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by **urhegyi** » Thu Dec 21, 2023 9:26 am

Code: Select all: 3.......2.8...6.4...1...5......29.8....5.3....7.61....5.....3...6.8...7...2.....1

by **P.O.** » Thu Dec 21, 2023 7:29 pm

all single-solution puzzles are solvable with one combination of 8 templates, so they all have a one-step solution.

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Code: Select all: 3.......2.8...6.4...1...5......29.8....5.3....7.61....5.....3...6.8...7...2.....1 #VT: (9 9 10 426 9 18 36 4 708) Cells: nil nil nil nil nil nil nil nil nil Candidates:nil nil nil nil nil nil nil nil nil 3 459 45679 1479 45789 14578 16789 169 2 279 8 579 12379 3579 6 179 4 379 24679 249 1 23479 34789 2478 5 369 36789 146 1345 3456 47 2 9 1467 8 34567 124689 1249 4689 5 478 3 124679 1269 4679 2489 7 34589 6 1 48 249 2359 3459 5 149 4789 12479 4679 1247 3 269 4689 149 6 349 8 3459 1245 249 7 459 4789 349 2 3479 345679 457 4689 569 1 (1 2 3 4 5 6 7 8): 1 instance 354.8176228735614.6.127453841572.683826543.17.736182545481.732616.8324757324658.1 .....1.........1....1.......1..............1.....1.......1.....1................1 ........22...........2.........2.....2.............2.........2......2.....2...... 3...........3............3.........3.....3.....3............3......3.....3....... ..4.............4......4...4............4............4.4.............4.....4..... .5...........5..........5....5.........5............5.5................5.....5... .......6......6...6..............6....6.........6.............6.6...........6.... ......7....7..........7.......7.............7.7............7..........7.7........ ....8.....8...............8.......8.8.............8.....8.........8...........8.. #VT: (1 1 1 1 1 1 1 1 708) Cells: (6 16 29 44 58 64) (10 22 38 52 62 69) (13 26 36 48 68 74) (3 24 28 41 54 56 70 76) (2 14 30 53 72 78) (8 19 34 39 63 77) (7 12 23 31 45 60 73) (5 27 37 51 57 79) nil SetVC: ( n5r1c2 n4r1c3 n8r1c5 n1r1c6 n7r1c7 n6r1c8 n2r2c1 n7r2c3 n3r2c4 n5r2c5 n1r2c7 n9r2c9 n6r3c1 n9r3c2 n2r3c4 n7r3c5 n4r3c6 n3r3c8 n8r3c9 n4r4c1 n1r4c2 n5r4c3 n7r4c4 n6r4c7 n3r4c9 n8r5c1 n2r5c2 n6r5c3 n4r5c5 n9r5c7 n1r5c8 n7r5c9 n9r6c1 n3r6c3 n8r6c6 n2r6c7 n5r6c8 n4r6c9 n4r7c2 n8r7c3 n1r7c4 n7r7c6 n2r7c8 n6r7c9 n1r8c1 n9r8c3 n3r8c5 n2r8c6 n4r8c7 n5r8c9 n7r9c1 n3r9c2 n4r9c4 n6r9c5 n5r9c6 n8r9c7 n9r9c8 n9r1c4 n9r7c5 ) 3 5 4 9 8 1 7 6 2 2 8 7 3 5 6 1 4 9 6 9 1 2 7 4 5 3 8 4 1 5 7 2 9 6 8 3 8 2 6 5 4 3 9 1 7 9 7 3 6 1 8 2 5 4 5 4 8 1 9 7 3 2 6 1 6 9 8 3 2 4 7 5 7 3 2 4 6 5 8 9 1

by **Cenoman** » Thu Dec 21, 2023 11:21 pm

A four step solution:

Code: Select all: +--------------------------+---------------------------+--------------------------+ | 3 c459* 467-59 | 1479 45789 14578 | 678-19 169* 2 | | 279* 8 b579* | 123-79 a5-379 6 | 179* 4 379* | | 267-49 249* 1 | 23479 34789 2478 | 5 369* 678-39 | +--------------------------+---------------------------+--------------------------+ | 146 d135-4 3456 | 47 2 9 | 1467 8 34567 | | 124689 12-49 4689 | 5 478 3 | 124679 12-69 4679 | | 2489 7 34589 | 6 1 48 | 249 235-9 3459 | +--------------------------+---------------------------+--------------------------+ | 5 149* 78 | 12479 4679 1247 | 3 269* 4689 | | 149* 6 f349* | 8 g35-49 125-4 | 249* 7 459* | | 78 e349* 2 | 3479 345679 457 | 4689 569* 1 | +--------------------------+---------------------------+--------------------------+

1. Look at b1379p2468: 16 cells, 8 digits
Digits 1 to 5 can be there at most twice: 1,3 in b37; 2,5 in b19, 4 in r8,c2.
Digits 6,7 can be there at most once: 6 in c8, 7 in r2
=> digit 9 must be there four times (once per box), and digits 1 to 7 must be there in all their possible units.

=>21 eliminations
-259 b1p37, -79 b2p34, -139b3p19, -49 r45c2, -69 r56c8, -49 r8c56, -9 b9p37

For a more 'classical' presentation, this rank0 logic is a SK-Loop:
(79=2|5)r2c13 - (25=4|9)r13c2 - (49=1|3)r79c2 - (13=4|9)r8c13 - (49=2|5)r8c79 - (25=6|9)r79c8 - (69=1|3)r13c8 - (13=7|9)r2c79 loop
(17 eliminations, same as above w/o 9 b1p37, b3p19)

2. (5)r2c5 = r2c3 - r1c2 = (5-3)r4c2 = r9c2 - r8c3 = (3)r8c5 => -3 r2c5; 9 placements, lcls

Code: Select all: +---------------------+------------------------+----------------------+ | 3 5 467 | 1479 4789 1478 | 678 69 2 | | 279 8 79 | 23 5 6 | 1 4 379 | | 267 49 1 | 23479 4789 2478 | 5 369 678 | +---------------------+------------------------+----------------------+ | 46 1 35* | 47 2 9 | 467 8 35* | | 4689 2 689 | 5 478 3 | 4679 1 4679 | | 489 7 35* | 6 1 48 | 249 235 49-35* | +---------------------+------------------------+----------------------+ | 5 49 78 | 12479 4679 1247 | 3 269 4689 | | 1 6 49 | 8 3 25 | 249 7 459 | | 78 3 2 | 479 4679 457 | 4689 569 1 | +---------------------+------------------------+----------------------+

3. UR(35)r46c39 using internals => -35 r6c9; 13 placements, lcls

Code: Select all: +---------------------+-----------------------+---------------------+ | 3 5 467 | 1479 4789 1478 | 678 69 2 | | 2 8 c79 | 3 5 6 | 1 4 79 | | 67 b49 1 | 2 a4789 A478 | 5 3 z678 | +---------------------+-----------------------+---------------------+ | 46 1 5 | 47 2 9 | 467 8 3 | | 4689 2 689 | 5 478 3 | 4679 1 4679 | | C489 7 3 | 6 1 B48 | 2 5 49 | +---------------------+-----------------------+---------------------+ | 5 49 c78 | 1479 4679 147 | 3 2 469-8 | | 1 6 49 | 8 3 2 | 49 7 5 | | D78 3 2 | 479 4679 5 | E4689 69 1 | +---------------------+-----------------------+---------------------+

4. Kraken row (8)r3c569
||(8-9)r3c5 = r3c2 - (9=78)r27c3
||(8)r3c6 - r6c6 = r6c1 - r9c1 = (8)r9c7
||(8)r3c9
=> -8 r7c9; ste

Thanks for the puzzle !

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