I’m not experienced at solving jigsaw killers, only having done one before; a simpler one which didn’t require a walkthrough. While starting to compile the Texas Jigsaw Killer archive I noticed that some solvers draw diagrams to label the Jigsaw Houses. In this walkthrough I’ve described them by their left-hand cell in their upper row, for example the first house is HR1C1, the next three HR1C4, HR1C7 and HR2C5. I hope this is clear enough; it saves cross-referencing between two diagrams.
Prelims
a) R34C4 = {16/25/34}, no 7,8,9
b) R34C9 = {16/25/34}, no 7,8,9
c) R67C1 = {16/25/34}, no 7,8,9
d) R67C6 = {18/27/36/45}, no 9
e) 21(3) cage at R6C4 = {489/579/678}, no 1,2,3
f) 29(4) cage at R4C1 = {5789}
g) 14(4) cage at R7C3 = {1238/1247/1256/1346/2345}, no 9
1. 31(5) cage at R1C5 contains 9, locked for R1
2. 45 rule on R12 3 outies R3C678 = 21 = {489/579/678}, no 1,2,3
2a. 45 rule on R12 1 outie R3C8 = 2 innies R2C67 + 4
2b. Min R2C67 = 3 -> min R3C8 = 7
2c. Max R2C67 = 5, no 5,6,7,8,9 in R2C67
3. 45 rule on R89 3 outies R7C234 = 8 = {125/134}, 1 locked for R7, clean-up: no 6 in R6C1, no 8 in R6C6
3a. R7C234 = {125/134}, CPE no 1 in R8C4 using HR6C2
4. 45 rule on R123 3 innies R3C459 = 6 = {123}, locked for R3, clean-up: no 1,2,3 in R4C4, no 1,2,3 in R4C9
4a. Max R3C5 = 3 -> min R4C56 = 11, no 1 in R4C56
5. 45 rule on R1234 1 outie R5C3 = 2 innies R4C78 + 6 -> R5C3 = 9, R4C78 = 3 = {12}, locked for R4, HR1C7 and 15(4) cage at R4C8, no 1,2 in R45C89
5a. R4C78 = 3 -> R5C89 = 12 = {48/57}, no 3,6
6. Naked triple {578} in R4C123, locked for R4, clean-up: no 2 in R3C4, no 2 in R3C9
6a. Naked triple {578} in R4C123, CPE no 5,7,8 in R12C1 using HR1C1
7. R3C5 = 2 (hidden single in R3), R4C56 = {39} (hidden pair in R4)
7a. 13(3) cage at R5C4 = {148/157/346}
7b. Killer pair 4,5 in 13(3) cage and R5C89, locked for R5
7c. Killer pair 4,7 in 13(3) cage and R5C89, locked for R5
7d. Killer pair 1,3 in R3C4 and 13(3) cage, locked for HR2C5 -> R4C56 = [93], clean-up: no 6 in R67C6
8. 9 in R7 only in 18(3) cage at R7C7 = {279/369} (cannot be {459} which clashes with R7C234), no 4,5,8
8a. Killer pair 2,3 in R7C234 and 18(3) cage, locked for R7, clean-up: no 4,5 in R6C1, no 7 in R6C6
8b. 8 in R7 only in R7C56, locked for HR6C4, no 8 in R6C4 + R8C5 + R9C456
9. 45 rule on R789 3 innies R7C156 = 19 = {478/568}
9a. 4 of {478} must be in R7C1 -> no 4 in R7C56, clean-up: no 5 in R6C6
10. 21(3) cage at R6C4 = {489/579/678}
10a. 9 of {489/579} must be in R6C4 -> no 4,5 in R6C4
11. 18(3) cage at R2C8 = {459/468/567}
11a. R3C8 = {789} -> no 7,8,9 in R2C89
12. 45 rule on R12 4 innies R2C6789 = 14 = {1256/1346/2345}
12a. 1,2,3 of {1346/2345} must be in R2C67 -> no 4 in R2C67
13. 45 rule on HR1C7 2 outies R1C56 = 10 = [19/37/46/64/82], no 5,7 in R1C5, no 1,5,8 in R1C6
14. 45 rule on R1 2 outies R2C45 = 1 innie R1C1 + 13
14a. Max R2C45 = 17 -> max R1C1 = 4
14b. Min R2C45 = 14, no 1,2,3,4, no 5 in R2C4
15. Law of Leftovers (LoL) using HR1C7 + HR3C9 + HR6C7, R23C7 and R48C6 must contain the same numbers, R4C6 = 3 -> R2C7 = 3, R3C7 = {456789} -> R8C6 = {456789}, no 1,2, clean-up: no 7 in R1C6 (step 13)
16. LoL using HR1C1 + HR3C1 + HR6C2, R23C3 and R48C4 must contain the same numbers, no 3 in R23C3 -> no 3 in R8C4
16a. Same LoL, R4C4 = {46} -> one of R23C3 must be {46}
16b. R1C56 (step 13) = [19/82] (cannot be {46} which clashes with one of R23C3), no 4,6
16c. Killer pair 1,2 in R1C56 and R2C6, locked for HR1C4
16d. Same LoL, no 2 in R23C3 -> no 2 in R8C4
16e. 2 in C4 only in R79C4, locked for HR6C4, no 2 in R69C6, clean-up: no 7 in R7C6
17. R1C56 (step 16b) = [19/82] -> 31(5) cage at R1C5 = {16789/25789} (other combinations don’t contain 1 or 2), no 4, 7,8 locked for R1, 7 also locked for HR1C7, no 7 in R3C8
17a. 4 in HR1C7 only in R2C89, locked for R2
18. 21(3) cage at R6C4 = {579/678} (cannot be {489} which clashes with R67C6), no 4
19. R2C45 = R1C1 + 13 (step 14)
19a. Max R2C45 = 16 (cannot be [98] which clashes with R1C56) -> max R1C1 = 3
19b. R1C1 = {123} -> R2C45 = 14,15,16 = {59/68/69/78/79}
20. 4 in R1 only in 27(5) cage at R1C2 = {14679/34569/34578} (cannot be {14589/24678} which clash with 31(5) cage at R1C5, cannot be {24579} which clashes with R1C6), no 2
20a. 9 of {14679/34569} must be in R2C4 -> no 6 in R2C4
21. Hidden killer pair 8,9 in R2C123 and R2C45 for R2, R2C45 contains one of 8,9 -> R2C123 must contain one of 8,9
21a. 2 in HR1C1 only in 18(4) cage at R1C1 = {1269/1278/2358} (cannot be {2367} which doesn’t contain 8 or 9)
21b. 6 of {1269} must be in R2C3 -> no 6 in R2C12
22. 2 in C3 only in R6789C3, locked for HR6C2, no 2 in R67C2 + R9C12
23. 2 in C9 only in R6789C9, CPE no 2 in R8C8
24. 3,9 in HR3C9 only in R367C9 + R8C89, CPE no 3,9 in R9C9
25. 3 in HR6C7 only in R679C8, locked for C8
26. LoL for HR1C1 + HR1C4 + HR1C7 + HR2C5 R34C19 must contain the same numbers as 13(3) cage at R5C4 + R6C5
26a. R34C9 = [16/34], 13(3) cage at R5C4 (step 7a) = {148/157/346}
26b. 13(3) cage = {148/157} => R34C9 = [16] (LoL) => R6C5 = 6 (LoL) or 13(3) cage = {346}
-> 6 in 13(3) cage + R6C5, locked for HR2C5, no 6 in R2C5 + R3C6
26c. When 13(3) cage = {148} => R34C9 = [16], R34C1 = [48] -> no 8 in R3C1
27. R1C1 = {123} -> R2C45 = 14,15,16 = {59/78/79} (step 19b) -> R1C1 + R2C45 = 1[95]/2{78}/3[97]
27a. 18(4) cage at R1C1 (step 22a) = {1269/1278/2358} = 2{19}6/[12]{78}/[32]{58} (cannot be 1{29}6 which clashes with R1C1 + R2C45 = 1[95], cannot be [21]{78} which clashes with R1C1 + R2C45 = 2{78}) -> no 2 in R2C2
27b. 2 in 18(4) cage only in R12C1, locked for C1, clean-up: no 5 in R7C1
28. R7C156 (step 9) = {478/568}
28a. R7C1 = {46} -> no 6 in R7C5
29. Consider combinations for 21(3) cage at R6C4 (step 18) = {579/678}
21(3) cage = {579} => R6C4 = 9 => no 9 in R9C6
or 21(3) cage = {678}, 8 locked for C5 => R1C5 = 1, R1C6 = 9 (step 16b) => no 9 in R9C6
-> no 9 in R9C6
29a. 9 in HR6C4 only in R69C4, locked for C4
30. 9 in R2 only in R2C12 -> 18(4) cage at R1C1 (step 21a) = {1269} (only remaining combination) -> R2C3 = 6, R2C89 = {45}, locked for R2 and HR1C7, no 5 in R1C789, R3C8 = 9 (cage sum), R2C45 = {78}, locked for R2
30a. R2C45 = {78} -> R1C1 = 2 (step 27), R1C6 = 9, R1C5 = 1 (step 16b), R2C6 = 2
30b. Naked triple {678} in R1C789, locked for R1
31. 21(3) cage at R6C4 (step 18) = {579/678} = 9{57}/7[68] (cannot be 6{78} which clashes with R2C5), no 6 in R6C4, no 8 in R6C5
31a. Killer pair 7,8 in R2C5 and R67C5, locked for C5
32. 13(3) cage at R5C4 (step 7a) = {148/157/346}
32a. 5 in {157} must be in R5C5 -> no 5 in R5C46
33. 6 in R3 only in R3C12, CPE no 6 in R5C2 and R8C2 using HR3C1
34. 45 rule on R6789 3 outies R5C127 = 11 = {128/236}
34a. {236} must be [326] (only remaining permutation, cannot be [632] which clashes with R78C1) -> no 6 in R5C1, no 3 in R5C2
35. R5C127 (step 34) = {128/236}
35a. Consider permutations for R34C9 = [16/34]
R34C9 = [16] => no 6 in R5C7 => R5C127 = {128}
or R34C9 = [34] => 13(3) cage at R5C4 = {346} (LoL, step 26b), locked for R5 => R5C127 = {128}
-> R5C127 = {128}, locked for R5
35b. 13(3) cage at R5C4 (step 7a) = {346} (only remaining combination), locked for R5 and HR2C5 -> R3C4 = 1, R4C4 = 6, R34C9 = [34]
35c. Naked pair {57} in R5C89, locked for HR3C9, no 5,7 in R67C9 + R8C89
36. 21(3) cage at R6C4 (step 18) = {579} (only remaining combination) -> R6C4 = 9, R67C5 = {57}, locked for C5 -> R2C45 = [78]
37. R2C67 = [23] = 5 -> R3C67 = 12 = [75], R6C5 = 5, R7C5 = 7, R1C4 = 4, R3C3 = 8, R5C4 = 3
38. 18(3) cage at R7C7 (step 8) = {369} (only remaining combination), locked for R7 -> R7C1 = 4, R6C1 = 3, R3C1 = 6, R3C2 = 4
38a. R7C6 = 8 (hidden single in R7), R6C6 = 1
39. R8C4 = 8 (hidden single in C4)
39a. 14(4) cage at R7C3 = {1238} (only remaining combination) -> R7C34 = [12], R8C3 = 3, R7C2 = 5, R9C4 = 5
40. R7C2 = 5 -> R8C12 = 9 = [72]
41. R1C23 = [35], R4C3 = 7, R4C2 = 8, R4C1 = 5
42. R5C12 = [81] = 9 -> R6C23 = 8 = [62]
and the rest is naked singles.
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