Prelims
a) 10(3) cage no 8,9
b) 10(4) cage = {1234}
c) 14(4) cages no 9
d) 27(4) cage no 1,2
e) 28(4) cage no 1,2,3
1. 45 rule on R1234 total of 8 ‘zero’ cells = 65 -> 3 ‘zero’ cells in N1 total 24, 3 ‘zero’ cells in N3 total 24 and 2 ‘zero’ cells in R4 total 17
1a. R1C12 + R2C1 = 24 = {789}, locked for N1
1b. R2C9 + R3C89 = 24 = {789}, locked for N3
1c. R4C19 = 17 = {89}, locked for R4
1d. Naked triple {789} in R123C1, locked for C1, 7 also locked for N1
1e. Naked triple {789} in R234C9, locked for C9, 7 also locked for N3
2. 22(4) cage at R1C9 = {4567} (only remaining combination) -> R4C8 = 7, R1C9 + R2C8 + R3C7 = {456}, locked for N3 and D/
2a. Naked quad {1234} in R1689C8, locked for C8 and 10(4) cage at R6C8, no 4 in R7C9
2b. 20(4) cage at R1C8 = {2369} (only remaining combination) -> R1C8 + R2C7 = {23}, locked for N3, R3C6 + R4C7 = [96], R3C89 = [87], R24C9 = [98], R4C1 = 9, R1C2 = 9 (hidden single in N1)
2c. 18(3) cage at R5C7 = {459} (only remaining combination), locked for R5 and N6
2d. Naked triple {123} in R6C789, locked for R6
2e. 4 in C8 only in R89C8, locked for N9
2f. R1C7 = 1, 17(4) cage at R1C7 = {1268/1358/1367} (cannot be {1457} because R4C6 only contains 2,3) -> R2C6 = {78}, R3C5 = {56}
2g. Naked pair {78} in R2C16, locked for R2
2h. Naked pair {23} in R26C7, locked for C7
2i. 10(3) cage at R5C1 = {127/136}, 1 locked for R5 and N4
2j. 1 in R4 only in R4C45, CPE no 1 in R3C4
2k. 8 in R5 only in R5C456, locked for N5
2l. R7C7 = 9 (hidden single on D\) -> R5C8 = 9 (hidden single in R5)
2m. R89C7 = {78} (hidden pair in C7)
2n. 22(4) cage at R6C7 = {2578} (only remaining combination) -> R6C7 = 2, R7C8 = 5, R1C8 + R2C7 = [23], R2C8 = 6
2o. 10(4) cage at R6C8 = {1234} -> R7C9 = 2
2p. R6C4 = 9 (hidden single on D/)
2q. R8C3 = 9 (hidden single in N7)
2r. 1 on D/ only in R7C3 + R8C2 + R9C1, locked for N7
2s. 18(4) cage at R6C2 = {1278/1368} (cannot be {1458/1467} because 4,5,6 only R6C2), CPE no 8 in R79C2
2t. Killer pair 2,3 in R4C6 and R7C3 + R8C2 + R9C1, locked for D/
2u. Naked pair {78} in R1C1 + R5C1, locked for D\, CPE no 7,8 in R1C5
2v. 45 rule on R5 3 innies R5C456 = 17 = {278/368}
2w. Killer pair 2,3 in R4C4 and R5C46, locked for N5
2x. 2 on D\ only in R2C2 + R3C3, locked for N1
2y. 28(4) cage at R6C3 = {4789/5689}, CPE no 8 in R7C3
2z. 1 in R3 only in R3C123, locked for N1
2aa. 8 in R7 only in R7C456, locked for N8
2ab. R9C5 = 9 (hidden single in C5)
2ac. Min R2C3 + R3C2 + R4C3 = 7 -> max R1C4 = 7
2ad. 8 in N2 only in R12C6, locked for C6
2ae. 1 in C6 only in R789C6, locked for N8
3. 17(4) cage at R6C5 = {1367/1457/2357/2456}
3a. 21(4) cage at R6C6 = {1479/1569/2469/3459} (cannot be {2379} because R6C4 only contains 4,5,6)
3b. Hidden killer pair 1,2 in 17(4) cage and 21(4) cage for N8, 17(4) cage contains one of 1,2 -> 21(4) cage must contain one of 1,2 = {1479/1569/2469}, no 3
4. 18(4) cage at R6C2 (step 2s) = {1278} (cannot be {1368} because R4C6 + R5C5 (hidden pair on D/) clashes with R5C456 (step 2v), no 3,6, 2 locked for N7 and D/, CPE no 7 in R7C2
4a. R4C6 = 3
4b. R5C456 (step 2v) = {278}, locked for R5 and N5
4c. R6C23 = {78} (hidden pair in N6), CPE no 7 in R7C3
4d. R7C3 = 1 -> R9C1 = 2
4e. Naked pair {78} in R68C2, locked for C2
4f. Naked pair {78} in R8C27, locked for R8
4g. 6 in N5 only in R6C56, locked for R6, CPE no 6 in R7C6
4h. 17(4) cage at R6C5 (step 3) = {2357/2456} -> R8C5 = 2
5. 45 rule on R6789 using R689 = {136} = 10, R9C5 = 9 -> remaining cells R6C1 + R7C12 + R8C1 = 18 = 4{36}5/5{346}, 3,6 locked for N7
5a. R8C2 + R9C3 = {78} (hidden pair in N7)
5b. Naked pair {78} in R9C37, locked for R9
5c. 28(4) cage at R6C3 = {4789/4589}, R9C2 = {45} -> no 4 in R7C4
6. 21(4) cage at R6C6 (step 3a) = {1569} (only remaining combination, cannot be {1479} which clashes with R7C6), 1,5,6 locked for C6
6a. 17(4) cage at R6C5 (step 4h) = {2357/2456}
6b. 4 of {2456} must be in R7C6 -> no 4 in R6C5 + R9C4
6c. Naked pair {56} in R6C56, locked for R6 and N5
6d. Naked pair {14} in R4C45, locked for R4, CPE no 4 in R3C4
6e. Naked pair {25} in R4C23, CPE no 5 in R3C2
6f. R6C1 = 4 -> R7C12 = {36}, R8C1 = 5 (step 5), 3,6 locked for R7
6g. Naked triple {478} in R7C456, 4 locked for N8
6h. R9C2 = 4 (hidden single in N7)
6i. R8C8 = 4 (hidden single in N9), placed for D\
6j. R4C4 = 1, placed for D\, R4C5 = 4
6k. R7C6 = 4 (hidden single in R6) -> 17(4) cage = {2456}
6l. R6C4 = 9, R7C5 + R9C3 = {78} -> R8C4 = 3 (cage total)
6m. Naked pair {78} in R12C6, locked for N2 and C6
6n. Naked pair {56} in R36C5, locked for C5 -> R12C5 = [31]
6o. R9C9 = 3 (hidden single on D\) -> R69C8 = [31], R68C9 = [16]
6p. R1C5 + R4C4 = [31] = 4 -> R2C4 + R3C3 = 10 = [46], 6 placed for D\
6q. R1C4 + R2C3 + R4C3 = [652] = 13 -> R3C1 = 1 (cage sum)
6r. R24C5 = [14] = 5 -> R1C6 + R3C4 = 9 = [72]
6s. R1C1 = 8, placed for D\
6t. R5C5 = 7, placed for D/
and the rest is naked singles, without using the diagonals.