SudokuSolver Forum

This is an optimised solution so some possible eliminations are not included. However, I try and do all the clean-up at the time. Please let me know of any corrections or clarifications needed.

Prelims

i. 12(2)n1: no 1,2,6
ii. 40(8)r1c3: no 5
ii. 8(2) cages at r1c6, r1c9, r2c6 (no 4,8,9)
iii. 22(3)r3c2: no 1,2,3,4
iv. 13(2)r3c6: no 1,2,3
v. 14(4)r3c8: no 9
vi. 10(2)n4: no 5
vii. 11(2)n5: no 1
viii. 42(7)r4c9: no 1,2
ix. 14(2)r6c3 = {59/68}
x. 5(2)n5 = {14/23}
xi. 5(2)n7 = {14/23}
xii. 8(2)n9: no 4,8,9
xiii. 11(2)r9c6: no 1

1. 9 in n3 only in the two 13(2) cages at r1c8 & r3c6 -> no 4 in r3c89 (Locking cages)
1a. and no 4 in r3c7 since it would block all 9s in n3 (Locking-out cages)
1b. no 9 in r3c6

2. 4 in n3 only in 13(2)r1c8 = {49} only: both locked for c8 & n3
2a. no 4 in r3c6

3. "45" on n5: 2 outies r3c4 + r7c6 = 5 = {14/23}

4. 13(3) cage at r2c1: the only combo with 9 is {139} but this is blocked by 5(2)n7 = [1/3..]
4a. no 9 in 13(3)

5. "45" on r12: 1 outie r3c5 + 1 = 1 innie r2c1
5a. no 8,9 in r3c5
5b. no 1,6 in r2c1
5c. 9 in r3 only in n1: 9 locked for n1 and 22(3)r3c2
5d. no 9 in r4c2
5e. no 3 in 12(2)n1

Now a fun recursive move: [Andrew points out this move can also be seen as a series of CPE moves - with the nice extra twist of the IOD to make it more fun!]
6. r3c5 sees all 1 in n1 through the 40(8)r1c3 -> no 1 in r3c5
6a. -> no 2 in r2c1 (IOD r12 = -1)
6b. -> r3c5 sees all 2 in n1
6c. etc...
6d. left with r2c1 = (58)
6e. r3c5 = (47)

7. "45" on c1: 3 outies r156c2 = 9 (no 7,8,9)
7a. can't be {126} because none in r1c2
7b. = {135/234}(no 6)
7c. must have 3: 3 locked for c2, n4 and 24(5)r5c1 -> no 3 in r7c1
7d. can't have both 4&5 -> no 4,5 in r56c2
7e. r1c1 = (78)
7f. no 7 in 10(2)n4

8. 12(2)n1 = [75/84] = [5/8..]
8a. Killer pair 5,8 with r2c1: both locked for n1
8b. 8 locked for c1

9. r3c23 must have 6 or 7
9a. {67} blocked from 13(2)r3c6
9b. 13(2) = {58} only: both locked for r3
9c. r3c7 = 8 (hsingle in n3)
9d. r3c6 = 5
9e. no 3 in r12c7
9f. no 6 in r9c7
9g. no 3 in r9c6

10. 22(3)r3c2 = {679} only since no 5/8 in r3c23

11. 8 in n4 only in c3: locked for c3
11a. no 6 in r6c3
11b. 8 is in the 10(2) or 14(2)r6c3 -> {46} blocked from 10(2) since it would leave no 8 for c3 (Locking-out cages)

12. 10(2)n4 = {19/28} = [1/2..]
12a. r56c2 = [3]{1/2}
12b. Killer pair 1,2: both locked for n4

13. 13(3)r2c1 needs 5/8 for r2c1 = {148/157/256}(no 3)({238} blocked by 2,3 only in r3c1)
13a. needs 1/2 -> r3c1 = (12)
13b. needs 4/6/7 -> r4c1 = (467)

14. 3 in c1 only in 5(2)n7 = {23}: both locked for c1 and n7

15. 2 & 3 in 45(9)r7c3 only in n8: locked for n8
15a. no 9 in r9c7
15b. no 2,3 in r3c4 (outies n5 = 5)

16. r3c1 = 1 -> r3c4 = 4 -> r7c6 = 1
16a. r3c5 = 7 -> r2c1 = 8 (IODr12 = -1)

on from there.

I'll rate my walkthrough for A226 at 1.5 because of step 18. Ed's solving path will have a slightly lower rating, determined by his steps 1 and 1a.

Prelims

a) R1C12 = {39/48/57}, no 1,2,6
b) R1C67 = {17/26/35}, no 4,8,9
c) R12C8 = {49/58/67}, no 1,2,3
d) R12C9 = {17/26/35}, no 4,8,9
e) R2C67 = {17/26/35}, no 4,8,9
f) R3C67 = {49/58/67}, no 1,2,3
g) R45C3 = {19/28/37/46}, no 5
h) R4C56 = {29/38/47/56}, no 1
i) R67C3 = {59/68}
j) R6C45 = {14/23}
k) R89C1 = {14/23}
l) R89C9 = {17/26/35}, no 4,8,9
m) R9C67 = {29/38/47/56}, no 1
n) 22(3) cage at R3C2 = {589/679}
o) 14(4) cage at R3C8 = {1238/1247/1256/1346/2345}, no 9
p) 42(7) cage at R4C9 = {3456789}, no 1,2
q) 40(8) cage at R1C3 = {12346789}, no 5
r) And, of course, both 45(9) cages = {123456789}

Steps resulting from Prelims
1a. 22(3) cage at R3C2 = {589/679}, CPE no 9 in R12C2, clean-up: no 3 in R1C1
1b. 9 in C9 only in R4567C9, locked for 42(7) cage at R4C9, no 9 in R567C8

2. 45 rule on C1 3 outies R156C2 = 9 = {135/234} (cannot be {126} because no 1,2,6 in R1C2), no 6,7,8,9, 3 locked for C2, clean-up: no 4,5 in R1C1

3. 13(3) cage at R2C1 and 5(2) cage at R8C1 form combined 18(5) cage which must contain 1,2, both locked for C1

4. Hidden killer pair 1,2 in 13(3) cage at R2C1 and 5(2) cage at R8C1 for C1, 5(2) cage must contain one of 1,2 -> 13(3) cage must contain one of 1,2 = {148/157/238/256} (cannot be {139/247} which clash with 5(2) cage), no 9

5. 45 rule on R12 1 innie R2C1 = 1 outie R3C5 + 1, no 1,6 in R2C1, no 8,9 in R3C5

6. 45 rule on N5 2 outies R3C4 + R7C6 = 5 = {14/23}

7. 9 in N3 only in R12C8 = {49} or R3C7 => R3C67 = [49] -> no 4 in R3C89 (locking cages)
7a. 4 in C9 only in R4567C9, locked for 42(7) cage at R4C9, no 4 in R567C8

8. 34(7) cage at R3C4 contains all except the pair in R4C56 -> 34(7) cage and R4C56 form a hidden 45(9) cage
8a. R3C4 “sees” all cells of the hidden 45(9) cage except for R6C5 -> R3C4 = R6C5 -> R36C4 form a hidden 5(2) cage
8b. Similarly R7C6 “sees” all cells of the hidden 45(9) cage except for R6C45 but from step 8a it cannot equal R6C5 -> R7C6 = R6C4 -> R6C5 + R7C6 form a hidden 5(2) cage
8c. These two hidden 5(2) cages must contain the same pair
[At this stage I don’t know how to use these “clones” and hidden cages, but they might be useful later.]

9. Just spotted a bit more for step 7, R12C8 = {49} or R3C7 => R3C67 = [49] -> no 4 in R3C7 (locking-out cages), clean-up: no 9 in R3C6
9a. 4 in N3 only in R12C8 = {49}, locked for C8 and N3, clean-up: no 4 in R3C6
9b. 8 in N3 only in R3C789, locked for R3, CPE no 8 in R4C7, clean-up: no 5 in R3C7
9c. 14(4) cage at R3C8 = {1238/1247/1256/1346/2345}
9d. 4 of {1247} must be in R4C7 -> no 7 in R4C7

10. 9 in R3 only in R3C23, locked for N1 and 22(3) cage at R2C2, no 9 in R4C2, clean-up: no 3 in R1C2
10a. 22(3) cage at R3C2 = {589/679}
10b. 8 of {589} must be in R4C2 -> no 5 in R4C2

11. R156C2 (step 2) = {135/234}
11a. R1C2 = {45} -> no 4,5 in R56C2
11b. R156C2 = {135/234}, 3 locked for N4 and 24(5) cage at R5C1, no 3 in R7C1, clean-up: no 7 in R45C3

12. 1,2,3 in C1 only in combined 18(5) cage (step 3) = {12348/12357}, no 6 -> 13(3) cage at R2C1 (step 4) = {148/157/238}
12a. Killer pair 7,8 in R1C1 and 13(3) cage, locked for C1

13. R45C3 = {19/28/46}, R67C3 = {59/68} -> combined cage R4567C3 = {19}{68}/{28}{59}/{46}{59}, 9 locked for C3
13a. R3C2 = 9 (hidden single in N1)
13b. 9 in N7 only in R7C13, locked for R7
13c. 9 in C9 only in R456C9, locked for N6

14. 9 in 45(9) cage at R5C7 only in R8C567, locked for R8
14a. 9 in 45(9) cage at R7C2 only in R9C45, locked for R9 and N8, clean-up no 2 in R9C67
14b. R8C7 = 9 (hidden single in R8)

15. 5 in N2 only in R123C6, locked for C6, clean-up: no 6 in R4C5, no 6 in R9C7

16. 9 in C6 only in R456C6, locked for N5, clean-up: no 2 in R4C6

17. 7 in N4 only in R4C12, locked for R4, clean-up: no 4 in R4C56

[Possibly a bit OTT but I now spotted …]
18. Consider the combinations for 22(3) cage at R3C2 = {589/679}
18a. 22(3) cage = {589} => R3C3 = 5 => R67C3 = {68}, locked for C3
or 22(3) cage = {679} => R3C3 + R4C2 = {67}, CPE no 6 in R456C3
-> no 6 in R45C3 -> R45C3 = {19/28}, no 4,6
18b. Killer pair 1,2 in R45C3 + R56C2, locked for N4
18c. Killer pair 8,9 in R45C3 and R67C3, locked for C3
[At this stage I saw a contradiction move to eliminate the {589} combination from the 22(3) cage but decided not to use it.]

19. R2C1 (step 12) = {148/157/238}
19a. 1 of {148/157} must be in R3C1, 8 of {238} must be in R4C1 -> no 4,5,7 in R3C1

20. Variable combined cage R3C3 + R3C67 = 5{67}/6[58]/7[58], 5 locked for R3

21. 4 in R3 only in R3C45, locked for N2
21a. R3C4 = R6C5 (step 8a) -> 4 must be in R36C5, locked for C5

22. 8,9 in N2 only in R12C45, locked for 40(8) cage at R1C3, no 8 in R2C2
22a. 8 in N1 only in R12C1, locked for C1

23. R2C1 (step 12) = {148/157} (cannot be {238} because R4C1 only contains 4,5,7) -> R3C1 = 1, clean-up: no 4 in R7C6 (step 6), no 4 in R89C1
[The puzzle is now cracked.]
23a. Naked pair {23} in R89C1, locked for C1 and N1
23b. 2,3 in N1 only in R1C3 + R2C23, locked for 40(8) cage at R1C3, no 2,3 in R12C45 + R3C5

24. 2,3 in 45(9) cage at R7C2 only in R789C4 + R9C5, locked for N8 -> R7C6 = 1, R3C4 = 4 (step 6), clean-up: no 7 in R12C7, no 1 in R6C5, no 8 in R9C7

25. R6C45 = [14] (hidden pair in N5)

26. R3C67 = [58] (cannot be {67} which clashes with R3C5), clean-up: no 8 in R4C2, no 3 in R12C7
26a. Naked pair {67} in R3C3 + R4C2, no 6,7 in R2C2 + R6C3, clean-up: no 8 in R7C3

27. R3C89 = {23} (hidden pair in R3), locked for N3 and 14(4) cage at R3C8, no 2,3 in R4C78, clean-up: no 6 in R12C6, no 5,6 in R12C9
27a. 14(4) cage at R3C8 = {1238/2345} -> R4C78 = [18/45]

28. Naked pair {17} in R12C9, locked for C9 and N3, clean-up: no 7 in R12C6
28a. Naked pair {23} in R12C6, locked for C6, clean-up: no 8 in R4C5
28b. 7 in 42(7) cage at R4C9 only in R567C8, locked for C8
28c. Naked pair {56} in R12C7, locked for C7, clean-up: no 6 in R9C6

29. Killer pair 2,3 in R3C9 and R89C9, locked for C9
29a. 3 in 42(7) cage at R4C9 only in R567C8, locked for C8 -> R3C89 = [23], clean-up: no 5 in R89C9
29b. Naked pair {26} in R89C9, locked for C9 and N9

30. 1 in N9 only in R89C8, locked for 45(9) cage at R5C7, no 1 in R5C7
30a. R4C7 = 1 (hidden single in N6), R4C8 = 8 (step 27a), clean-up: no 3 in R4C5, no 2,9 in R5C3
30b. Naked pair {15} in R89C8, locked for C8, N9 and 45(9) cage at R5C7, no 5 in R78C5

31. R4C56 = [56] (cannot be [29] which clashes with R4C3), R4C2 = 7, R4C1 = 4, R2C1 = 8 (step 23), R1C1 = 7, R1C2 = 5, R1C7 = 6, R1C6 = 2

and the rest is naked singles.

[Some of the later steps could have been a bit quicker if I’d remembered to make clean-ups using step 5.]