' controlled-bias samples sometime), with SER in the range 6.6-7.8 at a minimum (that is, the best case house is filled) and 8.9-9.2 at a maximum (worst case house is filled) - puzzle 2 is from the solution grid of a 20c 9.9 instead, chosen because of what it reduces to with the trick. For three-digit, there are just a ton of examples using the 17c solution grids, with the maximum ED found among them being 7.8/7.8/7.8 (with examples reducing to singles with the right relabel; puzzle 1 is a minimal from one of these), and the maximum ER/EP at 8.3/8.3/6.7.

by **mith** » Sat Mar 06, 2021 9:59 pm

(I was amused to see the recent postings in the bivalue conjecture thread, I actually started exploring this on the same day nazaz posted their counterexample, though I only saw it a couple days ago. I wouldn't be surprised if more examples could be found from my four-digit grids, though obviously not by filling a house entirely - you'll always end up with a second house only missing two digits.)

Website · by **denis_berthier** » Sun Mar 07, 2021 8:56 am

mith wrote:So, as I mentioned in the OP, this is not a new idea, and in fact has been discussed on these forums a couple years ago (if not earlier).

Code: Select all
.--------------.----------------.-----------------. | 9 8 6 | 5 3 124 | 1247 124 1247 | | 7 24 3 | 124 9 6 | 5 124 8 | | 24 5 1 | 24 7 8 | 9 6 3 | :--------------+----------------+-----------------: | 12 126 4 | 9 8 5 | 1267 3 127 | | 123 1236 9 | 124 1246 7 | 8 5 124 | | 8 7 5 | 3 1246 124 | 1246 124 9 | :--------------+----------------+-----------------: | 134 134 8 | 7 5 124 | 124 9 6 | | 5 14 7 | 6 124 9 | 3 8 124 | | 6 9 2 | 8 14 3 | 14 7 5 | '--------------'----------------'-----------------'

The key is to note that, after singles, the digits 124 only appear once each, while almost all of the other digits are placed. Whatever the solution, the digits 124 will appear in some order in other houses. Let's pick column 4, for example, and we'll relabel r235c4 as ABC. We don't know how the digits 124 map to ABC, but let's look at a grid with ABC in place of all the 124s:

Code: Select all
.-----------------.-------------.------------------. | 9 8 6 | 5 3 C | ABC7 ABC ABC7 | | 7 BC 3 | A 9 6 | 5 BC 8 | | AC 5 AC | B 7 8 | 9 6 3 | :-----------------+-------------+------------------: | ABC ABC6 ABC | 9 8 5 | ABC67 3 ABC7 | | AB3 AB36 9 | C AB6 7 | 8 5 AB | | 8 7 5 | 3 AB6 AB | ABC6 ABC 9 | :-----------------+-------------+------------------: | AB3C AB3C 8 | 7 5 ABC | ABC 9 6 | | 5 ABC 7 | 6 ABC 9 | 3 8 ABC | | 6 9 ABC | 8 ABC 3 | ABC 7 5 | '-----------------'-------------'------------------'

But this grid just solves with singles:

Code: Select all
.---------.---------.---------. | 9 8 6 | 5 3 C | 7 B A | | 7 B 3 | A 9 6 | 5 C 8 | | A 5 C | B 7 8 | 9 6 3 | :---------+---------+---------: | B C A | 9 8 5 | 6 3 7 | | 3 6 9 | C A 7 | 8 5 B | | 8 7 5 | 3 6 B | C A 9 | :---------+---------+---------: | C 3 8 | 7 5 A | B 9 6 | | 5 A 7 | 6 B 9 | 3 8 C | | 6 9 B | 8 C 3 | A 7 5 | '---------'---------'---------'

And now we can match this to the original puzzle to find C=1, A=4, B=2.

From a theoretical point of view, there is some interesting (pseudo-)mystery here:
- you start by deleting some information (stating that B=1or2or4 instead of B=2or4 in r3c4), which should make the puzzle harder
- but then you find a solution with Singles only whereas the original puzzle required whips[4].

As a result, there must be some hidden information input.

I have an answer, but I'll let you think about it for a while.

by **mith** » Mon Mar 08, 2021 10:25 pm

I'm not sure what you might mean by hidden information. Here's how I look at it:

The original puzzle had a unique solution. (This argument doesn't rely on uniqueness as a condition for working; it will prove or disprove uniqueness in the process.) There are 3 (in the case of EF1) or 4 (EF2) digits which only appear once, all in the same house, after basics.

For EF1, then, if we remove those 3 unique digits, what property does the puzzle we are left with have? If it was unique before, then clearly it now has 3! = 6 solutions - the filled digits were serving to disambiguate which permutation we actually had. Now, for those 6 solutions, what are the values of the empty cells in c4? 124, in some order. And, in fact, a different order for each of the solutions. We don't care about which is correct, yet. We just know that whatever permutation was in c3, there is a corresponding permutation in c4.

Effectively, what we are doing by putting ABC into c4, rather than 124 into c3, is looking at that correspondence from the opposite direction. We know 124 are in c4 in some order, but it's a "hard" problem (SER 8.3) to determine what that order is. On the other hand, with ABC in c4, it is easy to determine the order of ABC in c3. (Note that we are not actually losing information here! We're not going from r3c4 is 24 to r3c4 is B which could be 124; we're going from r3c4 is 24 to r3c3 is AC.)

Where we are gaining in solving power is that one direction of the mapping is easy, and the other direction is hard. But because it's a one-to-one correspondence between solutions, it doesn't matter which direction we choose. We can do it the hard way and find out that r235c4 is 421 in that order, or we can do it the easy way and find that r349c3 is CAB in that order (and therefore C=1, A=4, B=2). In some sense, we are finding all 6 solutions to the original puzzle (with 124 removed) simultaneously, and then using the order of 124 in c3 to determine which solution is correct.

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Re: Endor Fins 1/2 (SER 8.3/8.9)