Sudoku Multi-SectorLocked Sets

Multi-SectorLocked Sets(MSLS)

Multi-Sector Locked Sets (MSLS) is an algorithm that uses locks that span multiple sectors (rows, columns, and blocks). MSLS is an extension of Locked algorithms such as Locked Set, Fish.

While the Fish algorithm uses BaseSet/CoverSet in its explanation, here use Sector/Link in accordance with MSLS.

MSLS is the following algorithm:

Place several Sectors on the board and focus on the intersections of undetermined digits. Intersections where Sectors overlap three times are excluded.

Focus on the candidate digits at the intersections and find the links connecting them. There may be multiple ways to represent the links connecting the candidate digits.
For each set of links for a candidate digits, find the smallest set of links that covers them. Count the number of links at this time.

If the number of cells at the intersection and the total number of links are the same, the intersection becomes a Locked Set. In the following example, 5 cells = 5 links.

Candidate digits outside the intersection of the same Sector as the links are excluded.

ID:00_1 MultiSectorLockedSets .46..3.8. 72....... .8.7..3.. ..42.7... ..9.58.2. .....6..1 ..14396.. 4.....139 .976.1.4. Puzzle .46..3.8. 723...... .857..3.. ..42.7... 6.9.58.2. ....46..1 .5143967. 46..7.139 3976.1.4. Current Multi Sector Locked Set Size: 4-sectors, 5-Int.Cells, 5-Links sectors: r1 c16 b2 Digits: #12459 Links: r1#1 b2#2 c6#4 b2#5 r1#9 Exclude: r3c5#2 r2c4#5 r1c7#9

Proof of Multi-Sector Locked Sets (MSLS)

MSLS is proven as follows:

Premise:
- Set a set containing multiple sectors (let's call it A).
- Find the links connecting candidate digits at the undetermined intersection. (Hereafter, "undetermined" will be omitted.)
- Find the minimum cover link set for each candidate digit.
- The number of cells at the intersection and the total number of links in the cover link set should be equal.

Conclusion:
If |cells of the intersection| = |links of the cover link set|, then A is Multi-sector Locked Sets (MSLS).

Proof:
- Since the number of cells at the intersection and the total number of links are the same, the minimum number of links for each candidate digit is determined.
- Therefore, the minimum covered link set is one where all links are positive. (Each link must have at least one positive digit.)
- A is a Multi-sector Locked Set (MSLS).

While all links are valid, it's important to note that the candidate digits at the intersections are not uniquely determined. This is similar to, for example, Fish.

Multi-Sector Locked Sets (MSLS)

SK Loop

SK Loop is an 8th-order MSLS. It consists of two row sectors, two column sectors, and four block sectors.
The sector layout is as shown in the following diagram.
In SK Loop, there is a triple overlap of sectors. This area is treated as outside the intersection, and any candidate digits can be excluded.
Since SK Loop is MSLS, there is no special feature to the solution algorithm. Speed can be improved by specifying the sector set as 2-row,2-column,4-blocks.

LockedSet

SK Loop Example

Multi-SectorLocked Sets(MSLS) Example

Sample Multi-Sector Locked Sets (MSLS) are available in the data folder... and can be analyzed with GNPX(v6.2-). A portion of them is shown below.

ID:00_2 MultiSectorLockedSets .37..5.2. 56....... .4.6..3.. ..21.9... ..5.26.1. .....3..9 ..13647.. 7.....546 .268.7.3. Puzzle 137..5628 56..3.1.4 24.6..3.. .721.9... .95.26.1. .145.32.9 951364782 783...546 426857931 Current Multi Sector Locked Set Size: 4-sectors, 5-Int.Cells, 5-Links sectors: r23 c3 b3 Digits: #5789 Links: r3#5 b3#7 c3#8 r23#9 Exclude: (r2c4 r3c5)#9

ID:00_3 MultiSectorLockedSets .97..1.5. 86....... .1.7..8.. ..95.3... ..1.96.8. .....7..9 ..53249.. 1.....475 .481.5.6. Puzzle .97681.54 86.45.... 51473.8.6 7895.36.. 4.1296.8. ..68.7..9 675324918 1..968475 948175.6. Current Multi Sector Locked Set Size: 4-sectors, 3-Int.Cells, 3-Links sectors: r2 c8 b12 Digits: #239 Links: r2#2 r2#3 r2#9 Exclude: (r2c79)#2 (r2c79)#3

ID:00_11 MultiSectorLockedSets 6.7.9.... 1......9. 85.7.2136 ...416..8 9.8..7... ..1.8...5 .1.23846. ...1..2.. .......51 Puzzle 6.7.915.. 1......9. 859742136 5..4169.8 9.8527.1. ..1.8...5 715238469 ...1..2.. .......51 Current Multi Sector Locked Set Size: 5-sectors, 9-Int.Cells, 9-Links sectors: r89 c13 b8 Digits: #2345679 Links: r9#2 b7#3 r89#4 r8#5 r89#6 c5#7 b8#9 Exclude: r9c2#2 (r89c2)#3 (r89c2)#4

ID:00_21 MultiSectorLockedSets .4.8.9.63 ..61437.8 8.7.....9 52..91.8. ......... 7..25.9.4 .......4. ..962...5 3....58.. Puzzle .4.879.63 ..61437.8 837562419 52..91687 ....8.... 7..256934 ...93..4. ..962.375 3...1589. Current Multi Sector Locked Set Size: 4-sectors, 5-Int.Cells, 5-Links sectors: r45 c3 b5 Digits: #347 Links: r45#3 r45#4 r5#7 Exclude: r5c1#4

ID:00_32 MultiSectorLockedSets 263....8. ..5..6..1 9..8..... 7..2..1.. 32148.7.. ..67..3.8 5.8.174.. 6.7.....3 ..2...8.. Puzzle 2631...8. 875..62.1 9148.26.. 7892631.. 3214857.. 456791328 5.8.174.2 647.28.13 1.2...8.. Current Multi Sector Locked Set Size: 4-sectors, 5-Int.Cells, 5-Links sectors: r13 c6 b3 Digits: #34579 Links: r3#3 r1#4 b3#5 b3#7 r1#9 Exclude: r3c5#3 r1c5#4

ID:00_42 MultiSectorLockedSets ....49.3. 13.67..45 4...18.2. .......84 7.2...3.6 814...... 3........ ..1.25... 54.19..62 Puzzle 2.6549.3. 13.672.45 4.531862. ..3.5..84 752.8.3.6 814.36... 32..6.... ..1.25..3 54.193.62 Current Multi Sector Locked Set Size: 6-sectors, 5-Int.Cells, 5-Links sectors: r35 c469 b6 Digits: #1479 Links: r5#1 r5#4 c9#7 r5 c9#9 Exclude: (r17c9)#7 r7c9#9

ID:00_52 MultiSectorLockedSets ...9...84 .4...59.. .2948.5.. ..3.....9 5.1.6..4. ...8..316 19.24.6.5 ........2 2.6.7.4.. Puzzle .159..784 .4...592. .2948.56. 463...859 581369247 972854316 19.24.6.5 ..4.9.1.2 2.6.7.498 Current Multi Sector Locked Set Size: 5-sectors, 4-Int.Cells, 4-Links sectors: r27 c359 Digits: #1378 Links: r2#1 r2#3 c3#7 c3#8 Exclude: r2c4#1 r2c1#3

ID:00_61 MultiSectorLockedSets 3.....6.. 2..719..5 5..34.... ..49..3.7 ..3..45.. 7.51.3.9. .3..5.978 ......... .976..45. Puzzle 37.5..64. 246719835 5..34672. ..49.53.7 9.3.745.6 7.51.3294 63.45.978 45..97163 .9763.452 Current Multi Sector Locked Set Size: 5-sectors, 4-Int.Cells, 4-Links sectors: r49 c18 b8 Digits: #18 Links: r49#1 r49#8 Exclude: r4c2#1 (r4c25)#8

ID:00_71 MultiSectorLockedSets .1..295.. .596..4.2 72...13.6 ..4..2... 5.2.139.. ...954... .37...8.4 ......6.. .4.2....9 Puzzle 4163295.. 3596..412 728541396 ..4.62... 562.1394. ..395426. .3719.8.4 ...4..6.. .4.2....9 Current Multi Sector Locked Set Size: 4-sectors, 3-Int.Cells, 3-Links sectors: r4 c89 b6 Digits: #178 Links: b6#1 b6#7 c9#8 Triple overlap: r4c89 Exclude: r4c9#1 (r4c89)#7 (r14c9)#8

ID:00_81 MultiSectorLockedSets .5......2 ......6.8 286..9.3. ..4...... ..7214.8. 5.2..7... 6.8...9.3 ...563..7 3..798.16 Puzzle .5......2 ......6.8 286.7953. 8.4...... 967214385 5.2..7... 678...953 ...5638.7 3.5798.16 Current Multi Sector Locked Set Size: 4-sectors, 3-Int.Cells, 3-Links sectors: r23 c4 b8 Digits: #134 Links: c4#1 r2#3 c4#4 Exclude: r1c4#1 (r2c35)#3 r1c4#4

ID:01_10001 MultiSectorLockedSets .5.....3. 9.652.... .....1... 7...9..4. .....7... .2.368.17 .9.4.2.6. ....7..5. ......9.8 Puzzle .5..86.3. 936524..1 ....316.5 7...95.4. ....47... 429368517 .9.4.2.63 ...879.5. ...6.39.8 Current Multi Sector Locked Set Size: 4-sectors, 3-Int.Cells, 3-Links sectors: r458 c2 Digits: #168 Links: c2#1 c2#6 c2#8 Exclude: r9c2#1 r3c2#8

ID:01_10001 MultiSectorLockedSets .5.....3. 9.652.... .....1... 7...9..4. .....7... .2.368.17 .9.4.2.6. ....7..5. ......9.8 Puzzle .5..86.3. 936524..1 ....316.5 7...95.4. ....47... 429368517 .9.4.2.63 ...879.5. ...6.39.8 Current Multi Sector Locked Set Size: 5-sectors, 9-Int.Cells, 9-Links sectors: r13 c4 b15 Digits: #124789 Links: r1 c4#1 c4 b1#2 r3#4 r13#7 r3#8 c4#9 Exclude: (r1c7 r3c8)#7

ID:02_1 MultiSectorLockedSets .....71.. .8142.... ..9.3.4.. 9......53 5.......4 27......9 ..2.8.3.. ....4592. ..56..... Puzzle .....71.. .8142.5.. ..9.3.4.. 9......53 5.......4 27......9 ..278.365 .6734592. ..56.2.4. Current Multi Sector Locked Set Size: 3-sectors, 4-Int.Cells, 4-Links sectors: c13 b1 Digits: #3467 Links: b1#3 r1#4 c1#6 c1#7 Exclude: r1c2#3 r1c2#4

ID:02_14 MultiSectorLockedSets ....5.... .26..3..4 .4..2.9.. 7.3...... 46.2.9.71 ......6.8 ..9.3..8. 8..5..72. ....1.... Puzzle ....5.... .26.73.14 .4..2.9.. 783165... 465289371 .9.347658 ..9.3.18. 814596723 ....1.... Current Multi Sector Locked Set Size: 5-sectors, 8-Int.Cells, 8-Links sectors: r123 c8 b1 Digits: #1356789 Links: b1#1 r13#3 c1#5 c8#6 b1#7 c3#8 c1#9 Exclude: r9c8#6

ID:02_21 MultiSectorLockedSets ........9 ...682.3. ..5..9..6 ..1...8.. 7.3.6.9.1 ..4...5.. 1..3..2.. .5.716... 8........ Puzzle ..8....29 9.7682.35 ..5..9.86 5.1...8.. 7.3.65941 ..4...5.. 176398254 452716398 839..4617 Current Multi Sector Locked Set Size: 7-sectors, 8-Int.Cells, 8-Links sectors: r459 c14 b24 Digits: #1245689 Links: c4#1 c4 b4#2 c4#4 c4#5 b4#6 r5#8 r4#9 Exclude: r6c4#1 (r6c24)#2 r6c2#6

ID:02_22 MultiSectorLockedSets ..3..54.. .4..8.... 6....3... .6..24.7. 3..5.6..1 .5.91..2. ...1....2 ....3..5. ..94..8.. Puzzle ..3.954.. .4..81..9 69..43..5 961324578 3..576941 .5.918.2. ...1.7.92 ...839.54 ..94.28.. Current Multi Sector Locked Set Size: 5-sectors, 5-Int.Cells, 5-Links sectors: r267 c7 b1 Digits: #23567 Links: r2#2 c7#3 r2#5 c7#6 r2#7 Exclude: r2c4#2 r8c7#6 r2c4#7

ID:02_43 MultiSectorLockedSets ..1..5..3 .73...96. ....2..1. 3...76... 8.......7 ...24...6 .5..3.... .42...39. 6..1..8.. Puzzle ..1..5..3 .73...96. ....2..1. 3...76..9 8.......7 ...24...6 .58.3.6.. .42...39. 63.1..8.. Current Multi Sector Locked Set Size: 5-sectors, 8-Int.Cells, 8-Links sectors: r13 c78 b1 Digits: #2456789 Links: r1#2 r13#4 r3#5 b1#6 c7#7 c2#8 b1#9 Exclude: (r1c4 r3c469)#4 r3c9#5

ID:02_105 MultiSectorLockedSets ..7..1..4 .43...... 8..6...1. ...4.7.85 ...1.8... 75.3.9... .1...4..6 ......13. 6..2..9.. Puzzle 567.31..4 143..2... 8..645.1. ..14.7.85 4.6158... 7583.94.1 31...4..6 ..45.613. 6.521394. Current Multi Sector Locked Set Size: 4-sectors, 4-Int.Cells, 4-Links sectors: r36 c7 b6 Digits: #2367 Links: b6#2 c7#3 b6#6 c7#7 Exclude: (r5c89)#2 r7c7#7

ID:02_161 MultiSectorLockedSets .....1... ..25.3..6 4..8..1.2 .86.....1 ....9.... 7.....24. 1.8..2..7 3..7.94.. ...3..... Puzzle ...921..4 .12543..6 4..8..1.2 .86.....1 ....9.... 7.....24. 1.8..2..7 3.57.94.8 ...3..... Current Multi Sector Locked Set Size: 3-sectors, 4-Int.Cells, 4-Links sectors: r4 c9 b6 Digits: #3579 Links: b6#3 b6#5 r4#7 b6#9 Exclude: (r5c78)#3 (r5c78)#5 (r4c56)#7

ID:02_168 MultiSectorLockedSets ..3...7.. ...34.5.. ....89.63 .8....1.7 2.......8 3.4....5. 57.19.... ..1.56... ..6...2.. Puzzle ..3...78. 76834.5.. 125789463 689...1.7 257...3.8 314....5. 57219..3. .31256.7. ..6...215 Current Multi Sector Locked Set Size: 5-sectors, 6-Int.Cells, 6-Links sectors: r5 c468 b2 Digits: #124569 Links: c6#1 c6#2 r5#4 r1#5 c4#6 r5#9 Exclude: (r46c6)#2

SK Loop Example

SK Loop examples are available in the data folder... and can be analyzed with GNPX(v6.2-). A portion of them is shown below.

ID:00_29 MSLS_SK_Loop .9.3.6.72 ..21794.8 6.8.....9 56..24.9. ......... 8..95.7.6 .......8. ..526...7 9....86.. Puzzle .9.386572 352179468 678542..9 567824.9. 2.96.78.. 84.95.726 7.649528. .8526.9.7 92.7.86.. Current MSLS_SK_Loop Size: 8-sectors, 6-Int.Cells, 6-Links sectors: r59 c36 b4578 Digits: #13 Links: r56 b8#1 r56 b8#3 Triple overlap: r9c3 Exclude: (r5c89)#1 (r5c89)#3

ID:00_243 MSLS_SK_Loop 94...2.3. .3.4..6.9 ....9...2 86.7.5941 ....89... .92.4.7.. 38..1.... .5.2....3 .....6.94 Puzzle 94..6253. 23.45.6.9 6.5.9.4.2 863725941 ..4.8932. .92.4.7.. 3869142.. 459278163 .2.536894 Current MSLS_SK_Loop Size: 8-sectors, 10-Int.Cells, 10-Links sectors: r15 c68 b2356 Digits: #135678 Links: c468#1 c6#3 b6#5 r5#6 c6 b3#7 r1 c8#8 Exclude: (r36c4)#1 r6c9#5 r1c3#8

ID:00_253 MSLS_SK_Loop 6.1....5. 3...7.6.. ......74. ..7..42.. ....9.167 ....2.584 ..258.3.. 7.....42. 56..4.918 Puzzle 671439852 3...7.691 ..9..6743 ..76.4239 ....9.167 ..6.27584 ..2581376 718963425 563742918 Current MSLS_SK_Loop Size: 8-sectors, 11-Int.Cells, 11-Links sectors: r36 c36 b1245 Digits: #1234589 Links: r36#1 r3#2 r6#3 c3#4 r235#5 r3 c6#8 r6#9 Exclude: (r25c2)#5

ID:00_433 MSLS_SK_Loop 8.2..7.51 ......82. .1.2.593. .6.47.... .....23.. 28.....76 ...34...2 371..94.. ..8..6.9. Puzzle 842937651 .3...4827 .1.285934 .6347.2.. .....234. 284.93.76 ...34.7.2 371.2946. 4287.6.93 Current MSLS_SK_Loop Size: 8-sectors, 4-Int.Cells, 4-Links sectors: r67 c68 b5689 Digits: #158 Links: r46#1 r6#5 r4#8 Triple overlap: r7c68 Exclude: r4c1#1 r4c9#8

ID:00_481 MSLS_SK_Loop 23.1.87.5 9..3476.. .......14 4.2....5. ...7..48. .68..3... 8.9..5..7 756...... 12......9 Puzzle 234168795 915347628 687529314 472.8..53 391752486 5684.3.7. 849..5..7 756...8.. 12387.5.9 Current MSLS_SK_Loop Size: 8-sectors, 10-Int.Cells, 10-Links sectors: r68 c47 b5689 Digits: #123469 Links: r68 c7#1 r7 c9#2 r8#3 r8#4 c4#6 r4 c5#9 Triple overlap: r6c7 r8c4 Exclude: r6c7#1

ID:00_822 MSLS_SK_Loop .9173.24. 7..926... ..5.8..9. .7..48..2 9.2...85. .....913. .8.69...5 .4.....1. ..9.....8 Puzzle 891735246 734926581 625.8.793 .73.48962 9.2...854 4.82.9137 .8769.425 .468..319 ..9...678 Current MSLS_SK_Loop Size: 8-sectors, 11-Int.Cells, 11-Links sectors: r47 c25 b4578 Digits: #13567 Links: r4579#1 r7#3 r4 c25#5 r56#6 c5#7 Exclude: (r5c46 r9c146)#1

ID:00_877 MSLS_SK_Loop .3....8.. 9...1.532 5....8.67 ...5.4.7. ......2.. 4..17..9. 3..6...51 8.1.597.. .5..419.. Puzzle 13...5849 9..41.532 524938167 .1.5.4.78 ..58.3214 4..17..95 3..68.451 841359726 .5..41983 Current MSLS_SK_Loop Size: 8-sectors, 7-Int.Cells, 7-Links sectors: r19 c16 b1278 Digits: #267 Links: r1 b8#2 c3 b2#6 r19 c6#7 Triple overlap: r9c1 Exclude: (r246c3)#6 r9c1#7

ID:00_892 MSLS_SK_Loop ..6...5.. ..4.563.9 .9..7.24. ....8.... 23..6...5 .8.7..69. .4..37... 9.3..846. 7.8.9...2 Puzzle ..6...5.1 ..4.563.9 .9..7.246 ..9.8.72. 237.6.815 .8.7..69. .42.37958 9.3..8467 7.8.9.132 Current MSLS_SK_Loop Size: 8-sectors, 9-Int.Cells, 9-Links sectors: r57 c25 b4578 Digits: #124569 Links: r7 c25#1 c5#2 b5#4 c2#5 r7 c2#6 r5#9 Exclude: r2c2#1 r1c5#2 (r4c46 r6c6)#4

ID:00_1080 MSLS_SK_Loop .8.6.9... ...783..6 ....4.... 13.5.6.82 .57...96. 46.9.8.71 ....6.... 2..314... ...2.5.1. Puzzle .81659... .247831.6 ...142... 139576482 857421963 462938571 .1.867... 2..314... .4.295.1. Current MSLS_SK_Loop Size: 8-sectors, 12-Int.Cells, 12-Links sectors: r17 c28 b1379 Digits: #234579 Links: c7#2 r137#3 c9#4 r7 c8#5 r1 c2#7 r7 c28#9 Triple overlap: r17c8 Exclude: (r17c8 r3c137)#3 r7c8#5 r7c8#9

ID:00_1081 MSLS_SK_Loop .1.5.9... ...173..8 ....2.... 32.6.7.85 .68...43. 49.3.5.16 ....5.... 1..836... ...7.4.9. Puzzle 81.56937. 9..173..8 ...4281.9 321647985 568291437 497385216 ...952..1 1.9836... 2..714.93 Current MSLS_SK_Loop Size: 8-sectors, 12-Int.Cells, 12-Links sectors: r37 c27 b1379 Digits: #345678 Links: c3#3 r7 c2#4 r3 c27#5 r37 c7#6 r8 c1#7 r9#8 Triple overlap: r3c2 r7c27 Exclude: r7c2#4 r3c2#5 r7c7#6

ID:01_283 MSLS_SK_Loop .1...2.7. 2.......1 ..81.94.. 3.1.2.8.. ...8.3... ..6.4.2.3 ..97.43.. 1.......9 .5.6...2. Puzzle .1.3.2.7. 2..4....1 ..81.94.2 3.1.2.8.. ..28.3.1. ..6.412.3 .297143.. 1..235..9 .5369812. Current MSLS_SK_Loop Size: 8-sectors, 12-Int.Cells, 12-Links sectors: r37 c37 b1379 Digits: #3456789 Links: r3#3 c3#4 r7 c3 b3#5 r37 c7#6 r8 b1#7 r7#8 c7#9 Exclude: (r1c9 r2c8)#5 (r3c5 r5c7)#6 (r28c2)#7 r5c7#9

ID:01_514 MSLS_SK_Loop 8..67...5 ...4.36.. .6....... .2.1.9.58 5.......1 97.5.2.4. .......1. ..69.4... 2...58..7 Puzzle 8..671..5 1574.368. .6.8.517. 62.1.9758 5.87.6..1 971582346 ..526781. 7869145.. 21.358.67 Current MSLS_SK_Loop Size: 8-sectors, 12-Int.Cells, 12-Links sectors: r19 c19 b1379 Digits: #2349 Links: r1 c9#2 r1 c19#3 r1379#4 r19 c9#9 Exclude: (r3c3 r7c2)#4

ID:01_595 MSLS_SK_Loop .7...532. 5...3...6 1...62... 9.6...... .53...67. ......5.1 ...67...3 3...9...4 .672...1. Puzzle 67...532. 58273.1.6 13..62.57 916.57..2 253...67. 748.265.1 495671283 321598764 867243915 Current MSLS_SK_Loop Size: 8-sectors, 8-Int.Cells, 8-Links sectors: r26 c59 b2356 Digits: #13489 Links: c5#1 r6#3 r2#4 r15#8 r26 c9#9 Exclude: (r15c4)#8

ID:01_920 MSLS_SK_Loop 1...748.9 .7...9.2. 8........ 41..3.... 5..2.6..7 ....1..64 ........2 .3.6...1. 6.872...3 Puzzle 162.748.9 37...9.2. 8....2... 416.372.. 58.2461.7 .2..1..64 ..14.36.2 23.6...1. 6.8721..3 Current MSLS_SK_Loop Size: 8-sectors, 9-Int.Cells, 9-Links sectors: r59 c19 b4679 Digits: #345789 Links: r5#3 r9#4 r9 c9#5 c1#7 c9#8 r59 c1#9 Exclude: (r23c9)#5

ID:02_308 MSLS_SK_Loop .7.5.29.. ..3.8..6. ....1...4 ....358.9 ......... 5.724.... 2...5.... .1..9.2.. ..56.4.8. Puzzle 471562938 9.3487.6. 8..9137.4 6..135879 139876... 587249.1. 2.8.51.9. .1..982.. .95624.8. Current MSLS_SK_Loop Size: 8-sectors, 10-Int.Cells, 10-Links sectors: r29 c38 b1379 Digits: #1234567 Links: r29#1 r23#2 r9#3 r8#4 r2 c8#5 c3#6 r9#7 Exclude: r3c2#2 r5c8#5

ID:03_175 MSLS_SK_Loop ....1.... ......743 .2..6..98 ....5791. ..1...8.. .9528.... 31..9..5. 648...... ....4.... Puzzle ....1.5.. 1.6.2.743 .2..63198 ...65791. .61.3.8.5 .952813.. 312896457 648175239 ...342681 Current MSLS_SK_Loop Size: 8-sectors, 10-Int.Cells, 10-Links sectors: r35 c29 b1346 Digits: #2345678 Links: r5 c9#2 c2#3 r3 c9#4 b1#5 c9#6 r5 b1#7 c2#8 Exclude: r3c4#4 r1c3#7

ID:04_20 MSLS_SK_Loop ..1...45. 3....6... 7.9..43.6 .924.5... ......... ...3.192. 9.78..6.2 ...7....9 .18...7.. Puzzle 62193.45. 3....629. 7.9.243.6 .924.5..3 ...29.... ...3.192. 9.78.36.2 ...7.2..9 2186.973. Current MSLS_SK_Loop Size: 8-sectors, 9-Int.Cells, 9-Links sectors: r19 c48 b2389 Digits: #14578 Links: c48#1 r9 c8#4 r9 c4#5 r1#7 r1 c8#8 Exclude: r5c8#1 r5c8#4 r5c8#8

ID:05_5 MSLS_SK_Loop .3.17.2.. ........6 8.74.23.. ..1...8.2 3.......9 9.4...1.. ..92.67.3 7........ ..2.38.1. Puzzle 436175298 215389476 897462351 ..1...832 3.86..549 9.48.3167 1.92.6783 783...62. ..273891. Current MSLS_SK_Loop Size: 8-sectors, 9-Int.Cells, 9-Links sectors: r57 c14 b4578 Digits: #1245679 Links: r5#1 r5#2 r7#4 r4 b78#5 c1#6 r5#7 c4#9 Exclude: (r4c25 r8c5 r9c2)#5

ID:06_44 MSLS_SK_Loop ......1.. 4..2..3.. ...45.268 73...9... ...5.4... ...1...52 623.75... ..9..1..7 ..8...... Puzzle 382796145 456218379 917453268 735.29... 2.15.47.. 8.4137.52 623.75... 5493.1.27 178.425.. Current MSLS_SK_Loop Size: 8-sectors, 10-Int.Cells, 10-Links sectors: r47 c59 b5689 Digits: #134689 Links: c8#1 c9#3 c7#4 r4 c59#6 r47 c5#8 r7#9 Triple overlap: r47c9 Exclude: r4c9#6

ID:07_65 MSLS_SK_Loop .26....7. ...4.5..8 .5..7.2.. ......937 3.7...4.1 249...... ..2.4..8. 8..7.1... .7....15. Puzzle .26...57. 7..4256.8 .58.7.2.. ...2.4937 3.7...421 249..7865 ..2.4.78. 8..7.13.2 .7...215. Current MSLS_SK_Loop Size: 8-sectors, 14-Int.Cells, 14-Links sectors: r28 c19 b1379 Digits: #134569 Links: r2 c1#1 r2 c9#3 r8 c19#4 b7#5 c9 b7#6 r28 c19#9 Exclude: r4c1#1 r7c2#6 r8c5#9

SK_Loop Example (1-100) The data file contains over 8,000 examples. .9.3.6.72 ..21794.8 6.8.....9 56..24.9. ......... 8..95.7.6 .......8. ..526...7 9....86.. 9 "ID:00_29 MSLS_SK_Loop" 94...2.3. .3.4..6.9 ....9...2 86.7.5941 ....89... .92.4.7.. 38..1.... .5.2....3 .....6.94 9 "ID:00_243 MSLS_SK_Loop" 6.1....5. 3...7.6.. ......74. ..7..42.. ....9.167 ....2.584 ..258.3.. 7.....42. 56..4.918 9 "ID:00_253 MSLS_SK_Loop" 8.2..7.51 ......82. .1.2.593. .6.47.... .....23.. 28.....76 ...34...2 371..94.. ..8..6.9. 9 "ID:00_433 MSLS_SK_Loop" 23.1.87.5 9..3476.. .......14 4.2....5. ...7..48. .68..3... 8.9..5..7 756...... 12......9 9 "ID:00_481 MSLS_SK_Loop" ...45.832 582..1.97 3..8...6. 1..7..... ..3.....1 .76.1..2. ...98.35. 6....4... ..51.3..4 9 "ID:00_538 MSLS_SK_Loop" .92.6..74 .7..4...5 5....26.. .....674. 76.4..... .24..7..8 ..759.... ...71.8.9 9..628... 9 "ID:00_544 MSLS_SK_Loop" .6...753. 5....37.. ..7561.4. ..53...78 4..65.3.. ...2.8... ....3.1.. .36..28.. ..2.856.. 9 "ID:00_614 MSLS_SK_Loop" .89..7..6 21..638.. .3..4..1. ....9...8 ...4.2... ....8..93 .5..24.37 628..1..5 .4.....82 9 "ID:00_787 MSLS_SK_Loop" .9173.24. 7..926... ..5.8..9. .7..48..2 9.2...85. .....913. .8.69...5 .4.....1. ..9.....8 9 "ID:00_822 MSLS_SK_Loop" .3....8.. 9...1.532 5....8.67 ...5.4.7. ......2.. 4..17..9. 3..6...51 8.1.597.. .5..419.. 9 "ID:00_877 MSLS_SK_Loop" ..6...5.. ..4.563.9 .9..7.24. ....8.... 23..6...5 .8.7..69. .4..37... 9.3..846. 7.8.9...2 9 "ID:00_892 MSLS_SK_Loop" ...4...2. 4.2..1... 97.....46 1.3.549.. 7..62.3.. ......472 .4...9.5. 83.54.6.1 25....... 9 "ID:00_948 MSLS_SK_Loop" .8.6.9... ...783..6 ....4.... 13.5.6.82 .57...96. 46.9.8.71 ....6.... 2..314... ...2.5.1. 9 "ID:00_1080 MSLS_SK_Loop" .1.5.9... ...173..8 ....2.... 32.6.7.85 .68...43. 49.3.5.16 ....5.... 1..836... ...7.4.9. 9 "ID:00_1081 MSLS_SK_Loop" .9.6.7... ...192..3 ....3.... 84.7.6.39 .76...51. 31.9.8.67 ....6.... 7..825... ...3.9.4. 9 "ID:00_1083 MSLS_SK_Loop" .2.4.7... ...539..7 ....2.... 79.2.1.36 .65...42. 43.9.6.71 ....8.... 3..795... ...6.2.9. 9 "ID:00_1086 MSLS_SK_Loop" .6.8.5... ...312..5 ....6.... 52.6.4.19 .43...56. 81.2.9.43 ....7.... 1..523... ...4.6.2. 9 "ID:00_1087 MSLS_SK_Loop" .7.9.6... ...513..6 ....7.... 63.4.1.72 .45...91. 91.3.2.64 ....8.... 1..635... ...7.4.3. 9 "ID:00_1088 MSLS_SK_Loop" ...41..7. 57.2...4. ..697.5.. ......912 9.3...4.8 824...... ..8.917.. .6...3.94 .9..42... 9 "ID:00_1095 MSLS_SK_Loop" ...21..4. 34.6...2. ..594.3.. ......897 9.8...2.6 762...... ..7.519.. .5...9.32 .9..62... 9 "ID:00_1098 MSLS_SK_Loop" ..62..7.. ...68.... 7.13.59.2 ..8...435 .4.....7. 237...8.. 5.28.31.7 ....64... ..4..26.. 9 "ID:00_1121 MSLS_SK_Loop" ....64.8. 5...9.... ..67.21.. 2.8.1.6.. 63.2.9.18 ..5.4.2.9 ..39.18.. ....3...2 .6.85.... 9 "ID:00_1139 MSLS_SK_Loop" ....37.5. 3...6.... ..68.51.. 2.8.4.7.. 96.2.8.41 ..4.7.8.2 ..97.46.. ....8...9 .7.62.... 9 "ID:00_1141 MSLS_SK_Loop" ....79.3. 3...6.... ..73.89.. 4.6.3.5.. 15.9.6.42 ..8.2.3.6 ..56.38.. ....1...3 .3.74.... 9 "ID:00_1147 MSLS_SK_Loop" ....82.5. 8...4.... ..47.51.. 9.7.3.5.. 64.5.8.39 ..3.9.8.6 ..64.13.. ....5...7 .2.97.... 9 "ID:00_1148 MSLS_SK_Loop" .8..4.... .4.95..78 ....23... ..7.1..5. 8237.5146 .1..6.3.. ...18.... 79..32.8. ....7..3. 9 "ID:00_1161 MSLS_SK_Loop" 1..6..5.4 .6.3.2.1. 8..5..... .4.8.1235 ......... 3124.5.9. .....6..1 .3.1.9.8. 5.1..4..3 9 "ID:00_1174 MSLS_SK_Loop" .2..4.3.. ..196...2 7.8..249. ..7....6. 28.....31 .5....7.. .365..1.7 1...876.. ..4.1..8. 9 "ID:00_1224 MSLS_SK_Loop" ...81..3. 8..79.1.. .72.3.5.. .......95 547...821 29....... ..1.5.74. ..4.73..8 .5..48... 9 "ID:00_1302 MSLS_SK_Loop" ..3.19.4. 1..8..3.. .2...3..5 3.1.4..9. 4..9.6..1 .9..3.7.4 2..1...7. ..6..7..2 .5.62.9.. 9 "ID:00_1403 MSLS_SK_Loop" ..5...... .34.7.59. .7...4.13 ..1498... .8.6.2.4. ...3571.. 54.7...3. .96.4.27. ......4.. 9 "ID:00_1430 MSLS_SK_Loop" ..7..92.. .3.7...5. 1.5.8.6.3 5..2.3.9. ..6...8.. .2.6.8..7 6.2.9.7.8 .4...7.6. ..98..4.. 9 "ID:00_1587 MSLS_SK_Loop" 1...64.98 6..2..... ..78.16.. 8.1...46. 7.......1 .34...7.2 ..31.65.. .....2..3 48.57...6 9 "ID:00_1620 MSLS_SK_Loop" .6...783. 3149...52 7...4..9. 1......2. ..3...9.. .2......8 .4..7...3 65...3271 .371...8. 9 "ID:00_1664 MSLS_SK_Loop" .7.5.6.1. 6.3.1...4 .....3.8. 2.78.1..6 .5.....2. 9..6.23.7 .6.3..... 8...7.1.5 .9.1.5.3. 9 "ID:00_1866 MSLS_SK_Loop" 3....4..5 .9....38. .843.26.. 8.1.4.2.. ...6.3... ..7.1.9.6 ..39.156. .58....9. 6..4....7 9 "ID:00_1955 MSLS_SK_Loop" 1......8. ....1...2 .59...617 .4695.1.3 .8.....7. 9.3.2756. 361...85. 2...3.... .7......6 9 "ID:00_2020 MSLS_SK_Loop" 48.3..... ...5.4.8. ...16.234 8..937... 7.......9 ...685..3 945.16... .1.8.2... .....3.51 9 "ID:00_2152 MSLS_SK_Loop" 5.....8.. 79.3...1. .31.87.92 2.7..3.8. ......... .1.4..2.5 38.79.65. .7...1.28 ..9.....3 9 "ID:00_2275 MSLS_SK_Loop" .5.1...76 941...2.8 ..8.4.... ..97.5..1 ...4.1... 1..9.83.. ....1.5.. 7.4...913 31...4.8. 9 "ID:00_2286 MSLS_SK_Loop" ..9.21..5 53.6.72.. 12.5....3 ....529.. ......... ..486.... 9....5.87 ..27.6.49 7..93.6.. 9 "ID:00_2316 MSLS_SK_Loop" ..57..639 ...34...8 .8..95... .596..4.. 7.......5 ..8..971. ...91..4. 5...67... 217..48.. 9 "ID:00_2345 MSLS_SK_Loop" 7.52.8.1. 2...4.538 .......62 .7..2...1 ..9...3.. 4...5..9. 65....... 318.7...4 .4.1.36.5 9 "ID:00_2474 MSLS_SK_Loop" 2.8.4.691 ..1...... ..9..2..3 4.78.3.1. ..2...9.. .5.4.93.8 6..2..1.. ......7.. 723.1.8.5 9 "ID:00_2591 MSLS_SK_Loop" 1..7.234. 4......2. .2.15...9 ..65.9... 9..2.1..7 ...8.64.. 8...23.5. .3......4 .459.7..8 9 "ID:00_2778 MSLS_SK_Loop" 4.28..5.. .6..53.82 .3..6.1.. ..43...9. 9.......5 .7...64.. ..6.3..4. 74.92..5. ..9..57.8 9 "ID:00_2792 MSLS_SK_Loop" ..437..1. 6....5.3. 3..2416.5 ....39... 9.......4 ...65.... 7.6194..3 .9.8....6 .2..679.. 9 "ID:00_2987 MSLS_SK_Loop" 9..3.4..7 .8.....1. ......... ..96587.. .2.7.9.3. 17.4.2.69 7...2...3 3.1...6.2 .6.5.3.7. 9 "ID:00_3026 MSLS_SK_Loop" 2.91653.8 .1.....4. 8..294..1 ...4.9... .85...19. 6..871..2 ..8...9.. 17.....85 ....1.... 9 "ID:00_3041 MSLS_SK_Loop" 2.85314.9 .9.....1. 4..967..8 ...4.8... .16...24. 7..126..3 ..3...7.. 52.....86 ....1.... 9 "ID:00_3042 MSLS_SK_Loop" .5.6.2.3. 3.......9 .9.3.1.7. 9..5.6..1 67.....95 .8.....6. 73.2.4.18 ...8.3... .62...34. 9 "ID:00_3051 MSLS_SK_Loop" ..4...3.. 18.....64 5.......8 25.4.9.73 3.......6 ..7...9.. ...8.3... 64.1.7.85 8.52.61.9 9 "ID:00_3205 MSLS_SK_Loop" ...6.2... 236...485 ...3.8... 1.34859.2 6.......1 94.7.6.53 ..4.9.6.. 81.....94 ......... 9 "ID:00_3415 MSLS_SK_Loop" ..69.57.. 95.....62 4.1...9.3 ....6.... ...572... 6.53.48.7 8.74.93.6 .63...48. ....3.... 9 "ID:00_3461 MSLS_SK_Loop" ..73.89.. 23.....41 6.9...7.3 ....6.... ...819... 3.62.41.8 7.49.13.6 .13...27. ....3.... 9 "ID:00_3462 MSLS_SK_Loop" 4.......5 .728.431. .1.....4. .2.....8. 94.5.7.61 6..2.9..4 2.49.56.3 ......... 79.....58 9 "ID:00_3531 MSLS_SK_Loop" .2789351. ...7.6... ..9.1.2.. ..82.96.. .5.....3. 9...6...8 1.4...7.9 .629.814. .9.....6. 9 "ID:00_3606 MSLS_SK_Loop" .932.875. 4..5.3..6 .5.....8. 96.3.1.47 ..7...3.. 5..7.6..2 ......... .168.297. ...1.9... 9 "ID:00_3612 MSLS_SK_Loop" 6187.2354 .4.....1. 5...4...6 ...489... .8.....2. ......... 4...5...3 8...7...2 3756.1948 9 "ID:00_3695 MSLS_SK_Loop" 1467.2539 .5.....4. 9...6...2 ...276... .8.....9. ......... 8...3...5 4...2...8 6931.8427 9 "ID:00_3704 MSLS_SK_Loop" ..94.52.. 4..9.3..7 35.....49 ..45967.. ..1...4.. 5.7.1.3.8 ......... 2.3...5.4 9..748..3 9 "ID:00_3719 MSLS_SK_Loop" ...6.9... .8.5.2.9. 2..1.4..6 ......... .94...26. 5.74.69.3 .71...62. .4.....3. 3.29.87.4 9 "ID:00_3731 MSLS_SK_Loop" .3.8.1.2. 8..257..4 ...3.4... ..8...7.. ..35.86.. .2.1.3.8. 38.....96 1..4.6..5 6.5...8.1 9 "ID:00_3787 MSLS_SK_Loop" ...8.7... ...4.3... 8..9.1..3 7...8...1 9.1...4.8 3.5...6.7 .9.....7. 472...385 5..372..4 9 "ID:00_3853 MSLS_SK_Loop" 24.963.81 16.....93 3..4.5..2 ....4.... ...3.1... 5..298..4 79.....25 ..18.69.. ...5.9... 9 "ID:00_3910 MSLS_SK_Loop" 35.691.48 46.....13 2..7.4..6 ....5.... ...1.8... 8..946..2 68.....29 ..25.94.. ...8.2... 9 "ID:00_3915 MSLS_SK_Loop" ...4.7... .1..8..9. 54..9..76 6.2...4.8 1..8.4..7 854...932 ..37.26.. .6.....5. ...1.9... 9 "ID:00_3918 MSLS_SK_Loop" .4.627.5. ......... 7.8...2.6 .2.1.6.7. .54.3.68. 6..8.9..2 28.5.3.64 4.......3 3.......7 9 "ID:00_3971 MSLS_SK_Loop" .4.3.2.9. ....4.... .89...13. 9328.6417 7...1...9 ..8...2.. 8.5...3.1 .916.584. ....3.... 9 "ID:00_4012 MSLS_SK_Loop" ...4.862. .......7. 42...6..5 7.8...93. 9..3.1756 3.5...41. 65...9..7 .......8. ...7.456. 9 "ID:00_4026 MSLS_SK_Loop" ..631.9.. 4.1.87.5. ....62... 9....5.14 6.3....7. 7....6.32 ....73... 1.4.58.9. ..724.1.. 9 "ID:00_4174 MSLS_SK_Loop" 48.2.79.5 1.6...... ..95..... ..48.3.79 .3.6....2 ..71.4.38 ..87..... 6.2...... 71.9.24.3 9 "ID:00_4274 MSLS_SK_Loop" 2...41... .5...2..3 ..75.3642 ..489.... .7....9.. ..912.... ..17.5328 .8...9..7 7...38... 9 "ID:00_4284 MSLS_SK_Loop" .2561.9.. .9.....7. 6..5.9... 13.9.2... .6.384..5 85.7.1... 2..1.6... .8.....1. .1689.7.. 9 "ID:00_4309 MSLS_SK_Loop" ..3..26.. 47236...5 6.5.....3 3.4.8.9.. .....7... 7.8.2.1.. 8.7.....6 24961...8 ..6..42.. 9 "ID:00_4356 MSLS_SK_Loop" ..35.61.. .1.32...6 ....4.... 8.673..9. 321....7. 7.946..2. ....9.... .3.65...7 ..48.73.. 9 "ID:00_4431 MSLS_SK_Loop" .2..1..3. 157...2.. ..67.95.. ..4..59.1 .1.4..6.. ..3..74.2 ..89.13.. 932...1.. .7..5..9. 9 "ID:00_4454 MSLS_SK_Loop" 9...4.1.. ..583.4.. .....9235 2...73.8. ...2...1. 5...18.9. .....7326 ..249.8.. 7...2.9.. 9 "ID:00_4466 MSLS_SK_Loop" ...4237.. ....71..5 1.....6.3 8.279...4 ......96. 7.316...2 9.....5.6 ....56..9 ...9483.. 9 "ID:00_4480 MSLS_SK_Loop" 6.53...72 ....6...1 .924..6.. ..4...127 .6...5... ..3...586 .218..7.. ....9...3 5.91...64 9 "ID:00_4507 MSLS_SK_Loop" .4..3...7 7381...9. 16....2.3 .9.4.1... ....9.63. .5.7.3... 31....7.9 5879...4. .2..4...8 9 "ID:00_4538 MSLS_SK_Loop" .2..3...1 7341...5. 51....3.4 .7.6.3... ....4.63. .6.8.2... 69....1.5 3417...9. .5..9...3 9 "ID:00_4542 MSLS_SK_Loop" 8..47.... ..3...645 ..16.9... .723.1..9 9..84.... .847.2..6 ..75.4... ..8...274 1..28.... 9 "ID:00_4549 MSLS_SK_Loop" ..73.92.. ...6..184 ...85.... 18.92.3.. .......12 23.18.6.. ...56.... ...2..576 ..57.34.. 9 "ID:00_4555 MSLS_SK_Loop" ..59.21.. ...7..548 ...41.... 37.14.2.. .......39 29.35.4.. ...59.... ...8..651 ..42.19.. 9 "ID:00_4559 MSLS_SK_Loop" 5...3.7.. 14.57..8. .79.8.... 31.86...9 ..6...... 82.34...6 .52.1.... 93.72..6. 6...9.3.. 9 "ID:00_4627 MSLS_SK_Loop" 182...... .7..6.... 6.9..47.. .159.7246 9........ .463.2958 8.4..65.. .6..3.... 237...... 9 "ID:00_4642 MSLS_SK_Loop" 4..3..15. 395...... ...564..3 .7..4.28. ...2...3. .3..5.46. ...416..2 168...... 2..8..31. 9 "ID:00_4696 MSLS_SK_Loop" 3...756.2 .8....... ...3.278. 9154...2. 6.....1.. 8231...6. ...7.895. .6....... 5...213.6 9 "ID:00_4734 MSLS_SK_Loop" 2.59.3678 .....7.2. 1.7.6.9.. 3...2.8.. .....9... 9...7.5.. 4.9.5.3.. .....4.1. 5.17.6249 9 "ID:00_4823 MSLS_SK_Loop" .3....9.. 1.95.46.8 ..43.975. ..8....6. 46....... ..3....9. ..29.517. 5.78.23.9 .9....2.. 9 "ID:00_4870 MSLS_SK_Loop" ..25.3.8. .7.9..324 3...2.... .1..85.6. ....4.1.. .3..16.4. 6...9.... .2.4..693 ..56.2.7. 9 "ID:00_4895 MSLS_SK_Loop" ..2.8..61 .8.1..9.2 ....4.78. .4.937... 69.....3. .5.268... ....1.69. .6.8..5.3 ..4.9..27 9 "ID:00_4982 MSLS_SK_Loop" .91...27. 2.3..8... ...2.9..1 .2439.... 9...27.1. .1768.... ...8.2..4 7.6..3... .32...96. 9 "ID:00_5008 MSLS_SK_Loop" ..9...2.. 13..2..98 .5..9..6. 4.29.68.3 ...248... 8.15.79.4 .4..6..8. 72..8..39 ..8...6.. 9 "ID:00_5047 MSLS_SK_Loop" ......... .57...31. ..85.92.. 48.6.2.31 ..21574.. 67.8.3.25 ..34.51.. .16...54. ......... 9 "ID:00_5085 MSLS_SK_Loop" .4.893.1. ..6...8.. .81...34. 85.4.9.61 ......... 47.1.6.83 .14...75. ..8...4.. .9.748.3. 9 "ID:00_5090 MSLS_SK_Loop" .7.5.3.8. 3..2.6..5 ...9.8... .59.8.23. 7.......6 .81.6.75. ...6.1... 1..4.2..8 .3.8.7.6. 9 "ID:00_5133 MSLS_SK_Loop" .6.3.7.9. 7..4.2..3 ...8.9... .38.4.57. 6.......4 .95.7.63. ...7.1... 5..6.4..7 .7.2.5.6. 9 "ID:00_5135 MSLS_SK_Loop"