Ed wrote:
Congratulations
manu on finding a brilliant solution to this puzzle!! Love it.
I agree completely! It was brilliant to combine two different LoLs in that way! I'd got each of them, and the hidden cage, but hadn't thought to put them together.
Congratulations to
manu for finding a short solving path.
I had to work very hard to solve it. TJK 37 and TJK 38 both have the same score 1.60 (using SS v3.6.1). However this was a lot harder than TJK 37.
Thanks Ed for the pics, explaining
manu's breakthrough and your breakthrough, which I did find.
My walkthrough is fairly long, but I found some interesting steps.
Prelims
a) R12C1 = {17/26/35}, no 4,8,9
b) R23C9 = {18/27/36/45}, no 9
c) R45C1 = {19/28/37/46}, no 5
d) R56C9 = {19/28/37/46}, no 5
e) R89C9 = {19/28/37/46}, no 5
f) R9C12 = {19/28/37/46}, no 5
g) 24(3) cage at R2C2 = {789}
h) 9(3) cage at R2C3 = {126/135/234}, no 7,8,9
i) 20(3) cage at R4C7 = {389/479/569/578}, no 1,2
j) 8(3) cage at R6C6 = {125/134}
k) 21(3) cage at R7C8 = {489/579/678}, no 1,2,3
l) 14(4) cage at R7C5 = {1238/1247/1256/1346/2345}, no 9
1. 8(3) cage at R6C6 = {125/134}, 1 locked for C6
1a. 8(3) cage at R6C6 = {125/134}, CPE no 1 in R6C8
2. Naked triple {789} in 24(3) cage at R2C2, CPE no 7,8,9 in R1C2
3. 45 rule on R1234 2 innies R4C12 = 7 = {16/34}/[25], no 7,8,9, no 2 in R4C2, clean-up: no 1,2,3 in R5C1
4. 16(3) cage at R1C2 must contain one of 7,8,9
4a. Killer triple 7,8,9 in 16(3) cage and 24(3) cage, locked for NR1C1, clean-up: no 1 in R12C1
4b. 16(3) cage = {169/268/349/358/457} (cannot be {178} which clashes with 24(3) cage at R2C2, cannot be {259/367} which clash with R12C1)
5. 45 rule on NR1C6 + NR2C6 2 innies R1C6 + R3C5 = 12 = {39/48/57}, no 1,2,6
5a. 45 rule on NR1C6 + NR2C6 2 outies R12C5 = 11 = {29/38/47/56}, no 1
6. 45 rule on C6789 1 outie R4C5 = 1 innie R1C6 -> R4C5 = {345789}
6a. 1 in NR2C6 only in R3C78, locked for R3, clean-up: no 8 in R2C9
[Note. R1C6 + R3C5 = 12 (step 5), R4C5 = R1C6 (step 6) -> R34C5 = 12 may come in useful later.]
7. 45 rule on NR6C1 + NR6C2 1 outie R5C5 = 2 innies R7C5 + R9C4 + 1
7a. Min R7C5 + R9C4 = 3 -> min R5C5 = 4
7b. Max R7C5 + R9C4 = 8, no 8 in R7C5 + R9C4
8. 45 rule on C6789 4 outies R1234C5 = 23
[Alternatively, 23(4) cage at R1C5 and R1C6 = R4C5, step 6]
8a. 45 rule on NR6C1 + NR6C2 3 outies R589C5 = 15
8b. 45 rule on C5 2 remaining innies R67C5 = 7 = {16/25/34}, no 7,8,9
9. 45 rule on NR1C6 2 innies R1C6 + R4C9 = 1 outie R2C6 + 9, IOU no 9 in R4C9
9a. Max R1C6 + R4C9 = 17 -> max R2C6 = 8
10. 45 rule on NR1C1 1 outie R3C2 = 2 innies R2C3 + R4C1 + 3
10a. Max R2C3 + R4C1 = 6, no 6 in R2C3 + R4C1, clean-up: no 1 in R4C2 (step 2), no 4 in R5C1
10b. 1 in NR1C5 only in R2C4 + R5C2, CPE no 1 in R5C4
11. 45 rule on NR6C9 1 outie R7C8 = 2 innies R6C9 + R8C7 + 3
11a. Min R6C9 + R8C7 = 3 -> min R7C8 = 6
11b. Max R6C9 + R8C7 = 6, no 6,7,8,9 in R6C9 + R8C7, clean-up: no 1,2,3,4 in R5C9
12. 45 rule on NR6C9 2 outies R5C9 + R7C8 = 1 innie R8C7 + 13
12a. Max R5C9 + R7C8 = 17 -> max R8C7 = 4
12b. Max R8C7 = 4 -> min R67C7 = 14, no 1,2,3,4 in R67C7
13.
17(3) cage at R9C6 must contain one of 1,2,3,413a. Killer quad 1,2,3,4 in R6C9, R8C7, R89C9 and 17(3) cage, locked for NR6C9
13b. 21(3) cage at R7C8 = {579/678}, CPE no 7 in R9C8
14. Law of Leftovers (LoL) for R6789 two outies R5C89 must exactly equal two innies R6C67, R5C9 = R6C7 = {6789} (because no 6,7,8,9 in R6C6), R6C6 = R5C8 = {12345}
15. 45 rule on R6789 2 outies R5C59 = 1 innie R6C8 + 9, IOU no 9 in R5C5
15a. Max R5C59 = 17 -> max R6C8 = 8
15b. 9 in NR5C8 only in R5C9 + R7C78, CPE no 9 in R7C9
16. 9 in C5 only in R1234C5 (step 8) = 23 = {2489/2579/3479} (cannot be {3569} which clashes with R67C5)
16a. R12C5 = 11 (step 5a) -> R12C5 = {29/47}, no 3,5,6,8
17. LoL for C1234 four outies R1267C5 must exactly equal four innies R45C34, no 8 in R1267C5 -> no 8 in R45C34
18. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34, no 8 in R4C34 -> no 8 in R5C12, clean-up: no 2 in R4C1, no 5 in R4C2 (step 3)
19. 45 rule on C1234 2 outies R56C5 = 1 innie R9C4 + 8
19a. Max R56C5 = 14 -> max R9C4 = 6
20. 45 rule on C9 3 innies R147C9 = 16 = {268/358/367/457} (cannot be {169/259} because 1,2,9 only in R1C9, cannot be {178/349} which clash with the pair of 10(2) cages in C9), no 1,9
20a. 9 in C9 only in R56C9 = [91]
or R89C9 = {19}, 1 locked for C9 (locking cages), clean-up: no 8 in R3C9
20b. 1 in C9 only in R689C9, locked for NR6C9
21. 45 rule on NR1C1 4 innies R2C23 + R34C1 = 21 = {1389/1479/1578/2478} (cannot be {2379} which clashes with R12C1, cannot be {3459} because R2C2 + R3C1 only contain 7,8,9)
21a. R2C23 + R34C1 = {1389/1479/2478} (cannot be {1578} because 24(3) cage at R2C2 clashes with R45C1 = [19], killer combo clash in NR1C5), no 5
21b. 3 of {1389} must be in R2C3 (cannot be 1{89}3 because 24(3) cage at R2C2 clashes with R45C1 = [37], killer combo clash in NR1C5), no 3 in R4C1, clean-up: no 4 in R4C2 (step 3), no 7 in R5C1
22. Hidden killer pair 5,6 in R12C1 and 16(3) cage at R1C2 for NR1C1, R12C1 contains one of 5,6 -> 16(3) cage must contain one of 5,6, 16(3) cage (step 4b) = {169/268/358/457} (cannot be {349} which doesn’t contain 5 or 6)
23. R4C1 = {14}, R4C12 = 7 (step 3) -> R4C12 + R5C1 = [169/436], 6 locked for NR1C5
23a. R4C12 + R5C1 = [169/436], CPE no 6 in R5C34
23b. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34, when 6 is in R5C1 there must be 6 in one of R4C34 -> 6 in R4C2 or R4C34, locked for R4
[Taking step 21a a bit further]
24. R2C23 + R34C1 (step 21a) = {1479/2478} (cannot be {1389} because 7 from 24(3) cage at R2C2 and 9 from R45C1 = [19] clash with R12C5 using NR1C5, killer combo clash), no 3, 4,7 locked for NR1C1 and 24(3) cage at R2C2, no 7 in R3C3
24a. 4 in NR1C1 only in R2C3 + R4C1, CPE no 4 in R4C3
[And further still]
25. R2C23 + R34C1 (step 24) = {1479} (cannot be {2478} because R4C12 (step 3) = [43] blocks the only combination for 9(3) cage at R2C3 containing 2 in R2C3) -> R2C3 + R4C1 = {14}, locked for NR1C1, R2C2 + R3C1 = {79}, locked for NR1C1 and 24(3) cage at R2C2 -> R3C3 = 8, placed for NR1C5, clean-up: no 2 in R9C1
25a. Naked pair {14} in R2C3 + R4C1, CPE no 1 in R4C3
25b. 16(3) cage at R1C2 (step 22) = {268/358}, 8 locked for R1
25c. 1 in R1 only in R1C78, locked for NR1C6
25d. 8 in C1 only in R6789C1, locked for NR6C1
25e. Clean-up: no 4 in R1C6 + R3C5 (step 5), no 4,8 in R4C5 (step 6)
26. R12C5 (step 16a) = {29/47}, R1C6 + R3C5 (step 5) = {39/57}, R1C6 = R4C5 (step 5) -> R34C5 = {39/57}
26a. Killer pair 7,9 in R12C5 and R34C5, locked for C5
27. 9(3) cage at R2C3 = {126/135/234}
27a. R2C3 = {14} -> no 4 in R3C3
27b. 9(3) cage = {126/234} (cannot be {135} which clashes with R4C12 (step 3) = [43], killer combo clash), no 5, 2 locked for C3
27c. 1 in NR1
C1 only in R2C3 -> 9(3) cage = {126} = [126]
or in R4C1 -> R4C12 (step 3) = [16] (locking cages) -> 6 must be in R4C23, locked for R4
28. 15(3) cage at R2C4 = {159/249} (cannot be {357} because R4C2 = 6, R5C1 = 9 => R4C4 = 9 (LoL, step 18) blocks {357}), no 3,7, 9 locked for C4
28a. 3 in NR1C5 only in R3C3 + R45C2, CPE no 3 in R5C3
29. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34
29a. No 7 in R4C34 -> no 7 in R5C2
29b. R5C12 and R4C34 cannot be [69] because R23C3 = [12] clashes with R4C2 + R5C34 = [312] -> no 9 in R5C2
30. Variable hidden killer pair 1,5 in 15(3) cage at R2C4 and R5C2 for NR1C5, R5C2 cannot contain both of 1,5 -> 15(3) cage must contain at least one of 1,5 -> 15(3) cage (step 28) = {159} (only remaining combination, cannot be {249} which doesn’t contain 1 or 5), locked for C4
31. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34
31a. No 4 in R4C34 -> no 4 in R5C2
32. R12C5 = {47} (hidden pair in NR1C5), locked for C5 and 23(4) cage at R1C5, clean-up: no 5 in R1C6 + R3C5 (step 5), no 5 in R4C5 (step 6)
32a. Naked pair {39} in R34C5, locked for C5 and NR2C6
32b. 20(3) cage at R4C7 = {578} (only remaining combination), locked for R4
33. Killer pair 2,4 in R4C6 and 8(3) cage at R6C6, locked for C6
34. 1,2,4 in NR2C6 only in 22(5) cage at R3C6 = {12469} (only remaining combination), no 3,5,7 -> R3C6 = 6, R34C5 = [39], R3C3 = 2, placed for NR1C5, R1C6 = 9, placed for NR1C6, clean-up: no 3,6,7 in R2C9
35. Naked triple {159} in 15(3) cage at R2C4, 9 locked for NR1C5 -> R5C1 = 6, placed for NR1C5, R4C1 = 4, placed for NR1C1, R4C2 = 3, R24C3 = [16], R4C6 = 2, clean-up: no 2 in R12C1, no 5 in 8(3) cage at R6C6, no 4 in R6C9, no 7 in R9C1, no 4,6 in R9C2
35a. Naked pair {59} in R23C4, locked for NR1C5 -> R4C4 = 1, R5C2 = 1, clean-up: no 9 in R
9C1
35b. LoL for R6789 two outies R5C89 must exactly equal two innies R6C67, no 2,5 in R5C8, no 1 in R6C6, no 6 in R6C7
36. Naked pair {35} in R12C1, locked for C1 and NR1C1, clean-up: no 7 in R9C2
36a. Naked triple {268} in 16(3) cage at R1C2, locked for R1
37. Naked triple {134} in 8(3) cage at R6C6, 1 also locked for NR5C8
37a. Naked triple {134} in 8(3) cage at R6C6, CPE no 3,4 in R6C8
38. 1 in C5 only in R67C5 (step 8b) = {16}, locked for C5 and NR6C2
38a. 2 in C5 only in R89C5, locked for NR5C8 and 14(4) cage at R7C5, no 2 in R9C4
39. 2 in C5 only in 14(4) cage at R7C5 = {1238/1256} (cannot be {2345} because R7C5 only contains 1,6) -> R7C5 = 1, R6C5 = 6, R9C4 = {36}
39a. R8C6 = 1 (hidden single in C6), clean-up: no 9 in R9C9
39b.
Deleted, I did it earlier39c. 6 in C9 only in R789C9, locked for NR6C9
40. R6C5 = 6 -> 24(4) cage at R5C5 = {3678} (only remaining combination) -> R5C5 = 8, placed for NR4C3, R67C4 = {37}, locked for C4 and NR6C2 -> R9C4 = 6, placed for NR6C1, clean-up: no 2 in R6C9, no 4 in R8C9
41. R45C2 = [31] = 4 -> R5C34 = 11 = [74/92]
41a. Naked pair {79} in R5C39, locked for R5 -> R5C6 = 5
42. R5C6 = 5, 3 in R5 only in R5C78 -> 18(4) cage at R5C5 = {2358} (only remaining combination) -> R5C7 = 2, R56C8 = [38], both placed for NR5C8, R67C6 = [34], R6C9 = 1, R5C9 = 9, placed for NR5C8, R5C34 = [74], 7 placed for NR4C3
42a. Naked pair {25} in R89C5, locked for NR5C8
42b. R6C7 = 9 -> R78C7 = 9 = [63], 3 placed for NR6C9, clean-up: no 7 in R89C9
43. 16(3) cage at R1C2 = [682], R8C4 = 8, clean-up: no 2 in R9C9
44. R67C4 = [73], R6C1 = 2, placed for NR6C1, R9C2 = 9, placed for NR6C1, R9C1 = 1
45. Naked pair {45} in R6C23, locked for NR6C2 -> R7C23 = [29]
46. R8C9 = 6 (hidden single in R8), R9C9 = 4, clean-up: no 5 in R23C9
46a. R23C9 = [27], 7
placed for NR1C6, R7C9 = 5 (hidden single in R7), placed for NR6C9, R1C9 = 3, R12C1 = [53]
46b. R1C9 = 3 -> R12C8 = 10 = [46]
and the rest is naked singles, without using the nonets.