SudokuSolver Forum

3x3::k:3584:3584:3584:7445:6402:4355:4355:4355:3844:5121:5121:7445:7445:6402:6402:6402:6917:3844:7445:7445:7445:4360:6665:5130:6402:6917:3844:5121:4360:4360:4360:6665:5130:5130:6917:4620:5121:2061:2061:7694:6665:5130:6917:6917:4620:5121:7694:7694:7694:6665:4112:4112:4112:4620:3601:7694:5138:5138:6665:4112:5139:2580:2580:3601:7694:5138:3335:3335:5139:5139:3083:3083:3601:5647:5647:5647:3335:5139:3078:3078:3078:

Prelims

a) R5C23 = {17/26/35}, no 4,8,9
b) R7C89 = {19/28/37/46}, no 5
c) R8C89 = {39/48/57}, no 1,2,6
d) 20(3) cage at R7C3 = {389/479/569/578}, no 1,2
e) 22(3) cage at R9C2 = {589/679}

1a. 22(3) cage at R9C2 = {589/679}, 9 locked for R9
1b. 45 rule on C1234 1 innie R8C4 = 6 -> R89C5 = 7 = {25/34}
1c. 45 rule on N9 2 innies R78C7 = 11 = {29/38/47}/[65], no 1, no 5 in R7C7
1d. 45 rule on N9 2 outies R89C6 = 9 = {18/27} (cannot be {45} which clashes with R89C5)
1e. 45 rule on C6789 2 outies R12C5 = 12 = {39/48/57}, no 1,2,6
1f. 1,6 in C5 only in 26(5) cage at R3C5 = {12689/13679/14678}, no 5
1g. 45 rule in R9 3 innies R9C156 = 11 = {128/137/146/236/245}
1h. 2 of {128} must be in R9C5, 6 of {236} only in R9C1, 2 of {245} must be in R9C6 -> no 2 in R9C1
1i. 3 of {137} must be in R9C5, 6 of {236} only in R9C1 -> no 3 in R9C1
1j. 45 rule on N7 2 outies R79C4 = 2 innies R78C2 + 11
1k. Max R79C4 = 17 -> max R78C2 = 6, no 6,7,8,9 in R78C2
1l. Min R78C2 = 3 -> min R79C4 = 14, no 3,4 in R7C4
1m. R79C4 = {59/79/89} (cannot be {78} which clashes with R89C6), 9 locked for C4 and N8
1n. 45 rule on C1 2 innies R13C1 = 1 outie R2C2 + 11
1o. Min R13C1 = 12, no 1,2
1p. Max R13C1 = 17 -> max R2C2 = 6
1q. 12(3) cage at R9C7 = {138/147/237/246} (cannot be {156} which clashes with 22(3) cage at R9C2, cannot be {345} which clashes with R8C89), no 5
1r. 5 in N9 only in R8C789, locked for R8, clean-up: no 2 in R9C5
1s. 5 in N9 only in R78C7 = [65] or R8C89 = {57} -> R78C7 = {29/38}/[65] (cannot be {47}, locking-out cages), no 4,7
1t. 1 in N9 only in R7C89 = {19} or 12(3) cage at R9C7 = {138/147} -> R7C89 = {19/28/46} (cannot be {37} which clashes with 12(3) cage at R9C7 = {138/147}, blocking cages), no 3,7 in R7C89
1u. R9C156 = {137/146/236/245} (cannot be {128} which clashes with 12(3) cage at R9C7), no 8, clean-up: no 1 in R8C6

[Now to get more from N8 and N9]
2a. R78C7 (step 1s) = {29/38}/[65], 12(3) cage at R9C7(step 1q) = {138/147/237/246}
2b. Consider combinations for R89C6 (step 1d) = {27}/[81]
R89C6 = {27} with R78C7 = {38}, locked for N9 => 12(3) cage at R9C7 = {147/246}
or R89C6 = {27} with R78C7 = [65], R8C89 = {39/48} => 12(3) cage at R9C7 = {147/237/246} (cannot be {138} which clashes with R8C89)
or R89C6 = [81]
-> 12(3) cage at R9C7 = {147/237/246}, no 8
2c. 8 in R9 only in 22(3) cage at R9C2 = {589}, 5 locked for R9, clean-up: no 2 in R8C5
2d. Killer pair 3,4 in R9C5 and 12(3) cage at R9C7, locked for R9
2e. Naked pair {34} in R89C5, locked for C5 and N8, clean-up: no 8,9 in R12C5
2f. Naked pair {57} in R12C5, locked for N2 and 25(5) cage at R1C5, 7 locked for C5
2g. 2 in R9 only in R9C6789, CPE no 2 in R78C7, clean-up: no 9 in R78C7
2h. R7C89 (step 1t) = {19/28/46}
2i. Consider placement for 1 in N8
1 in R7C56 => 1 in N9 only in 12(3) cage at R9C7 = {147}, 4 locked for N9, 1,7 locked for R9 => R9C6 = 2
or R9C6 = 1 => 1 in N9 only in R7C89 = {19}
-> R7C89 = {19/28}, no 4,6, R9C6 = {12}, clean-up: no 2 in R8C6
2j. Killer pair 1,2 in R9C6 and 12(3) cage, locked for R9
2k. Consider combinations for R7C89 = {19/28}
R7C89 = {19}, 1 locked for R7 and N9 => R9C6 = 1 (hidden single in R9), R8C6 = 8, R7C5 = 2
or R7C89 = {28}, locked for R7 => R7C5 = 1
-> R7C5 = {12}
2l. Killer pair 1,2 in R7C5 and R7C89, locked for R7
2m. R79C4 = R78C2 + 11 (step 1j)
2n. Min R78C2 = 4 -> min R79C4 = 15, no 5 in R79C4
2o. R7C6 = 5 (hidden single in N8) -> R6C678 = 11 = {128/137/146/236}, no 9
2p. R79C4 = [79]/{89} = 16,17 -> R78C2 = 5,6 = [32/41/42], R8C2 = {12}
2q. 20(3) cage at R7C3 = {389/479}, no 6,
2r. 6 in N7 only in R79C1, locked for C1
2s. 14(3) cage at R7C1 = {167} (only remaining combination) -> R8C1 = 1, R79C1 = {67}, 7 locked for C1 and N7, R8C2 = 2, clean-up: no 6 in R5C3
2t. Consider placements for R7C5 = {12}
R7C5 = 1 => R7C89 = {28}, locked for N9 => R78C7 = [65]
or R7C5 = 2 => R9C6 = 1, R8C6 = 8 => R78C7 = [65]
-> R78C7 = [65], R79C1 = [76], clean-up: no 7 in R8C89
2u. 20(3) cage at R7C3 = {389} (only remaining combination), 3 locked for C3 and N7 -> R7C2 = 4, clean-up: no 5 in R5C2
2v. Naked pair {89} in R79C4, 8 locked for C4 and N8 -> R8C6 = 7, R9C6 = 2, R7C5 = 1, clean-up: no 9 in R7C89
2w. Naked pair {28} in R7C89, 8 locked for R7 and N9 -> R7C34 = [39], R8C3 = 8, R9C4 = 8, clean-up: no 4 in R8C89
2x. Naked pair {39} in R8C89, 3 locked for R8 and N9 -> R89C5 = [43]
2y. 45 rule on C9 3 innies R789C9 = 12 = {129/138/237} (cannot be {147} because 1,4,7 only in R9C9 and, of course, R789C9 cannot have the same combination at 12(3) cage at R9C7, combo crossover clash), no 4

3a. 20(5) disjoint cage at R2C1 = {12359/12458/23456} (cannot be {12368} because 1,6 only in R2C2)
3b. 1,6 only in R2C2 -> R2C2 = {16}
3c. 20(5) disjoint cage = {12359/12458/23456}, 5 locked for C1
3d. 45 rule on R1 3 innies R1C459 = 14 = {158/167/257/347/356} (cannot be {149/239/248} because R1C5 only contains 5,7), no 9
3e. 1 of {158/167}, 2 of {257} must be in R1C4 -> no 1,2 in R1C9
3f. 14(3) cage at R1C1 = {149/158/239/347} (cannot be {167} which clashes with R2C2, cannot be {257} which clashes with R1C5, cannot be {248} because 2,4 only in R1C3, cannot be {356} which clashes with R1C459), no 6
3g. 8 of {158} must be in R1C1 -> no 8 in R1C2
3h. 17(3) cage at R1C6 = {269/278/368/458/467} (cannot be {179/359} which clash with 14(3) cage at R1C1), no 1
3i. 1 in R1 only in R1C234, CPE no 1 in R2C3 + R3C23
[Maybe the later part of step 2 and step 3 were Ed’s “easy middle”]

4a. R78C2 = [42] -> 30(6) cage at R5C4 = {234579/234678} (cannot be {124689} because R56C4 must contain at least one of 3,5,7), no 1
4b. 30(6) cage at R5C4 = {234579} (only remaining combination, cannot be {234678} with R56C4, R6C23 = [86] which clashes with R6C678, step 2o), no 6,8, 9 locked for R6 and N4
4c. 30(6) cage = {234579}, CPE no 3 in R6C6
4d. 9 in C1 only in R123C1, locked for N1
4e. R5C23 = {17}/[62] (cannot be [35] which clashes with 30(6) cage since that only leaves R6C4 for 3,5 so ALS clash), no 3,5
4f. 20(5) disjoint cage at R2C1 (step 3a) = {12458/23456} (cannot be {12359} = [91]{235} which clashes with R56C23, killer ALS clash), no 9, 4 locked for C1
4g. 45 rule on N1 2 innies R2C12 = 2 outies R12C4 + 2
4h. Min R12C4 = 3 -> min R2C12 = 5
4i. R5C12 = {17} (cannot be [62] which would give R2C1 = 2 (hidden single in C1), R2C2 = 1 which only total 3), locked for R5 and N4
4j. 30(6) cage = {234579} -> R6C4 = 7
4k. 30(6) cage = {234579}, CPE no 3,5 in R5C1
4l. 6 in N4 only in R4C23, locked for R4
4m. 17(4) cage at R3C4 = {1268/2456}, no 3

5a. Naked pair {59} in R69C3, locked for C3
5b. R13C1 = R2C2 + 11 (step 1n)
5c. Consider combinations for 14(3) cage at R1C1 (step 3f) = {149/158/239/347}
14(3) cage at R1C1 = {149/158}, 1 locked for N1 => R2C2 = 6
or 14(3) cage at {239} = [932] => R3C1 = 8, R13C1 = 17 => R2C2 = 6
or 14(3) cage = {347} = [374] => R25C2 = [61]
-> R2C2 = 6, R13C1 = 17 = {89}, 8 locked for C1 and N1
[Cracked, at last; the rest is fairly straightforward.]
5d. R4C3 = 6 (hidden single in N4)
5e. 14(3) cage = {149/158/239} (cannot be {347} which doesn’t contain one of 8,9), no 7
5f. 7 in N1 only in 29(6) cage at R1C4 = {134579/234578} -> R3C2 = 5, R4C2 = 8, R9C23 = [95], R6C23 = [39], R5C4 = 5, R1C2 = 1 -> R1C13 = 13 = [94], R3C1 = 8
5g. R4C23 = [86] = 14 -> R34C4 = 3 = {12}, locked for C4 -> R12C4 = [34]
5h. R1C459 = 14 (step 3d), R1C4 = 3 -> R1C59 = 11 = [56], R1C6 = 8, R2C5 = 7, R23C3 = [27], R2C1 = 3
5i. Naked pair {27} in R1C78, 2 locked for N3
5j. R12C5 = [57] = 12 -> R2C67 + R3C7 = 13 = {139/148} -> R3C7 = {34}, R2C67 = [18]/{19}, 1 locked for R2
5k. R1C9 = 6 -> R23C9 = [54/81], no 3,9
5l. 3 in C6 only in R45C6, locked for 20(4) cage at R3C6, no 3 in R4C7
5m. 20(4) cage = {2369/3467} (cannot be {1379} because R345C6 = {139} clashes with R2C6), no 1, 6 locked for C6
5n. 2,7 only in R4C7 -> R4C7 = {27}
5o. Naked pair {27} in R14C7, locked for C7
5p. 3 in R3 only in R3C78, CPE no 3 in R5C7
5q. R3C7 = 3 (hidden single in C7) -> R2C67 = {19}, 9 locked for R2
5r. 8 in C7 only in R56C7, locked for N6
5s. 18(3) cage at R4C9 = {279/459}, no 1,3, 9 locked for C9 -> R8C89 = [93]
5t. R2C7 = 9 (hidden single in N3) -> R2C6 = 1, R67C6 = [45] = 9 -> R6C78 = 7 = [16], R59C7 = [84], R2C8 = 5, R23C9 = [81], R3C8 = 4
5u. R23C8 + R5C7 = [548] = 17 -> R45C8 = 10 = [73]

and the rest is naked singles.

1. Innies c1234 -> r8c4 = 6
(I skip the first 'easy' part. Andrew's Step 1 & 2, but I did it quite differently).
-> r7 = [7439{15}6{28}]
r8 = [1286475{39}]
r9 = [6{59}832{147}]

2. Remaining Innies c5 = r12c5 = +12(2) = {57}
-> r7c56 = [15]
Also (57) in c4/n5 in r456c4

3. IOD c1 -> r13c1 = r2c2 + 11
-> r2c2 is Max 6
Whatever is in r2c2 goes in c1 in r789c1
-> r2c2 from (16)

4. Whatever is in r2c4 is in n1 in r1c123
Whatever is in r1c4 is in n1 in r2c12
IOD n1 -> r2c12 = r12c4 + 2
Since r12c2 are both from (1234) -> r2c12 has one value from (1234) which = r1c4, and one value from (3456) which = r2c4 + 2

5. The values in r3c123 go in n2 in r12c56
Since r12c5 = {57} -> At least one of (57) in r3c123

6. Trying r2c2 = 1 ...
That puts r1c4 = 1 and r13c1 = +12(2)
-> 2 not in 14(3)n1 -> 2 not in r2c4 -> r2c14 = [53]
But this puts 7 (Step 5) and 2 in 29(6) in n1 which is impossible. (29(6) cannot contain all of (1237)).
-> r2c2 = 6
-> r13c1 = {89} and r2c4 = 4
-> 14(3)n1 = [914] and r3c1 = 8
-> r1c4 = r2c1 from (23)

7. r2456c1 = {2345} with r2c1 from (23)
-> (45) in r456c1
-> 8(2)n4 = [71] ((26) already both in c2)

8! (689) in n4 in r46c23
17(4)r3c4 -> at most one of (89) in r4c23
Since r78c2 = [42] and r8c4 = 6 -> r6c23 cannot be {89} (Would put r56c4 = +7(2))
-> One of (89) in r4c23 and the other in r6c23
-> 17(4)r3c4 cannot also contain a 7
-> (HS 7 in c4) r6c4 = 7
-> r6c678 from {128), {146}, or {236}
-> At least one of (68) in r4c23
-> 9 in r6c23
-> r4c2 = 8
Since r4c3 not from (13) -> 5 not in r4c4 -> (HS 5 in c4) r5c4 = 5
-> 30(6)r5c4 = [53942]
-> r4c23 = [86] and r34c4 = {12}
-> r1c4 = r2c1 = 3 and r456c1 = {245}
Also r9c23 = [95]

9. Remaining Innies r1 = r1c59 = +11(2) = [56]
-> 17(3)r1 = [8{27}]
2 in n2 in r3c12
-> 29(6)r1c4 = [324857]

10. r2c6 only from (19)
Remaining IOD c6 -> r2c6 + r6c6 = r4c7 + 3
Since 3 already in r6 -> r2c6 not same as r4c7
Trying r2c6 = 9 puts r3c8 = 9, r8c9 = 9, and r4c7 = 9 contradicting previous statement.
-> r2c6 = 1

11. r23c9 = [81] or [54]
-> 25(5)r1c5 = [57193]
-> r34c4 = [21] and r3c56 = {69}

12. Remaining cells c6 = r3456c6 = {3469} = +22(4)
-> r6c6 = r4c7 + 2
From previous placements only solution is r345c6 = {369}, r6c6 = 4 and r4c7 = 2
-> 16(4)r6c6 = [4165]
-> (HS 8 in c7) r5c7 = 8
-> (HS 8 in n3) 15(3)n3 = [681]
-> 27(5)r2c8 = [54783]
Also (HS 8 in n5) r6c5 = 8
-> 26(5)c5 = [69281]
etc.

1. Innies c1234 -> r8c4 = 6
-> r89c5 from {25} or {34}
Outies n9 = r89c6 = +9(2) from {18} or {27}

2. (34) on r9?
Neither of (34) in 22(3)r9
12(3)r9 cannot be {345} since one of (345) in 12(2)n9 -> at most one of (34) in 12(3)n9
Innies r9 = r9c156 = +11(3) - at most one of (34)
-> One of (34) in r9c145 and one in r9c789

3. Since one of (34) in h+11(3)r9 -> 8 not in h+11(3)r9
-> r89c6 not [18]
Also r89c6 cannot be [81] since that leaves no place for both (18) in n9
-> r89c6 = {27}
-> r89c5 = {34}
-> 1 in n8 only in r7c56
-> 1 in n9 in 12(3)n9
-> 9 in n9 in 12(2)n9 = {39}
-> 2 in n9 in 10(2)n9 = {28}
-> 7 in n9 in 12(3)n9 = {147}
-> r78c7 = [65]
Also r89c6 = [72] and r89c5 = [43]
Also 8 in n8 only in r9c4
-> r9c23 = {59}
-> r9c1 = 6
-> r78c1 = [71]
Also 2 in n7 in r8c2
Also 4 in n7 in r7c2 (20(3) cannot be {479})
-> 20(3)r7c3 = [398]
-> r7c56 = {15}

This is how I got started.

Preliminaries from SudokuSolver
Cage 8(2) n4 - cells do not use 489
Cage 12(2) n9 - cells do not use 126
Cage 10(2) n9 - cells do not use 5
Cage 22(3) n78 - cells do not use 1234
Cage 20(3) n78 - cells do not use 12

1. "45" on c1234: 1 innie r8c4 = 6
1a. -> r89c5 = 7 = {25/34}

2. 22(3)r9c2 = {589/679} = 5 or 6
2a. 9 locked for r9

3. 12(3)n9: {345} blocked by 12(2)n9 which needs one of those
3a. {156} blocked by 22(3) needs one of those (step 2)
3b. = {138/147/237/246}(no 5)
3c. note: if it has 8 must also have 1

4. "45" on n9: 2 outies r89c6 = 9
4a. but {45} blocked by h7(2)r89c5 needs one of them
4b. = {18/27}
4c. however [81] blocked since both the 8 and the 1 are both forced into the 10(2)n9 (step 3c)
4d. -> no 8 in r8c6, no 1 in r9c6

5. "45" on r9: 3 innies r9c156 = 11
5a. hidden killer pair 3,4 in r9 since 12(3)n9 can only have one of
5b. -> h11(3) must have one of
5c. = {137/146/236/245}(no 8)
5d. -> no 1 in r8c6 (h9(2))

6. r89c6 = {27} only
This leads to the easy middle and quickly results in wellbeback's end step 1.