1. D, E and G are adjacent. D:E = 1:2, D:G = 3:4, E:G = 2:3 -> D:G:E = 4:3:2, D includes both 2 and 3-cell cages, E,G only 3-cell cages -> D = 12,16, G = 9,12, E = 6,8
1a. 45 rule on N3 4(3+1) comprise 3 cells of D cages and one of a G cage. 3*D + E + G = 36+6+9 = 51 isn’t enough to give 4(3+1) outies -> D = 16, G = 12, E = 8
1b. D:N = 1:2 -> N = 8 (only 2-cell cage)
2. C:G = 2:3, G = 12 -> C = 8 or 18, B:C = 2:3 -> B = 12 (cannot be 24 because only 2-cell cage)
2a. B:F = 4:5 -> F = 15
2b. A:F = 2:5 -> A = 6
2c. A:H = 2:5 -> H = 15
2d. G:J = 3:4 -> J = 16 (cannot be 9 because one J is in a 4-cell cage)
2e. G:K = 4:5 -> K = 15
2f. H:L = 4:5 -> L = 12
2g. L:Q = 2:3, J:Q = 1:2, L = 12, J = 16 -> Q = 8
3. M includes 4 and 5-cell cages so min 15, max 30, M:R = 1:3 with R a 2-cell cage -> M = 3*R -> M = 15,18,21,24,27,30, R = 5,6,7,8,9,10
3a. M:S = 1:3 with S a 3-cell cage -> M = 3*S with min S = 6, min M = 18, S = R
3b. M:U = 2:3 with U a 2-cell cage -> M = 3*U/2 with max U = 16 -> max M = 24
3c. R:T = 1:3 with T a 3-cell cage -> T = 3*R -> T = M
3d. P:S = 1:2, M:P = 2:3, M:S = 1:3 -> P = 2*S
3e. M,T = 18,21,24, R,S = 6,7,8, P,U = 12,14,16
4. 45 rule on the whole grid should now determine which values of C, M, P, R, S, T and U are correct.
Total of cages already fixed A + B + 4*D + E + F + 2*G + H + 2*J + K + 3*L + N + Q = 247
Cages not yet fixed = C + 3*M + 2*P + R + S + T + U = C + (3*3 + 2*2 + 1 + 1 + 3 + 2)*R (or S) = C + 20*R (or S)
Total of cages not yet fixed = 405-247 = 158 -> C = 18, R = S = 7, P = U = 14, M = T = 21
Now to solve the Killer
Prelims
a) R1C34 = {39/48/57}, no 1,2,6
b) R1C67 = {79}
c) R2C23 = {69/78}
d) R34C1 = {69/78}
e) R34C9 = {79}
f) R5C12 = {39/48/57}, no 1,2,6
g) R5C89 = {17/26/35}, no 4,8,9
h) R67C1 = {39/48/57}, no 1,2,6
i) R67C9 = {59/68}
j) R78C2 = {79}
k) R8C45 = {16/25/34}, no 7,8,9
l) R9C67 = {59/68}
m) 6(3) cage at R1C1 = {123}
n) 8(3) cage at R1C8 = {125/134}
o) 8(3) cage at R8C1 = {125/134}
p) 7(3) cage at R8C9 = {124}
q) 21(3) cage at R9C3 = {489/579/678}, no 1,2,3
r) 14(4) cage at R6C6 = {1238/1247/1256/1346/2345}, no 9
Steps resulting from Prelims
5a. Naked pair {79} in R1C67, locked for R1, clean-up: no 3,5 in R1C34
5b. Naked pair {48} in R1C34, locked for R1
5c. Naked pair {79} in R34C9, locked for C9, clean-up: no 1 in R5C8, no 5 in R67C9
5d. Naked pair {68} in R67C9, locked for C9, clean-up: no 2 in R5C8
5e. Naked pair {79} in R78C2, locked for C2 and N7, clean-up: no 6,8 in R2C3, no 3,5 in R5C1, no 3,5 in R6C1
5f. Naked pair {79} in R1C7 + R3C9, locked for N3
5g. Naked triple {123} in 6(3) cage at R1C1, locked for N1
5h. Naked triple {124} in 7(3) cage at R8C9, locked for N9
5i. 8(3) cage at R1C8 = {125/134}, 1 locked for N3
5j. 8(3) cage at R8C1 = {125/134}, 1 locked for N7
5k. 1 in R3 only in R3C456, locked for N2
5l. 1 in R7 only in R7C456, locked for N8, clean-up: no 6 in R8C45
5m. 1 in C3 only in R456C3, locked for N4
5n. 1 in C7 only in R456C7, locked for N6, clean-up: no 7 in R5C8
5o. 6 in N7 only in R789C3, locked for C3
[That must be the most Steps resulting from Prelims that I’ve ever done.]
6. R1C5 = 6 (hidden single in R1)
6a. R1C5 = 6 -> R2C45 = 12 = {39/57} (cannot be {48} which clashes with R1C4)
6b. Killer pair 7,9 in R1C6 and R2C45, locked for N2
7. 21(3) cage at R9C3 = {579/678} (cannot be {489} which clashes with R9C67), no 4, 7 locked for N8
8. 45 rule on N3 4(3+1) outies R123C6 + R4C9 = 23, R1C6 + R4C9 = {79} = 16 -> R23C6 = 7 = {25/34}
8a. Killer pair 3,5 in R2C46 and R23C6, locked for N2
9. 45 rule on R1234 2 outies R5C67 = 13 = {49/58/67}, no 1,2,3
10. 45 rule on R1 (using R1C5 = 6), 2 outies R2C19 = 3 = {12}, locked for R2, clean-up: no 5 in R3C6 (step 8)
11. 12(3) cage at R3C7 = {246/345}, no 8, CPE no 4 in R2C6, clean-up: no 3 in R3C6 (step 8)
11a. 12(3) cage = {246} (cannot be {345} which clashes with R2C6) -> R3C6 = 2, R2C6 = 5 (step 8), R2C78 = {46}, locked for R2 and N3, R2C2 = 8, placed for D\, R2C3 = 7, R1C34 = [48], clean-up: no 7,8 in R4C1, no 4 in R5C1, no 9 in R9C7
11b. Naked pair {38} in R3C78, locked for N3
11c. Naked pair {69} in R34C1, locked for C1, clean-up: no 3 in R5C2, no 3 in R7C1
12. 5 in R3 only in 16(4) cage at R3C2 = {1258/1357/1456/2356}, no 9
12a. R3C3 = 5, placed for D\, R3C2 = 6
12b. R3C23 = [65] = 11 -> R4C23 = {14/23}
13. R34C1 = [96], R34C9 = [79], R1C67 = [79]
14. 45 rule on R9 2 outies R8C19 = 5 = [14/32/41], no 2,5 in R8C1
14a. 45 rule on R9 4 innies R9C1289 = 10 = {1234}, no 5
15. 8(3) cage at R8C1 = {134} (only remaining combination), locked for N7, clean-up: no 8 in R6C1
15a. 2 in R9 only in R9C89, locked for N9
15b. 3 in R9 only in R9C12, locked for N7
15c. Naked pair {14} in R8C19, locked for R8, clean-up: no 3 in R8C45
15d. Naked pair {25} in R8C45, locked for R8 and N8
16. R5C9 = 3 (hidden single in C9), R5C8 = 5, R5C2 = 4, R5C1 = 8, R67C1 = [75], clean-up: no 1 in R4C3 (step 12b)
16a. Naked pair {23} in R4C23, locked for R4 and N4 -> R6C2 = 5
16b. R6C2 = 5 -> R57C3 = 11 = [92], 2 placed for D/, R4C23 = [23], R6C3 = 1, R5C6 = 6, R5C7 = 7, R5C5 = 1, placed for both diagonals, R5C4 = 2, R8C45 = [52], R1C9 = 5, R3C45 = [14], R4C4 = 7, placed for D\, clean-up: no 8 in R9C7
17. R5C6 = 6 -> R4C56 = 9 = [54], 4 placed for D/, R4C78 = [18], R3C78 = [83], R6C3 = 3, placed for D\, R1C1 = 2, placed for D\
and the rest is naked singles, without using the diagonals.
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