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by **P.O.** » Fri Feb 11, 2022 3:21 pm

Code: Select all: 5 chains (maxdepth 5) in 3 grid states for me. . . 3 6 . . . . 8 . 8 . . . 5 . . . 7 . . . . . 1 . . 1 . . . . 6 . . 5 . . 7 . 3 . . 9 . . 4 . 2 . . 8 . . 4 . . . . . 3 . . . 5 . . . 4 . . . . . 6 8 . . . . 9 ..36....8.8...5...7.....1..1....6..5..7.3..9..4.2..8..4.....3...5...4.....68....9

by **jco** » Fri Feb 11, 2022 5:52 pm

Code: Select all: .------------------------------------------------------------------------------. | 259 129 3 | 6 12479 1279 | 24579 2457 8 | | 269 8 1249 | 13479 12479 5 | 24679 23467 23467 | | 7 269 2459 | 349 2489 d38-29 | 1 23456 2346 | |--------------------------+-------------------------+-------------------------| | 1 239 289 | 479 4789 6 | 247 2347 5 | | f2568 26 7 | 145 3 e18 | 246 9 1246 | | 3569 4 59 | 2 1579 179 | 8 1367 1367 | |--------------------------+-------------------------+-------------------------| | 4 1279 1289 | 1579 125679 1279 | 3 125678 1267 | |ga38-29 5 1289 |b1379 12679 4 | 267 12678 1267 | | 23 1237 6 | 8 1257 c1237 | 2457 12457 9 | '------------------------------------------------------------------------------'

1. Loop (8-3)r8c1 = r8c4 - r9c6 = (3-8)r3c6 = (8)r5c6 - r5c1 = (8)r8c1 => -29 r8c1, -29 r3c6
---

Code: Select all: .------------------------------------------------------------------------------. | b259 129 3 | 6 12479 e1279 | 24579 2457 8 | | b269 8 1249 | 13479 12479 5 | 24679 23467 23467 | | 7 c269 c2459 |d349 d2489 38 | 1 23456 2346 | |--------------------------+-------------------------+-------------------------| | 1 239 289 | 479 4789 6 | 247 2347 5 | | 2568 26 7 | 145 3 18 | 246 9 1246 | | a3569 4 5-9 | 2 1579 A179 | 8 1367 1367 | |--------------------------+-------------------------+-------------------------| | 4 g1279 g1289 | 1579 125679 f1279 | 3 125678 1267 | | 38 5 h1289 | 1379 12679 4 | 267 12678 1267 | | 23 1237 6 | 8 1257 1237 | 2457 12457 9 | '------------------------------------------------------------------------------'

2. (9*)r6c6 = [(9)r6c1 = r1c12 - r3c23 = r3c45 - r1c6 =* r7c6 - r7c23 = r8c3] => -9 r6c3 [8 placements & basics]
---

Code: Select all: .--------------------------------------------------------------------. | 5 129 3 | 6 12479 1279 | 49 27 8 | |a29-6 8 1249 | 13479 12479 5 |e46-9 2367 23467 | | 7 269 249 | 349 2489 38 | 1 5 2346 | |----------------------+----------------------+----------------------| | 1 239 289 | 479 4789 6 | 27 237 5 | | 268 26 7 | 5 3 18 |d46 9 1246 | |b369 4 5 | 2 179 179 | 8 c1367 c1367 | |----------------------+----------------------+----------------------| | 4 1279 1289 | 179 5 1279 | 3 12678 1267 | | 38 5 1289 | 1379 6 4 | 27 1278 127 | | 23 1237 6 | 8 127 1237 | 5 4 9 | '--------------------------------------------------------------------'

3. (9)r2c1 = (9-6)r6c1 = (6)r6c89 - (6)r5c7 = (6)r2c7 => -6 r2c1, -9 r2c7; lclste

Thanks for the puzzle!
EDIT: I noticed a mistake in step 2 (due to a mistake in coloring) and made the correction.

Hidden Text: Show

Website · by **denis_berthier** » Sat Feb 12, 2022 7:52 am

.

Code: Select all: Resolution state after Singles and whips[1]: +----------------------+----------------------+----------------------+ ! 259 129 3 ! 6 12479 1279 ! 24579 2457 8 ! ! 269 8 1249 ! 13479 12479 5 ! 24679 23467 23467 ! ! 7 269 2459 ! 349 2489 2389 ! 1 23456 2346 ! +----------------------+----------------------+----------------------+ ! 1 239 289 ! 479 4789 6 ! 247 2347 5 ! ! 2568 26 7 ! 145 3 18 ! 246 9 1246 ! ! 3569 4 59 ! 2 1579 179 ! 8 1367 1367 ! +----------------------+----------------------+----------------------+ ! 4 1279 1289 ! 1579 125679 1279 ! 3 125678 1267 ! ! 2389 5 1289 ! 1379 12679 4 ! 267 12678 1267 ! ! 23 1237 6 ! 8 1257 1237 ! 2457 12457 9 ! +----------------------+----------------------+----------------------+ 226 candidates.

The puzzle is in W4: Show

P.O. wrote:5 chains (maxdepth 5) in 3 grid states for me.

As the puzzle is in W4, it makes sense to try to reduce the number of steps, by allowing slightly longer chains, of length 5 (my interpretation of "maxdepth 5" when chains are implemented by BFS.
Indeed, it's relatively easy to find a solution in 5 steps in W4:

biv-chain[4]: r8n3{c1 c4} - c6n3{r9 r3} - c6n8{r3 r5} - c1n8{r5 r8} ==> r8c1≠9, r8c1≠2
biv-chain[4]: c6n3{r3 r9} - r8n3{c4 c1} - c1n8{r8 r5} - c6n8{r5 r3} ==> r3c6≠9, r3c6≠2
whip[3]: b7n9{r8c3 r7c2} - c6n9{r7 r1} - r3n9{c4 .} ==> r6c3≠9
singles ==> r6c3=5, r1c1=5, r3c8=5, r9c7=5, r7c5=5, r5c4=5, r8c5=6, r9c8=4
whip[1]: r5n4{c9 .} ==> r4c7≠4
whip[3]: c7n6{r2 r5} - r6n6{c9 c1} - c1n9{r6 .} ==> r2c7≠9
singles ==> r1c7=9, r1c5=4, r4c4=4
finned-x-wing-in-columns: n6{c7 c1}{r2 r5} ==> r5c2≠6
w1-tte

The last step could also be:
z-chain[3]: b1n6{r3c2 r2c1} - c7n6{r2 r5} - c2n6{r5 .} ==> r3c2≠9, r5c2≠6, r3c9≠6, r3c2≠2, r2c1≠6
with z-candidates = n6r3c2
w1-tte

by **P.O.** » Sat Feb 12, 2022 1:12 pm

i found this puzzle interesting because its first state has no singles no intersections no subsets and no fishes and trying to shorten its path i found only two (sort of) symmetric looping chains to begin the resolution process:
c6n3{r3 r9} - r8n3{c4 c1} - c1n8{r8 r5} - c6n8{r5 r3} => r3c6 <> 2,9
r8n3{c1 c4} - c6n3{r9 r3} - c6n8{r3 r5} - c1n8{r5 r8} => r8c1 <> 2,9
then:
r6c3{n5 n9} - b7n9{r7c3r8c3 r7c2} - c6n9{r7 r1} - c7n9{r1 r2} - b1{r1c1r2c1r3c2}{n2n5n6} => r3c3 r5c1 r6c1 <> 5
singles: ( r9c8b9 n4 r8c5b8 n6 r7c5b8 n5 r9c7b9 n5 r1c1b1 n5 r3c8b3 n5 r6c3b4 n5 r5c4b5 n5 )
intersection: r4n4{c4c5} => r4c7 <> 4
c1n9{r2 r6} - r6n6{c1 c8c9} - c7n6{r5 r2} - c7n9{r2 r1} - c6n9{r1 r7} - c2n9{r7 r3} => r1c2 r2c3 r3c3 <> 9
c1n9{r2 r6} - r6n6{c1 c8c9} - c7n6{r5 r2} => r2c1 <> 6
ste.

Denis, it is a BFS algorithm, it starts with a xor relationship between two candidates or group of candidates and by convention its numbering starts at 0

Code: Select all: a < -- xor -- > b 0 | (1 --------- 1 --------- 1) | | | | | (2 --- 2) (2 --- 2) (2 --- 2) | | | | | | | | | (3-3) (3-3) (3-3) (3-3) (3-3) (3-3) depth 3 = 4 links -----------------------------------

by **jco** » Sat Feb 12, 2022 1:31 pm

After the Loop that removes 2,9 from two cells (first chain), there is a large fish

Code: Select all: .--------------------------------------------------------------------. | 259 129 3 | 6 12479 1279 | 24579 2457 8 | | 269 8 124-9 | 13479 12479 5 | 24679 23467 23467 | | 7 *269 *2459 | *349 *2489 38 | 1 23456 2346 | |---------------------+-----------------------+----------------------| | 1 *239 *289 | *479 *4789 6 | 247 2347 5 | | 2568 26 7 | 145 3 18 | 246 9 1246 | | 3569 4 5-9 | 2 1579 179 | 8 1367 1367 | |---------------------+-----------------------+----------------------| | 4 *1279 *1289 | 1579 125679 1279 | 3 125678 1267 | | 38 5 @1289 | *1379 *12679 4 | 267 12678 1267 | | 23 1237 6 | 8 1257 1237 | 2457 12457 9 | '--------------------------------------------------------------------'

Finned Franken Jellyfish (9) r348b7 \ c2345 efr8c3 => -9 r26c3
When solving the puzzle, I noticed some large fish on 9s, but did not even try to catch it.
For this puzzle, the additional elimination provided by this fish is not useful.

by **P.O.** » Sat Feb 12, 2022 3:06 pm

hi JCO, it is true that there exist many varieties of fish, lot of which i am unable to spot, but if this one need some previous 9 eliminations to be found it is not in the first state of the grid.

by **jco** » Sat Feb 12, 2022 3:17 pm

Hi P.O..

Agreed. That fish is not available at the start.
It is available (after just 4 eliminations) right after my first move (Loop)

Loop (8-3)r8c1 = r8c4 - r9c6 = (3-8)r3c6 = (8)r5c6 - r5c1 = (8)r8c1 => -29 r8c1, -29 r3c6

or after your two chains:
c6n3{r3 r9} - r8n3{c4 c1} - c1n8{r8 r5} - c6n8{r5 r3} => r3c6 <> 2,9
r8n3{c1 c4} - c6n3{r9 r3} - c6n8{r3 r5} - c1n8{r5 r8} => r8c1 <> 2,9

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