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SumoCueV1=11J0=10J0+1J0=13J0=3J1+4J2=13J2=11J2+7J2+0J0=8J0+10J0+3J1=15J1=11J1+6J1=15J2=10J2=10J0+18J3=9J0+13J1+13J1+14J1=5J2+16J2+17J4=10J3=8J3+20J3=18J5+30J1+30J5+24J5=11J2+34J4+27J3+28J3=11J3+38J5+30J5=10J5+41J4=9J4=7J4=7J3+45J6=13J5=22J5+48J7+48J5=9J4+43J4+44J4=12J3=17J6+47J6=10J7=13J7+58J7+51J8=5J4+61J8+54J6+55J6=5J7+57J7+58J7=14J8=7J8+69J8=10J8=6J6+72J6+65J6=11J6+75J7+68J7=16J8+78J8+71J8

Prelims

a) R12C1 = {29/38/47/56}, no 1
b) R1C23 = {19/28/37/46}, no 5
c) R12C4 = {49/58/67}, no 1,2,3
d) R1C56 = {12}
e) R12C7 = {49/58/67}, no 1,2,3
f) R1C89 = {29/38/47/56}, no 1
g) R2C23 = {17/26/35}, no 4,8,9
h) R23C6 = {29/38/47/56}, no 1
i) R23C8 = {69/78}
j) R23C9 = {19/28/37/46}, no 5
k) R3C12 = {19/28/37/46}, no 5
l) R34C3 = {18/27/36/45}, no 9
m) R34C7 = {14/23}
n) R45C1 = {19/28/37/46}, no 5
o) R45C2 = {17/26/35}, no 4,8,9
p) R4C89 = {29/38/47/56}, no 1
q) R5C34 = {29/38/47/56}, no 1
r) R5C67 = {19/28/37/46}, no 5
s) R56C8 = {18/27/36/45}, no 9
t) R56C9 = {16/25/34}, no 7,8,9
u) R6C12 = {16/25/34}, no 7,8,9
v) R67C3 = {49/58/67}, no 1,2,3
w) R67C7 = {18/27/36/45}, no 9
x) R78C1 = {39/48/57}, no 1,2,6
y) R78C2 = {89}
z) R78C4 = {19/28/37/46}, no 5
aa) R7C89 = {14/23}
bb) R89C3 = {14/23}
cc) R89C6 = {59/68}
dd) R8C78 = {16/25/34}, no 7,8,9
ee R89C9 = {19/28/37/46}, no 5
ff) R9C12 = {15/24}
gg) R9C45 = {29/38/47/56}, no 1
hh) R9C78 = {79}
ii) 22(3) cage at R6C4 = {589/679}

1. Naked pair {79} in R9C78, locked for R9 and NR7C7, clean-up: no 2 in R6C7, no 5 in R89C6, no 1,3 in R89C9, no 2,4 in R9C45
1a. Naked pair {68} in R89C6, locked for C6, clean-up: no 3,5 in R23C6, no 2,4 in R5C7
1b. Killer pair 6,8 in R9C45 and R9C6, locked for R9, clean-up: no 2,4 in R8C9
1c. Naked pair {68} in R8C69, locked for R8 and NR7C7 -> R78C2 = [89], both placed for NR6C2, clean-up: no 1,2 in R1C3, no 1,2 in R3C1, no 4,5 in R6C3, no 1,3 in R6C7, no 3,4 in R7C1, no 1,2,4 in R7C4, no 4 in R8C1, no 2 in R8C4, no 1 in R8C78, no 3 in R9C5
1d. Killer pair 2,4 in R8C78 and R9C9, locked for NR7C7, clean-up: no 5,7 in R6C7, no 1,3 in R7C8

2. Naked pair {12} in R1C56, locked for R1, clean-up: no 8,9 in R1C3, no 9 in R1C89, no 9 in R2C1
[As with the other early Texas Jigsaw Killers, I’m deliberately not using CPEs, which aren’t necessary for this puzzle but might shorten the solving path.]
2a. R1C89 = {38/56} (cannot be {47} which clashes with R1C23), no 4,7
2b. Killer pair 3,6 in R1C23 and R1C89, locked for R1, clean-up: no 5,8 in R2C1, no 7 in R2C4, no 7 in R2C7

3. R9C12 = {15} (cannot be {24} which clashes with R9C9), locked for R9 and NR6C2, clean-up: no 2,6 in R6C1, no 7 in R7C1, no 4 in R8C3, no 6 in R9C45
3a. R9C4 = 3, placed for NR6C2, R9C5 = 8, R9C6 = 6, placed for NR6C5, R8C6 = 8, R8C9 = 6, R9C9 = 4, placed for NR7C7, R9C3 = 2, placed for NR6C2, R8C3 = 3, placed for NR6C5, R8C1 = 7, placed for NR6C2, R7C1 = 5, placed for NR3C2, R9C12 = [15], clean-up: no 4 in R1C1, no 7 in R1C2, no 5 in R1C8, no 4,6 in R2C1, no 6 in R2C2, no 3 in R3C2, no 4,6,7 in R3C3, no 3,9 in R45C1, no 3 in R45C2, no 6,7 in R4C3, no 5,7 in R4C8, no 8 in R5C3, no 6,8,9 in R5C4, no 1,3 in R56C9, no 4 in R6C1, no 6 in R6C2, no 6,8 in R6C3, no 4 in R6C7
[So many clean-ups; apologies if I’ve missed any.]

4. R6C1 = 3, R6C2 = 4, placed for NR6C2, R7C3 = 6, R6C3 = 7, placed for NR4C4, R1C3 = 4, R1C2 = 6, both placed for NR1C1, R2C1 = 2, R1C1 = 9, R3C1 = 8, placed for NR1C1, R3C2 = 2
4a. Naked pair {15} in R23C3, locked for C3 and NR1C1 -> R1C4 = 7, placed for NR1C1, R2C4 = 6, placed for NR1C5, R2C2 = 3, R2C3 = 5, R3C3 = 1, R4C3 = 8, R5C3 = 9, R5C4 = 2, placed for NR4C4, R56C9 = [52], both placed for NR3C9, R7C8 = 4, placed for NR3C9, R7C9 = 1, R7C7 = 3, R6C7 = 6, placed for NR3C9, R7C4 = 9, placed for NR6C5, R8C4 = 1, R6C5 = 5, R6C4 = 8, R6C6 = 9, R6C8 = 1, placed for NR3C9, R5C8 = 8, R5C7 = 7, placed for NR3C9, R5C6 = 3, R3C7 = 4, R4C7 = 1, placed for NR4C4, R9C78 = [97], R2C78 = [89], R2C9 = 7, R3C9 = 3

and the rest is naked singles, without using the nonets.

SumoCueV1=24J0+0J0=14J1+2J1+2J1=12J1=12J2+6J2=17J3+0J0=15J0+10J1+2J1=10J1+5J1=10J2+15J2+8J3+0J0=5J0=5J0+20J4+13J1=12J2+23J2+23J2+8J3=24J5+19J0=15J0+29J4+29J4=15J2=6J2+33J3=16J3+27J5+27J5+27J5=12J4+39J6+32J6=8J6+42J3+35J3=14J5=12J5+46J5=24J4+48J4+32J6=21J6+51J6+35J3+45J7=9J5+46J5+48J4+48J4+51J6+51J6=18J6+61J3+45J7+55J7=8J7=10J7=25J4+67J8=12J8+61J8=7J8=13J7+72J7+65J7+66J7+67J8+67J8+69J8+71J8+71J8

Prelims

a) R12C6 = {39/48/57}, no 1,2,6
b) R1C78 = {39/48/57}, no 1,2,6
c) R2C23 = {69/78}
d) R23C5 = {19/28/37/46}, no 5
e) R2C78 = {19/28/37/46}, no 5
f) R34C2 = {14/23}
g) R3C34 = {14/23}
h) R4C78 = {15/24}
i) R5C45 = {39/48/57}, no 1,2,6
j) R5C78 = {17/26/35}, no 4,8,9
k) R78C2 = {18/27/36/45}, no 9
l) R89C3 = {17/26/35}, no 4,8,9
m) R89C4 = {19/28/37/46}, no 5
n) R89C7 = {39/48/57}, no 1,2,6
o) R9C12 = {49/58/67}, no 1,2,3
p) 7(3) cage at R8C9 = {124}
q) 14(4) cage at R1C3 = {1238/1247/1256/1346/2345}, no 9

1. Naked triple {124} in 7(3) cage at R8C9, locked for NR8C6, clean-up: no 8 in R89C7

2. 45 rule on R123 1 outie R4C2 = 1, R3C2 = 4, both placed for NR1C1, clean-up: no 6 in R2C5, no 1 in R3C4, no 5 in R4C78, no 5,8 in R78C2, no 9 in R9C1
2a. Naked pair {23} in R3C34, locked for R3, clean-up: no 7,8 in R2C5
2b. Naked pair {24} in R4C78, locked for R4
[As with the other early Texas Jigsaw Killers, I’m deliberately not using CPEs, which aren’t necessary for this puzzle but might shorten the solving path.]

3. 12(3) cage at R3C6 = {156} (only remaining combination), locked for R3 and NR1C7, clean-up: no 7 in R1C78, no 4,9 in R2C78, no 4,9 in R2C5
3a. Killer pair 3,8 in R1C78 and R2C78, locked for NR1C7

4. 45 rule on R1234 3 innies R4C169 = 23 = {689} -> R4C6 = 9, placed for NR1C7, R4C19 = {68}, locked for R4, clean-up: no 3 in R12C6, 3 in R1C78
4a. Naked pair {48} in R1C78, locked for R1 and NR1C7 -> R4C7 = 2, placed for NR1C7, R4C8 = 4, placed for NR1C9, clean-up: no 4,8 in R2C6
4b. Naked pair {37} in R2C78, locked for R2 -> R12C5 = [75], placed for NR1C3, clean-up: no 8 in R2C23, no 3 in R2C5
4c. Naked pair {69} in R2C23, locked for R2
4d. R4C6 = 9 -> R56C6 = 6 = {24}, locked for C6 and NR5C5, clean-up: no 8 in R5C4, no 6 in R5C8

5. 14(4) cage at R1C3 = {1238/1346}, 1 locked for NR1C3 -> R2C5 = 2, R3C5 = 8, both placed for NR1C3, R2C1 = 8, R2C9 = 1, placed for NR1C9, R2C4 = 4, R4C1 = 6, placed for NR4C1, R4C9 = 8, clean-up: no 8 in R5C5, no 7 in R5C7, no 6 in R89C4, no 5,7 in R9C2

6. Naked pair {24} in R89C9, locked for C9 and 7(3) cage at R8C9 -> R9C8 = 1
6a. R4C9 = 8 -> R56C9 = 8 = {35}, locked for C9 and NR1C9, clean-up: no 3,5 in R5C7
6b. R2C9 = 1 -> R13C9 = 16 = [97], placed for NR1C9 -> R5C8 = 2, R5C7 = 6, placed for NR5C5, R56C6 = [42], R7C9 = 6, R3C1 = 9, placed for NR1C1, R2C23 = [69], clean-up: no 3 in R78C2, no 7 in R9C1

7. R1C12 = {25} (hidden pair in R1), locked for NR1C1 -> R3C34 = [32], R4C3 = 7, clean-up: no 1,5 in R89C3, no 8 in R89C4
7a. Naked pair {26} in R89C3, locked for C3 and NR7C1 -> R8C2 = 7, placed for NR7C1, R7C2 = 2, R1C12 = [25], R1C3 = 1, clean-up: no 3 in R89C4
7b. Naked pair {19} in R89C4, locked for C4 and NR7C1 -> R9C2 = 8, R9C1 = 5
7c. Naked pair {34} in R78C1, locked for C1, R6C1 = 7 (cage sum)

8. R4C1 = 6, R5C1 = 1 -> R5C23 = 17 = [98], R6C2 = 3, R56C9 = [35], R67C3 = [45]

9. Naked pair {35} in R4C45, locked for NR3C4 -> R5C4 = 7, placed for NR3C4, R5C5 = 5, R4C45 = [53]

10. R67C4 = [68] = 14 -> R68C5 = 10 = {19}, locked for C5, R1C45 = [36]

and the rest is naked singles, without using the nonets.

SumoCueV1=10J0+0J0+0J1=10J1=19J1=13J1=21J1+6J2+6J2=22J0+9J0+9J0+3J1+4J1+5J1=11J2+15J2+15J2=15J0+18J0+18J0+3J3+4J1+5J3=14J2+24J2+24J2=17J4=14J0=13J4=9J3+30J3+30J3=19J5=10J2=13J5+27J4+28J4+29J4=21J3+39J3+39J3+33J5+34J5+35J5+27J6+28J4+29J4=19J4+48J3+48J5+33J5+34J5+35J7=20J6+54J6+54J6=20J4=7J8=17J5=17J7+60J7+60J7=6J6+63J6+63J6+57J8+58J8+59J8=20J7+69J7+69J7=18J6+72J6+72J8+57J8+58J8+59J8=10J8+78J7+78J7

Prelims

a) 10(3) cage at R1C1 = {127/136/145/235}, no 8,9
b) 10(3) cage at R1C4 = {127/136/145/235}, no 8,9
c) 19(3) cage at R1C5 = {289/379/469/478/568}, no 1
d) 21(3) cage at R1C7 = {489/579/678}, no 1,2,3
e) 22(3) cage at R2C1 = {589/679}
f) 11(3) cage at R2C7 = {128/137/146/236/245}, no 9
g) 9(3) cage at R4C4 = {126/135/234}, no 7,8,9
h) 19(3) cage at R4C7 = {289/379/469/478/568}, no 1
i) 10(3) cage at R4C8 = {127/136/145/235}, no 8,9
j) 21(3) cage at R5C4 = {489/579/678}, no 1,2,3
k) 19(3) cage at R6C4 = {289/379/469/478/568}, no 1
l) 20(3) cage at R7C1 = {389/479/569/578}, no 1,2
m) 20(3) cage at R7C4 = {389/479/569/578}, no 1,2
n) 7(3) cage at R7C5 = {124}
o) 6(3) cage at R8C1 = {123}
p) 20(3) cage at R8C7 = {389/479/569/578}, no 1,2
q) 10(3) cage at R9C7 = {127/136/145/235}, no 8,9

1. Naked triple {123} in 6(3) cage at R8C1, locked for R8 and NR6C1 -> R8C5 = 4, placed for NR7C5, R79C5 = {12}, locked for C5 and NR7C5
1a. 10(3) cage at R9C7 = {136/145/235} (cannot be {127} which clashes with R9C5), no 7
1b. 9(3) cage at R4C4 = {126/135/234}
1c. 6 of {126} must be in R4C5 -> no 6 in R4C46

2. 20(3) cage at R8C7 = {569/578}, 5 locked for R8 and NR6C9
2a. 5 in NR7C5 only in R9C3467, locked for R9

3. 22(3) cage at R2C1 = {589/679}, 9 locked for R2 and NR1C1

4. 45 rule on R7 3 innies R7C456 = 8 = {125/134}, 1 locked for R7
4a. 5 of {125} must be in R7C4, no 5 in R7C6
4b. Killer pair 4,5 in 20(3) cage at R7C1 and R7C456, locked for R7

5. 20(3) cage at R7C4 = {389/479/569/578}
5a. R7C4 = {345} -> no 3,5 in R9C4
5b. 17(3) cage at R7C6 = {269/278/359/368/458/467} (cannot be {179} which clashes with 20(3) cage), no 1
5c. R7C5 = 1 (hidden single in R7), R9C5 = 2
5d. 10(3) cage at R9C7 (step 1a) = {136/145}, 1 locked for NR6C9

6. 17(3) cage at R7C7 = {269/278} (cannot be {368} which clashes with 20(3) cage at R8C7), no 3, 2 locked for R7 and NR6C9
6a. R7C456 (step 4) = {134} (only remaining combination), locked for R7

7. 20(3) cage at R7C4 = {389/479}, no 6, 9 locked for C4 and NR7C5

[In this position, Law of Leftovers (LoL) would normally be used here but, since the SS score is fairly low, I’ll continue using simpler steps.]

8. 45 rule on NR6C9 3 innies R6C9 + R9C89 = 8 = {134}
8a. 10(3) cage at R9C7 (step 5d) = {136/145}
8b. 5,6 only in R9C7 -> R9C7 = {56}

9. 45 rule on NR7C5, 2 innies R9C37 = 2 outies R7C46 + 1, R7C46 = {34} = 7 -> R9C37 = 8 = [35]
9a. Naked pair {14} in R9C89, locked for R9 and NR6C9 -> R6C9 = 3, R45C9 = 10 = {19/28/46}, no 5,7
9b. R6C1 = 4 (hidden single in NR6C1), R45C1 = 13 = {58/67}, no 1,2,3,9
9c. R138C1 = {123} (hidden triple in C1)
9d. 17(3) cage at R7C6 (step 5b) = {368/467}, 6 locked for C6
9e. 10(3) cage at R4C8 = {127/136/235} (cannot be {145} which clashes with R9C8), no 4

10. 45 rule on NR1C1 3 innies R1C12 + R4C2 = 8 = {125/134}, 1 locked for NR1C1
10a. Killer pair 2,3 in R1C12 + R4C2 and R3C1, locked for NR1C1
10b. 10(3) cage at R1C1 = {127/136/145} (cannot be {235} because R1C12 + R4C2 cannot have 3 and one of 2,5), 1 locked for R1
10c. 6,7 of {127/136} must be in R1C3, 1 of {145} must in R1C1 -> no 1 in R1C3
10d. 1 in R1 only in R1C12, locked for NR1C1

11. 45 rule on NR1C8 3 innies R1C89 + R4C8 = 20 = {389/479/569/578}, no 1,2

12. 45 rule on C4 3 innies R456C4 = 15 = {168/258/267/456} (cannot be {348} which clashes with R7C4, cannot be {357} which clashes with 20(3) cage at R7C4), no 3

13. 13(3) cage at R4C3 doesn’t contain 3 -> no 9 in 13(3) cage
13a. 9 in NR4C1 only in 14(3) cage at R4C2, locked for C2
13b. 14(3) cage = {149/239}, no 5,6,7,8
13c. 14(3) cage + R8C2 must contain 1, locked for C2

14. R1C1 = 1 (hidden single in R1)
14a. R1C12 + R4C2 (step 10) = {125/134} -> R14C2 = [34/43/52], no 2 in R1C2
14b. R1C1 = 1 -> R1C23 = 9 = [36/45/54], no 2,7 in R1C3

15. 9 in R4 only in R4C79, locked for NR4C7, clean-up: no 1 in R4C9 (step 9a)

16. 45 rule on NR4C7 2 remaining innies R67C6 = 1 outie R4C8 + 6
16a. Min R4C8 = 3 -> min R67C6 = 9, no 2 in R6C6
16b. Max R67C6 = 12 -> max R4C8 = 6
16c. R1C89 + R4C8 (step 11) = {389/569/578} (cannot be {479} because R4C8 only contains 3,5,6), no 4 in R1C89

17. 9(3) cage at R4C4 = {126/135} (cannot be {234} which clashes with R4C2), no 4, 1 locked for R4 and NR3C4

18. 19(3) cage at R6C4 = {289/569/578}
18a. 9 of {569} must be in R6C5 -> no 6 in R6C5

19. 1 in R3 only in 14(3) cage at R3C7, locked for NR1C8
19a. 14(3) cage at R3C7 = {149/158/167}, no 2,3
19b. 11(3) cage at R2C7 = {236/245}, no 7,8, 2 locked for R2
19c. Killer pair 5,6 in 22(3) cage at R2C1 and 11(3) cage, locked for R2

20. 11(3) cage at R2C7 (step 19b) = {236/245} + R4C8 = {356} must contain 5, locked for NR1C8
20a. 14(3) cage at R3C7 (step 19a) = {149/167}, no 8
20b. Killer pair 4,6 in 11(3) cage and 14(3) cage, locked for NR1C8

21. 21(3) cage at R1C7 = {489/678}, 8 locked for R1
21a. 4,6 only in R1C7 -> R1C7 = {46}
21b. Killer pair 4,6 in 10(3) cage and R1C7, locked for R1

22. 10(3) cage at R4C8 (step 9e) = {136/235} (cannot be {127} because R4C8 only contains 3,5), no 7, 3 locked for C8
[CPE for 3 could be used here, but I’m not using CPEs for the early Texas Jigsaw Killers.]

23. 9(3) cage at R4C4 (step 17) = {126} (cannot be {135} which clashes with R4C8) -> R4C5 = 6, placed for NR3C4, R4C46 = {12}, locked for R4 and NR3C4, clean-up: no 7 in R5C1 (step 9b), no 4,8 in R5C9 (step 9a)

24. R6C4 = 6 (hidden single in C4), placed for NR4C1, clean-up: no 7 in R4C1 (step 9b)
24a. Naked pair {58} in R45C1, locked for C1 and NR4C1
24b. 19(3) cage at R6C4 = {568} (only remaining combination) -> R6C56 = {58}, locked for R6

25. 21(3) cage at R5C4 = {489/579}, 9 locked for R5 and NR3C4
25a. Killer pair 5,8 in R5C1 and 21(3) cage, locked for R5
25b. Killer pair 5,8 in 21(3) cage and R6C5, locked for NR3C4

26. R1C12 + R4C2 (step 10) = {134} (only remaining combination), locked for NR1C1, 3,4 locked for C2 and NR1C1
26a. Naked pair {12} in R58C2, locked for C2 -> R6C2 = 9

27. 10(3) cage at R1C4 = {145} (only remaining combination, cannot be {127} which clashes with R4C4, cannot be {235} because 2,5 only in R1C4) -> R1C4 = 5, placed for NR1C3, R2C4 = 1, R3C4 = 4, placed for NR3C4, R7C4 = 3, R7C6 = 4, placed for NR4C7
27a. R7C4 = 3, R89C4 = 17 = {89}, locked for C4 and NR7C5

28. R5C4 = 7, placed for NR3C4, R5C56 = 14 = {59}, locked for R5 and NR3C4 -> R6C5 = 8
28a. Naked triple {379} in 19(3) cage at R1C5, locked for C5 and NR1C3 -> R12C6 = [28]
28b. 22(3) cage at R2C1 = {679} (only remaining combination), locked for R2 and NR1C1 -> R2C5 = 3

29. R1C2 = 3 (hidden single in R1), R1C3 = 6 (cage sum), R1C7 = 4, R1C89 = 17 = {89}, locked for R1 and NR1C8, R2C7 = 2, R6C7 = 7, R45C7 = 12 = [93], R4C9 = 8, R5C9 = 2 (cage sum), placed for NR4C7, R6C8 = 1

and the rest is naked singles, without using the nonets.

SumoCueV1=11J0=19J0=22J0+2J0+2J1+2J2=28J2=10J2+7J2+0J0+1J0+1J0=14J1+12J1+12J2+6J2+7J2+7J3+0J0+0J0=20J1+20J1+20J1=13J4+6J2+6J2=12J3=16J5=14J5=12J1=9J1+30J4+23J4+23J3=15J3+26J3+27J5+28J5+29J1=10J4+39J4+39J4=8J6+34J3=13J3=6J5+28J5=13J5+47J4=16J4+49J6+42J6+34J3+44J3+45J5=12J7+55J7+47J4=16J6+58J6+58J6=20J8+61J8=30J5+63J7+55J7=13J7+66J6+66J6=16J8+69J8+61J8+63J7+63J7+55J7=17J7+75J6+75J8+75J8+69J8+61J8

Prelims

a) R34C9 = {39/48/57}, no 1,2,6
b) R45C1 = {79}
c) R45C3 = {39/48/57}, no 1,2,6
d) R4C45 = {18/27/36/45}, no 9
e) R56C7 = {17/26/35}, no 4,8,9
f) R56C9 = {49/58/67}, no 1,2,3
g) R67C1 = {15/24}
h) R6C56 = {79}
i) 19(3) cage at R1C2 = {289/379/469/478/568}, no 1
j) 20(3) cage at R3C3 = {389/479/569/578}, no 1,2
k) 10(3) cage at R5C4 = {127/136/145/235}, no 8,9
l) 11(4) cage at R1C1 = {1235}
m) 28(4) cage at R1C7 = {4789/5689}, no 1,2,3
n) 10(4) cage at R1C8 = {1234}
o) 12(4) cage at R7C2 = {1236/1245}, no 7,8,9
p) 30(4) cage at R8C1 = {6789}

1. 4 in C1 only in R67C1 = {24}, locked for C1 and NR4C1

2. Naked pair {79} in R45C1, locked for C1 and NR4C1
2a. Naked pair {68} in R89C1, locked for 30(4) cage at R8C1
2b. Naked pair {79} in R89C2, locked for C2 and NR7C2
2c. 45 rule on NR4C1 2 innies R6C3 + R8C1 = 9 = [18/36]
2d. 5 in NR4C1 only in 14(3) cage at R4C2, locked for C2
2e. 14(3) cage = {158/356}

3. Naked triple {135} in R123C1, locked for 11(4) cage at R1C1 and NR1C1, R3C2 = 2, placed for NR1C1
3a. 45 rule on NR1C1 2 innies R1C34 = 15 = {69/78}, no 4
3b. R1C34 = 15 -> R1C56 = 7 = {16/25/34}, no 7,8,9
3c. 19(4) cage at R1C2 = {469/478}, 7,9 only in R2C3 -> R2C3 = {79}
3d. 19(4) cage = {469/478}, 4 locked for C2
3e. Killer pair 6,8 in 19(3) cage at R1C2 and 14(3) cage, locked for C2

4. 12(4) cage at R7C2 = {1236/1245}, 1,2 locked for NR7C2

5. 45 rule on NR7C8 2 innies R9C67 = 9 = {18/27/36/45}, no 9
5a. R9C67 = 9 -> R9C45 = 8 = [35/53/62]
5b. R9C67 = {18/27/45} (cannot be {36} which clashes with R9C45), no 3,6

6. 45 rule on NR2C9 2 innies R2C9 + R4C7 = 5 = {14/23}

7. 45 rule on C9 2 innies R12C9 = 1 outie R7C8
7a. Min R12C9 = 3 -> min R7C8 = 3
7b. Max R12C9 = 7 -> max R7C8 = 7

8. 45 rule on C89 1 innie R3C8 = 1 outie R8C7 + 4, no 4 in R3C8, R8C7 = {12345}

9. 45 rule on R789 2 innies R7C14 = 11 = [29/47]
9a. R6C3 = {13}, R7C4 = {79} -> R6C4 must also be odd = {13579}
9b. Naked pair {79} in R6C56, locked for R6, clean-up: no 1 in R5C7, no 4,6 in R5C9

10. 45 rule on R123 2 innies R3C69 = 11 = {38/47}/[65], no 1,9, no 5 in R3C6, clean-up: no 3 in R4C9
10a. R3C1 = 1 (hidden single in R3)

11. 45 rule on R3 2 remaining innies R3C78 = 11 = [47/56/65], no 8,9, no 7 in R3C7
11a. R3C78 = 11 -> R12C7 = {89}, locked for C7 and NR1C6, clean-up: no 1 in R9C6 (step 5)

12. 7 in R1 only in R1C34 (step 3b) = {78}, locked for R1 and NR1C1 -> R12C7 = [98], R2C3 = 9, clean-up: no 3 in R45C3
12a. Naked pair {46} in R12C2, locked for C2
12b. 14(3) cage at R4C2 = {158} (only remaining combination), locked for C2 and NR4C1 -> R6C3 = 3, R7C2 = 3, placed for NR7C2, R8C1 = 6, R9C1 = 8, placed for NR7C2, clean-up: no 5 in R5C7, no 5 in R9C5 (step 5a), no 1 in R9C67 (step 5)
12c. Killer pair 2,5 in R9C45 and R9C67, locked for R9
12d. Killer pair 7,8 in R1C3 and R45C3, locked for C3

13. R6C3 = 3 -> R67C4 = 10 = [19], placed for NR3C6 -> R6C5 = 7, placed for NR3C6, R6C6 = 9, placed for NR5C7, clean-up: no 2,8 in R4C4, no 8 in R4C5, no 7 in R5C7
13a. 10(3) cage at R5C4 = {235} (only remaining combination), locked for R5 and NR3C6 -> R5C7 = 6, R6C7 = 2, both placed for NR5C7, R9C5 = 3, placed for NR5C7, R9C4 = 5 (step 5a), R8C4 = 4, both placed for NR7C2, clean-up: no 7 in R4C3, no 8 in R6C9

14. R4C45 = {36} (only remaining combination) -> R4C4 = 3, R4C5 = 6, placed for NR3C6, 10(3) cage at R5C4 = [253]
14a. Naked pair {48} in R34C6, locked for C6 and 13(3) cage at R3C6 -> R4C7 = 1, R2C9 = 4 (step 6), both placed for NR2C9, R12C2 = [46]
14c. R2C4 = 7 -> R2C56 = 7 = [25], 5 placed for NR1C6

15. Naked pair {17} in R78C6, locked for C6 and NR5C7 -> R9C6 = 2, R9C7 = 7 (step 5), both placed for NR7C8

16. R3C7 = 4, R7C7 = 5, R7C56 = 11 = [47]

17. R3C6 = 8, 20(3) cage at R3C3 = [569], R3C8 = 7, R3C9 = 3, R4C9 = 9

and the rest is naked singles, without using the nonets.

SumoCueV1=18J0+0J0=11J1+2J1=18J1=16J1+5J1=12J2+7J2=18J3+0J0=16J0+2J1+4J1+5J1=10J2+15J2+7J2+9J3+9J3+11J0+11J0+4J1=23J2+23J2+15J2=12J4=14J3+27J3+27J3=11J0+30J0+23J2=18J4+33J4+26J4=13J3+36J3=22J5=14J6+30J0=12J7+41J7+33J4+26J4+36J3+38J5+38J5+39J6+39J6+41J7=12J7=15J4+52J4=11J5+54J5=12J5+56J5=15J6+58J6+51J7+51J7+52J4+54J5=12J8+56J5=21J8=12J8+58J6=16J6=21J7+70J7+64J8+64J8+66J8+66J8+67J8+67J8+69J6+69J6+70J7

sudokuEd
This is a real toughie. See what you mean Mike and Para. First attempt couldn't get anywhere, so had to put the creative hat on.

Seems like I've been able to make some progress with some interesting techniques. Can someone check through to make sure its all valid? [EDIT: Rewritten from step 22 + new marks pic. Thanks Mike for finding the mistake. EDIT2: problem with new step 27 and other typos fixed: thanks Para and Mike] Add some more too if you like . I probably won't be able to look at it again till the weekend.

Seems like we need to add Killer LoL to our jargon to account for the effect of cage combinations on LoL. I also use 'overlap' to describe the congruence between combinations. Might need to find a different name so it is not confused with the normal 'overlap' technique. Names and the need for them is not my forte.....


Texas Jigsaw Killer 18
1. 23(3)r3c6 = {689}: all locked for n3

2. "45" r123: r4c6 - 7 = r3c9
2a. r3c9 = {12}
2b. r4c6 = {89}
2c. 6 in n3 only in r3: 6 locked for r3

3. Already resorting to nibbles.
3a. no 1 r2c1. Here's how.
3b. 1 in r2c1 -> r3c12 = {89}
3c. But this clashes with r3c67.

4. Killer LoL (KLoL) chain r123: no 2 r2c1. Here's how.
4a. LoLr123: 4 innies r2c1 + r3c129 = 4 outies r4c456 + r5c5
4b. only one 2 is possible in outies -> only one 2 is possible in innies.
4c. 2 in r2c1 -> r3c12 = {79} -> r4c6 = 9 -> r3c9 = 2 (step 2)
4d. but this means two 2's in r123 innies. Not possible.

5.KLoL r123: no {567} combo in 18(3)r4c6. Here's how.
5a.LoLr123 same as step 4a. Outies must have 8/9 (r4c6) -> innies must have 8/9
5b. ->18(3)r2c1 {567} combo. blocked

6. KLoL r123: no 2 r3c12. Here's how.
6a. 2 in r3c12 -> rest of 18(3) = {79}(no 8)
6b. 2 in r3c12 -> r3c9 = 1 -> r4c6 = 8 (step 2)
6c. But we know from LoLr123 that there can be no 8 in outies when 2 in r3c12.

7. 22(3)r5c3 = 9{58/67}
7a. 9 locked for n6(r5c3)

8. "45" c12: r4c3 + 4 = r6c2
8a. r4c3 = 1..5

9. "45" r89: 3 innies r8c136 = 8 = h8(3)r8
9a. = 1{25/34}
9b. 1 locked for r8


10. 1 in n9(r8c2) only in r9: 1 locked for r9

11. 21(3)r8c8 = {489/579/678} = [4/7..]

12. KLoL r89: no {457} combo in 16(3)r8c7. Here's how.
12a. LoL r89: 6 innies r8c1367 + r9c78 = 6 outies r56c67 + r7c78
12b. and both = 24 ie h24(6)r89innies = 24(6)r89outies
12c. 6 outies must have 1,2 & 3 for n8(r5c6)
12d. -> 6 innies must have 1,2 & 3
12e. note: 6 innies cannot have any repeats since the 6 outies are all in the same nonet.
12f. -> when h8(3)r8 = {125}, 16(3)r8c7 must be 3....
12g. -> when h8(3)r8 = {134}, 16(3) must be 2...
12h. -> {457} blocked from 16(3)

13. KLoL r89: no {358} combo. in 16(3)r8c7
13a. since the 6 outies in LoL r89(step 12) are all in the same nonet, there can be no repeats in the 6 innies
13b. since h8(3)r8 = {125/134} = [3/5..]
13c. -> {358} combo blocked from 16(3)
13d. -> the valid combinations in the h24(6)r89innies are
i. {125/349} = {123459}
ii. {125/367} = {123567}
iii. {134/259} = {123459}
iv. {134/268} = {123468}
13e. = 123{459/468/567} = [4/7..] not both.

14. KLoL r89: {147} combo blocked from both 12(3) cages in n8(r5c6): clash with 21(3)r8c8 = 4/7..] step 11 (thanks Mike for this much easier way).

15. 16(3)r8c7 = {259/268/349/367}
15a. ->{456} blocked from 15(3)r7c5
15b. ->{239/356} blocked from 14(3)r5c4

16. "45" r789: r6c7 + 3 = r7c9
16a. min r7c9 = 4, max r6c7 = 6

17. "45" c1234: r6c5 - 2 = r4c4
17a. max r4c4 = 7, min. r6c5 = 3

18. "45" c789: r135c7 = 19 = h19(3)c7
18a. no 1

19. LoLc789: 4 innies r189c7 + r9c8 = 4 outies r3456c6
19a. no 1 in innies -> no 1 in r56c6 [edit typo]

20. 1 in n8 only in 12(3)r6c7 = {129/138/156}(no 4,7)

21. 12(3)r5c6 = {237/246/345}(no 8,9)

22. KLoL Over-lap c789: no {259} combo. in 16(3)r8c7. Here's how.
22a. Lol c789: 4 innies r189c7 + r9c8 = 4 outies r3456c6
22b. the 12(3)r5c6 has 2/4 cells in the outies.
22c. the 16(3)r8c7 has 3/4 cells in the innies
22d. -> the valid combination in the 12(3) must overlap with 2 of the candidatates from the 16(3) combination
(IF that combo has at most 1 of 6/8/9 corresponding to r34c6 in outies: this is what I missed first WT)
22e. since 12(3) combinations are {237/246/345} and since {259} combo in 16(3)r8c8 has ONLY 1 of 6/8/9 AND does not have 2 candidates overlapping with 12(3)
22f. ->{259} combo blocked

23. 16(3)r8c8 = {268/349/367}(no 5)

24. KLoL + (double)Overlap c789: r56c6 no 5 [edit out invalid part]. Here's how.
24a...d (same reasoning as steps 22a..d)
24e. the 12(3) combinations are {237/246/345}
24f. the {349/367} combo's from 16(3) must have a double-overlap candidates with the 12(3) -> r56c6 = {34/37}
24g. for the {268} combo from 16(3), r3456c3 = {68}2{3/4/7} -> r56c6 = {23/24/27}(no 5)
24h. finally, the {268} combo from 16(3) could also double-overlap with 12(3) -> r56c6 = {26}
24h. In summary: r56c6 = {23/24/26/27/34/37}(no 5)

25. LoLc789: no 5 in outies -> no 5 in r1c7

26. "45" c6789: 3 innies r789c6 = 13 = h13(3)c6 [edit:typo]
26a. = {139/148/157/256} others blocked by r56c6 (step 24h.)

27. {148} combo blocked from h13(3)c7. Here's how. [edit: true but flawed reason: see Mikes preamble to step 37]
27a. 15(3)r7c5 = {159/168/249/258/267/357} ({348} blocked by 16(3)r8c7)

[edit: following steps are still valid]
28. h13(3)c6 = {139/157/256}(no 4,8)
28a. r78c6 + r9c6 = [913]/{15}[7]/[751]/{25}[6]/[625]
28b. ({13}[9]/[931]/{17}[5]/{56}[2] blocked by combinations unavailable in 15(3))
28c. no 3 r78c6, no 2 or 9 in r9c6 [edit:typos]

29. 15(3)r7c5 = [591]/9{15}/[375]/8{25}/[762] = {159/258/267/357} = [5/7..]
29a. r7c5 = {35789}

30. 14(3)r5c4: {257} blocked by 15(3)





Richard (rcbroughton)
Hi Ed

haven't picked up all the techniques for Jigsaw Killers yet. I might need to review the tutorials.

Picking up from you marks pic,

31. 45 Rule on R789. r7c9 = r6c7+3 - eliminates 7 from r7c9

32. 45 on r123. outies total 19.
32a when r4c6=8 r4c9<>8 = {29}/[38]/{47}/{56}
32b when r4c6=9 r4c9<>9 = [19]/{37}/{46} (can't be {28} because of 12(3) cage)
32c. r4c9 can't be 8 and r5c9 can't be 1

33. 45 on c789 - innes total 19
33a. r1c7 can't be 2

34. 45 on r1234 outies r5c58 minus innies r34c9 equals 5
34a no combos with 1 allowed :
out [71] - in {12} - 2 1's in nonet
out [81] - in {13} - 2 1's in nonet

35. 45 on c6789 - outies r789c5 total 14
35a. no combo with 8 at r9c5

36. 45 on c9 - outies r68c8 minus innies r12c9 equals 3 - innies limited to total of 11,10,9,8,7 or 5.
36a. 11 - 8 - no 1 in r6c8
36b. 7 - outies = [19] - but innies can only be {34} - {34} and 1 blocked by 12(3) r3459
36c. 5 - outies = [17]/[26]/[35] - but [17] blocked because r1c8 needs to be 7 to fit 12(3)
36d. -> no 1 at r6c8


Mike (mhparker)
Hi guys,

sudokuEd wrote "27b. Each combo in the h13(3)c7 must have 2 cells from the 15(3). No available combination in 15(3) has 2 of {148} -> {148} not possible"How about {168} - that has 2 of {148}?

Fortunately, h13(3)c6 = {148} implies 15(3)r7c5 = {168} and vice versa.
Due to observation in step 37a below, h13(3)c6 = {148} and 15(3)r7c5 = {168} together block all possible combinations for 16(3)r8c7
Therefore, we can indeed deduce that h13(3)c6 <> {148} and 15(3)r7c5 <> {168}.
The walkthrough is therefore not invalidated.

Here are some further steps...

37. Because 16(3)r8c7 maps to 3 of r3456c6 (LoL c789), of which r9c6 is a peer...
37a. ...r9c6 cannot duplicate any digit in 16(3)r8c7
37b. -> {367} combo for 16(3)r8c7 blocked by h13(3)c7 (see step 28)
37c. -> no 7 in 16(3)r8c7

38. 16(3)r8c8 = {268/349}
38a. -> blocks {248} combo from 14(3)r5c4
38b. -> 14(3)r5c4 = {149/158/167/347} (no 2)

39. LoL c6789: outies r56c4+r67c5 = innies r129c6+r1c7
39a. no 2 in outies -> no 2 in innies r12c6

40. LoL r89: outies r56c67+r7c78 = innies r8c1367+r9c78
40a. no 7 in innies -> no 7 in outies (r5c67+r6c6)
40b. -> 12(3)r5c6 = {246/345} = {(5/6)..}
40c. -> 4 locked for n8

41. {156} combo for 12(3)r6c7 blocked by 12(3)r5c6 (step 40b)
41a. -> 12(3)r6c7 = {129/138} (no 5,6)

42. LoL c789 (see step 19)
42a. no 7 in outies -> no 7 in innie r1c7

43. 7 no longer available for h19(3)c7 (step 18)
43a. -> no 3 in r15c7

44. LoL r789: outies r5c34+r6c2345 = innies r7c789+r8c89+r9r9
44a. no 2 in outies -> no 2 in innies r7c78
44b. -> no 8,9 in r7c9

45. LoL r789: outies r5c467+r6c4567 = innies r7c12349+r8c13
45a. no 9 in innies -> no 9 in outies
45b. -> 14(3)r5c4 = {158/167/347} (no 9)
45c. -> no 7 in r4c4 (step 17)

46. {357} combo for 15(3)r7c5 blocked by 14(3)r5c4 (step 45b)
46a. -> 15(3)r7c5 = {159/258/267} (no 3)

47. 15(3)r7c5 must contain one of {12} in r78c6
47a. -> no 1 in r9c6, otherwise h13(3)c6 cage total can't be reached (see step 26)

48. LoL c1234: outies r4589c5+r9c6 = innies r1c3+r1256c4
48a. no 9 in innies -> no 9 in outies r89c5

49. h19(3)c7 (step 18) = {289/469/568} (3,7 unavailable)
49a. -> if 16(3) = {268}, 6 must go in r9c8
49b. -> no 6 in r89c7, and no 2,8 in r9c8

50. n9(r8c2) confined to r89
50a. -> remaining cells in r89 (r8c136789+r9c789) must contain the digits 1..9 (no duplicates)
50b. -> r8c9 <> r9c8
50c. -> r89c89+r7c8 (innies, LoL c89) contain 5 different digits
50d. -> r34c67+r2c7 (outies, LoL c89) contain (same) 5 diffent digits
50e. -> no 6,8,9 in r4c7 (since these digits are already taken in 23(3)r3c6
50f. -> no 1 in r4c8
50g. no 2 in innies (step 50c)
50h. -> no 2 in outies r24c7

51. 6 in c7 now only in h19(3)c7 (step 18)
51a. -> h19(3)c7 = {469/568}
51b. -> no 2 in r5c7

52. LoL c89: outies r56c6+r4567c7 = innies r1239c8+r12c9
52a. no 8 in innies -> no 8 in outie r7c7
52b. 8 in 12(3)r6c7 now only in r7c8
52c. -> no 3 in r7c8

53. no 6 in r9c5 (due to {124} unavailable in r9c6 and 1 unavailable in r8c5)

54. no 8 in r12c6 (no permutations for 16(3)r1c6 w/ 8 in r12c6)
54a. h13(3)c7 (step 28) blocks {16} combo in r12c6
54b. -> no 6 in r12c6 (no remaining permutations available for 16(3)r1c6)

55. 8 in c6 locked in n3(r1c8)
55a. -> no 8 in r3c7

marks pic after step 55a:




Para
I have no vision today. Can only see these 2 steps. Maybe better tomorrow evening.

56. 15(3) at R7C5 only has 6 in R7C6 -> R7C6: no 7

57. Hidden13(3) at R78C6 + R9C6 = [913]/{15}[7]/{25}[6]/[625] : {3/5...}/{3/6/7}/{1/6/9}
57a. 16(3) at R1C6: {358} blocked, R12C6 would be {35}, blocked by hidden 13(3)
57b. 16(3) at R1C6: {367} blocked, R12C6 would be {37}, R1C7 = 6 -> R3C6 = 6(hidden single), blocked by hidden 13(3)
57c. 16(3) at R1C6: {169} blocked, R12C6 = {16}, R1C7 = 9 -> R34C6 = {89} or R12C6 = {19}, R1C7 = 6 -> R3C6 = 6, either way would force {169} in C6, but blocked by hidden 13(3)
57d. Conclusion: 16(3) at R1C6 = {17}[8]/{57}[4]/{349}: no 6


Mike (mhparker)
Thanks Para.

Here's the next small batch of moves (still got that sinking feeling about this puzzle though ). Hopefully, one of us will have a "Eureka!" moment:

58. 11(3)r1c3 = {128/137/146/236/245}
58a. {137} and {146} both blocked by 16(3)r1c6 (step 57d)
58b. -> 11(3)r1c3 = {128/236/245}
58c. -> no 7, 2 locked for n2

59. Nishio: if r6c7 = 3, then
59a. 3 in n7(r5c4) forced into r9c8 (cannot go in r5c4 due to LoL r789)
59b. -> 3 in n3(r1c8) forced into c9
59c. but this would leave nowhere to place the 3 in n5(r3c9)
59d. -> no 3 in r6c7
59e. -> no 6 in r7c9 (step 16)

60. 18(3)r4c7 = {189/279/369/459/378/468/567}
60a. {459} blocked by r7c9
60b. -> only combo with 4 is {468}
60c. {68} only in r45c8 -> no 4 in r45c8

61. h15(3)r7 (at r7c789 = innies r789) = [384]/{19}[5]
61a. -> {519} blocked from 15(3)r4c8
(would require 9 in both r6c8 and r7c7, leaving nowhere to place 9 in c9)
61b. -> no 1 in r6c9

62. 15(3)r4c8 = {249/258/267/348/357/456}
62a. {258} blocked because of r789 i/o difference "pincer" (r7c9 = 5 -> r6c7 = 2)
62b. only other combo with 8 is {348}
62c. but when [348}, 8 must go in r6c9, otherwise clash w/ h15(3)r7 (step 61)
62d. Conclusion: no 8 in r6c8

63. 12(3)r7c3 = {138/147/237/246/345} ({156} blocked by 22(3)r5c3)
63a. either r7c9 = 5, or r7c789 = [384] -> {345} combo blocked for 12(3)r7c3
63b. Either way, no 5 in r7c34

New marks pic after step 63b:




Mike (mhparker)

TJK18 - Never-Ending Story

Found some more, although it doesn't make the puzzle any less impossible!

I can't even see how it can be solved using hypotheticals...
The reason for this pessimism is that it's possible to fill up over two-thirds of the grid with a near solution (consisting of almost all the wrong digits), including every cell outside of r12345c12345. Therefore, any contadiction of this "near-solution" must involve pulling in at least one cell from this region. Unfortunately, there's very little we have to go on there. The 2 locked in 11(3)r1c3 doesn't help (both solution and near-solution have 2 in the outies if doing LoL on c1234 or c12345), and the innie/outie difference on c12 (step 8) is too weak to have a significant short range effect in the absence of a massive hypothetical.


64. 21(3)r8c8 cannot contain both of {89}
64a. -> r456c9 must contain at least one of {89}
64b. -> 18(3)r4c7 cannot contain both of {89}
64c. -> no 1 in r4c7

65. 1 in n5(r3c9) now locked in 12(3)r3c9 -> not elsewhere in c9
65a. 12(3)r3c9 = {129/138/147/156}
65b. 8 only in r5c9 -> no 3 in r5c9

For reference, here's the near-solution I was talking about above:



Hopefully, someone can prove me wrong. Otherwise I'll be considering having "TJK18" engraved on my tombstone!

R.I.P.


Para
Hi

Just a little bit more. Only 2 digits eliminated in three steps.

66. 18(3) at R1C5 = {189/369/459/567}: {378/468} blocked by 16(3) at R1C6

67. Hidden13(3) at R78C6 + R9C6 = [913]/{15}[7]/[256]/[625]: [526] blocked by h15(3) at R7C789(needs one of {58}): R78C6 = [52] -> R7C5 = 8

68. LOL R89: R8C6789 + R9C789 = R5C3 + R6C23 + R7C1234: all different digits outies are in same house
68a. 21(3) at R8C8 = {579} -> 16(3) at R8C7 = {268} - > R7C7 = 1 -> R7C56 = {59}
68b. 21(3) at R8C8 = {678} -> h 15(3) in R7C789 = {19}[5]
68c. Conclusion: 5 in R7 locked for R7C569: R7C12: no 5


Mike (mhparker)
Another tiny increment:

65b. sub-step added (elimination of 3 in r5c9) - see corrected walkthrough above

...

69. r7c123456 = h30(6)r7 = {125679/234678}
69a. {59} only in r7c56
69b. -> no 1 in r7c6


Para

With this pace i think we are gonna go on till step 200.
But here's a bit more again.

70. 12(3) at R8C5 = {138/147/156/237/246/345}
70a. {345} blocked, this is how:
70aa. R9C6 = 3 -> R89C5 = {45} -> R7C5 = 5: 2 5's in C5
70ab. R9C6 = 5 -> R89C5 = {34} -> R7C5 = 7 -> 18(3) at R1C5 = {189} -> 16(3) at R1C6 = {457} -> R12C6 = {57}: 2 5's in C6
70b. 12(3) at R8C5 = {138/147/156/237/246}: only place for 1 is R9C5 -> R9C5: no 5
70c. 12(3) at R8C5 = {138/147/156/237/246}: {1/2...}

71. 12(3) at R8C2: {129} blocked by 12(3) at R8C5 -> no 9 in 12(3) at R8C2
71a. 9 in N9 locked in 21(3) at R8C4 = {489/579}: no 6; {4/5...}/{4/7...}
71b. 12(3) at R8C2 = {138/156/237/246}: {147/345} blocked by 21(3) at R8C4
71c. 12(3) at R8C5 = {138/156/237/246}: {147} blocked by 21(3) at R8C4


Richard (rcbroughton)
I share Mike's pessimism. This one is a stinker.

72. 45 rule on c9 - innies total 33
72a. 2 in r6c9 means r7c9=4 - r1289c9=27 = {37}{89} or {57}{69}
72b. {37}{69} - blocked because can't make cage sum in 21(3)r8c8
72c. {57}{69} - blocked because can't make cage sum in 12(3)r1c8

73. 15(3)r6c8 - only combo with 9 is {249} - no 9 at r6c8

74. 45 rule on r1234 r5c58 = r34c9 + 5
74a. Innies total 10,8,7,6,5,4,3 - outies total 15,13,12,11,10,9,8
74b. 10 - 15 - requires 2 9's in nonet r3c9
74c. 8 - 13 - can't have 13=[67] in outies
74d. 7 - 12 - can't have {66} in outies
74e. 6 - 11 - can't have 6 in outies because need 6 in r5c9
74f. 5 - 10 - no 6 at r5c5 (missing 4 at r5c8)
74g. 4 - 9 - outies can't be [63] as it makes two 3's in nonet r3c9
74h. 3 - 8 - outies can't be [62] as it makes two 2's in nonet r39
74i. (phew) no 6 at r5c5

Lots of work for not very much


Para

Just a little bit....

75. 15(3) in R6C8 = {249/348/357/456}(needs 4 or 5 in R7C9): {4/7...}
75a. 12(3) in R3C9 = {129/138/156}({3/6/9}: {147} blocked by 15(3) -->> no 4,7
75b. 18(3) in R4C7 = {279/378/468/567}: {369} blocked by 12(3)

76. LOL R789: R5C467 + R6C4567 = R7C12349 + R8C13: All cells in innies in same house except R7C9, which contains only {45} so only {45} can appear twice in the outies.
76a. 12(3) at R5C6 = {246/345} = {3/6..}
76b. 14(3) at R5C4 = {158/167/347}: {356} blocked by LOL R789 + 12(3) at R5C6

77. R5C4: no 5, this is how.
77a. R5C4 = 5 -> R6C45 = {18} -> R6C7 = 2 -> 12(3) at R5C6 = {345} except no room left for 5 in 12(3)


Mike (mhparker)

TJK18: The next action-packed episode

More candidates than usual gone this time. Fun, isn't it? .

78. 18(3)r2c1 = {189/369/378/459/468}
78a. {459} blocked. Here's how:
78b. LoL r123: r2c1+r3c129 = r4c456+r5c5
78c. outies include 11(3) cage
78d. -> 3 of the digits in innies must add up to 11
78e. -> if r3c9 = 1, 2 of 18(3)r2c1 must sum to 10 and...
78f. ...if r3c9 = 2, 2 of 18(3) must sum to 9
78g. now, if 18(3) = {459}, then 14(3)r4c1 = {167} (only non-conflicting combo)
78h. -> r3c9 = 1 (hidden single c9)
78i. but this contradicts assertion in step 78e
78j. -> 18(3)r2c1 <> {459}
78k. -> 18(3)r2c1 = {189/369/378/468} (no 5) = {(3/8)..}
78l. -> {238} combo blocked from 13(3)r5c1

79. LoL r123 (see step 78b): no 5 in innies -> no 5 in outies
79a. -> no 5 in 11(3)r4c4, no 7 in r6c5 (step 17)
79b. -> 11(3)r4c4 = {128/137/146/236} = {(1/6)..}
79c. -> {169} combo blocked for 16(3)r2c3

80. no 4 in r2c1 (requires {68} in r3c12 -> conflict w/ r3c67)

81. r3c679 (innies r123) = h16(3)r3 = {169/268} = {(1/8)..}
81a. -> r3c12 <> {18}
81b. -> no 9 in r2c1

82. LoL r12: r1c89+r2c1789 = r3c345+r4c45+r5c5
82a. no 9 in innies -> no 9 in outies r3c345

83. 9 in c4 locked in n9
83a. -> no 9 in r9c3

84. 18(3)r1c5 = {189/369/459/567} (step 66)
84a. {69} only in r12c5
84b. -> no 3 in r12c5

85. no 2 in r2c3 (would require one of {69} in r3c34 - unavailable)

86. CPE: r7c4 sees all 7's in n7(r5c4)
86a. -> no 7 in r7c4

87. no 1 in r7c3 (would require 8 in r7c4 -> clash w/ r7c789, which needs 1 of {18} (step 61))

88. 14(3)r5c4 = {158/167/347} (step 45b)
88a. {17} only in r56c4
88b. -> no 6 in r56c4

89. {345} combo blocked from r456c4 (innies c1234) = h(12)c4. Here's how:
89a. 5 only within 14(3)r5c4 -> 14(3) = {158}
89b. -> can't get 2 of {345} in r56c4
89c. -> h(12)c4 = {138/147/156/237} ({246} blocked because {26} only in r4c4)
89d. -> h(12)c4 = {(1/2)..}
89e. -> r12c4 <> {12}
89f. -> no 8 in r1c3

New marks pic after step 89f:




Para

Morning all, here's last nights moves from when internet went down here. That's 4 digits in 6 moves.

90. 18(3) at R2C1 = {189/369/378/468} = {1/6/7..}
90a. 14(3) at R4C1: {167} blocked by 18(3)

91. 14(3) at R4C1: {356} blocked; Here's how:
91a. 14(3) = {356}: R4C45 and R4C78 need {47} but can't contain both
91aa. 11(3) at R4C7 = {137}({146 blocked cause no 6 available) -> R5C5 = 3; R4C4 = 7; R4C3 = 1-> R6C5 = 3(step 17): 2 3's in C5
91b. 14(3) = {149/158/239/248/257/347}: no 6

92. 14(3) at R4C1: {347} blocked; here's how
92a. 14(3) = {347} -->> R2C7 = 7(hidden single) -->> R23C8 = {12} -->> naked pair {12} in R3C89(nowhere else in R3), which leaves no options for 18(3) at R2C1
92b. 14(3) = {149/158/239/248/257}: {1/2..}

93. R5C5: no 8
93a. R5C5 = 8 -> R4C45 = {12}: clashes with 14(3) cage at R4C1

94. outies R1234: R5C589 = 17: R8C59 = 15/13/12/11/10/9/8, so R8C8: no 3

95. 14(3) at R4C1: {158} blocked; this is how
a.14(3) = {158} -> R4C6 = 9 -> R3C9 = 2("45" R123): no place for 1 in C9
b. 14(3) = {149/239/248/257} = {2/4..}
c. 13(3) at R5C1: {247} blocked by 14(3)

So at a pace of eliminating an average of 1 digit per move how many moves do we have to make before we are done?


sudokuEd

First: a very big thankyou to Mike for getting around my mistake way back forever ago. A really neat move it was too. Para noticed the same mistake. Thanks to you both.

I admire everyones perserverence with this. I'm glad i can finally add some more.

96. no 6 r6c5. Here's how.
96a. "45" c1234: 3 outies r456c5 = 13 = h13(3)r4c5
96b. = {148}/{17}[5]/{238}/[625]/[634]/[643] ({{27}[4]/{34}[6} blocked by combo's in 11(3))
96c. -> no 6 r6c5
96d.-> no 4 r4c4 (I/O c1234)

97. 14(3)r5c4 = {158/347}

98. {258} combo. blocked from 15(3)r7c5. Forces 3 into both 14(3) and 16(3) in this nonet.
98a. 15(3)r7c5 = {159/267}(no 8)
98b. -> no 6 r9c6 (I/O c6789)
98c. 15(3)r7c5:6 only in r7c6 -> no 2 r7c6

99. 12(3)r8c5 = {138/156/237}(no 4)({246} blocked by r9c6)
99a. 1 only in r9c5 -> no 5 r8c5

100. from step 67. Hidden13(3) at R78C6 + R9C6 = [913]/[517]/[625]
100a. -> r7c5 + r9c6 =
i.[5][3]
ii.[9][7]
iii.[7][5]
100b. -> r789c5 =
i.[581]/[5]{27}
ii.[9]{23}
iii.[761]
100c. = [2/5/6..]

101. from step 96b. h(13)r456c5 = {148}/{17}[5]/{238}/[634]/[643] ([625] blocked by r789c5)(step 100c.)

102. {17}[5] blocked from r456c5. Here's how.
102a. r456c5 = {17}[5] forces 9 into both r789c5 (step 100b.) and 18(3)r1c5
102a. h(13)r456c5 = {148}/{238}/[634]/[643] (no 5 or 7 r456c5)
102b. -> no 3 r4c4 (I/O c1234)

103. 14(3)r5c4 = {158/347}.
103a. {15} only in r56c4 -> no 8 r56c4

104. 12(3)r8c5 = {138/156/237} = [1/3..]
104a. -> 12(3)r8c2 = {156/237/246}(no 8) ({138} blocked by other 12(3))

105. 11(3)r4c4 = {128/146/236} = [6/8..]
105a. -> {268} blocked from 16(3)r2c3
105b. -> {468} blocked from 18(3)r1c1

106.from step 89c. h(12)r456c4 = [1]{47}/[615]/[2]{37}
106a. no 1 r6c4

107. weak links on 1s in r7 & 12(3)r6c7
107a. -> no 1 r8c1 (1 in r6c7 -> 1 in r7 in n6r4c3 -> no 1 in 8c1: 1 in r7c1 -> no 1 in r8c1))

108. 11(3)r7c1 {128} blocked: r7c12 = {18} -> 12(3)r6c7 = {138}:but 2 8s r7
108a. = {137/146/236/245}(no 8) = [3/4..]

109. 12(3)r7c3 = {138/147/237/246}(no 5)({345} clashes with 11(3) step 108a)
i. {138}
ii. [714] only. 7 in r7c3 -> 15(3) r7c5 = {159} with 1 in r8c6 -> no 1 in r8c3
iii. [7]{23}
iv. {26}[4]. 6 in r7 -> 15(3)r7c5 = {59}[1] -> r7c9 = 4 -> no 4 possible in r7c34
109a. -> no 4 r7c34





Mike (mhparker)

Light at the end of the tunnel

Thanks Ed, that made a big difference.

A couple of overlooked moves first...

110. no 5 in r8c6 (from permutations for h13(3) listed in step 100)

111. 2 in r7 locked in n6(r5c3)
111a. -> no 2 in r8c13

Now to get down to business...

112. LoL r123 (see step 78b): no 7 in outies -> no 7 in innies r2c1+r3c12
112a. -> 18(3)r2c1 = {189/369/468} = {(4/9)..}
112b. 6 only in r2c1 -> no 3 in r2c1

113. 14(3)r4c1 = {149/239/248/257} (step 95b)
113a. {149} blocked by 18(3)r2c1 (step 112a)
113b. -> 14(3) = {239/248/257} (no 1)
113c. 2 locked for r4 and n4

114. h12(3)r456c4 (step 106) = {147/156} (no 3)
114a. 1 locked for c4

115. CPE: r5c5 sees all 1's in c4
115a. -> no 1 in r5c5

116. LoL r1234: r2c1+r3c12+r4c123 = r5c589+r6c89+r7c9
116a. no 1 in outies -> no 1 in innies r3c12

117. 18(3)r2c1 = {369/468}
117a. 6 locked, only in r2c1
117b. -> r2c1 = 6 \:D/

117 moves to make a placement! (never would have thought first to go would be a cell in the top-left of the grid). Time for a handover (marks pic to follow).


Para

Ok a bit of overlap with Mike. Here is what he left out that i did get.

118.First some clean up: R6C2: no 5(i/o C12); R6C5: no 4(I/O c1234)

119. 11(3) in R4C4 = {128/146/236}: 2 only in R5C5 -> R5C5: no 3
119a. Needs one of {24}: has to go in R5C5 -> R4C5: no 4

120. 14(3) at R5C4 = {158/347}; R6C5 needs {3/8} so nowhere else in 14(3) -> R56C4: no 3

More later


Para

Here's some more. I can't see that 6 opening things up but i might be overlooking something.

121. 6 in N2 locked for R1
121a. 6 in N1 locked for R4
121b. 6 in N1 locked in 11(3) at R4C3 -->> 11(3) = {164/632}: no 8; {3/4..}
121c. Clean up: R5C9: no 5

122. R3C12 = {39/48}: {8/9..}
122. Killer Triple {689} in R3C1267: locked for R3

123. 18(3) at R1C5 = {189/369/459/567}
123a. {89} only in R12C5 -> R12C5: no 1
123b. 6 only in R1C5 -> R1C5: no 7

124. 16(3) at R2C3 = {178/259/358/457}
124a. {89} only in R2C3 -> R2C3: no 1,3

125. colouring on 1's from C4: R3C5 <> 1
125a. R4C4 = 1 -> R3C9 = 1: R3C5 <> 1
125b. R5C4 = 1 -> R1/2C6 = 1: R3C5 <> 1

126. one more colouring 1's from C4: R9C1 <> 1
126a. R4C4 = 1 -> R9C5 = 1: R9C1 <> 1
126b. R5C4 = 1 -> R6C1 = 1: R9C1 <> 1

127. 1 in 12(3) at R8C2 only in R9C2 -->> R9C2: no 5

128. "45" R1234 : R5C589 = 17 = [2]{69}/[278]/[458]/[476]: R5C8: no 2,8; R5C9: no 2
128a. 12(3) at R3C9 = [219/156/138]: R4C9: no 9

Time for other things right now. Maybe later tonight some more. Leave something for me


Richard (rcbroughton)

A couple of quick ones

129. 2 in n5 (r3c9) only in r3c9/r6c8 - > no 2 in r3c8

130. from 125b. 18(3) r1c5 - no 1, so {189} no longer valid. No 8 r12c5

131. 8 in c5 locked at r6r8 -> CPE no 8 at r8c7

132. 10(3) ar r2c7 = {127}/{145}/{235} (6 has gone)
132a. 2 only at r2c8 -> no 3,7 r2c8

133. 12(3) r3c9 = [219]/[138]/[156] - > no 9 at r4c9

134. 18(3)r4c7 ={378}/{468}/{567} - no 9
134a. 8 only at r4c8 -> no 3 at r4c8
134b. 6 only at r5c8 -> no 5 at r4c8

135. 9 now locked at r56c9 for c9 - nowhere else in c9


Richard (rcbroughton)

Just had to post this one - might just have cracked it!!

136. 45 on r4. Innies r4c456789 = 31
only combinations are:
{135679}, {145678}. {134689}
(can't place {235678}, {234679}. {234589}, {125689}. {124789})
136a. {135679}/{134689} - 9 must be at r4c6
136b. {145678} - r4c45={16} r4c6={8} r4c9={5} - so r4c78={47} but it can't be {47} as you can't make 18(3).
136c. r4c6 = 9
136d. r3c7= 6
136e. r3c6 = 8


Mike (mhparker)

Thanks, Richard!

Unfortunately, you beat me to it with your last move. Here's another way of peeling the proverbial onion:

136. r4c789 cannot contain both of {45} due to r7c9
136a. only other place for {45} on r4 is 14(3)r4c1
136b. -> 14(3)r4c1 = {(4/5)..} = {248/257} (no 3,9)
136c. -> hidden single in r4 at r4c6 = 9
136d. -> r3c67 = [86]

As requested, I think we should leave the rest for Para...


Para

You could at least have provided me with a marks pic .


Para

I get to do all singles, i am so happy. I think i can handle that.

137. I/O dif R123: R3C9 = 2 -> R45C9 = [19]
137a. R4C45 = [63]; R5C5 = 2; R6C5 = 8
137b. R56C4 = [15](last possible combi); R789C6 = [625]; R7C5 = 7
137c. R89C5 = [61]; R7C13 = [51](hidden 8(3) cage); R6C1 = 1(hidden)
137d. R7C7 = 2; R7C9 = 5(hidden); R9C9 = 6(hidden); R5C7 = 5(hidden)
137e. R1C6 = 8(hidden 19(3) in C7); R3C8 = 1(hidden); R7C7 = 1(hidden); R7C8 = 9
137f. R1C3 = 6; R6C2 = 6; R5C8 = 6; R4C8 = 8; R8C9 = 8(all hidden)
137g. R4C7 = 4; R8C8 = 7; R6C89 = [37]; R56C6 = [34]; R56C3 = [79]
137h. R9C8 = 4; R9C3 = 8
137i. R2C2 = 8; R5C1 = 8; R7C4 = 8(all hidden); R5C2 = 4; R7C123 = [423]
137j. R8C247 = [349]; R9C12 = [27]; R9C47 = [93]; R4C123 = [752]
137k. R3C12 = [39]; R1C12 = [91]; R12C6 = [71]; R2C78 = [72]
137l. R12C4 = [23]; R12C9 = [34]; R1C8 = 5; R123C5 = [495]; R23C3 = [54]; R3C4 = 7

And we have this:

Just a quick post to pick up on one of the moves Ed made in the walkthrough...


Well spotted, Ed! Of course, what you found here was (to use the official jargon) an Alternating Inference Chain (AIC), consisting of both strong and weak links. To be more specific, it is a special case of an AIC referred to as a "grouped x-cycle" (grouped because at least one of the nodes in the chain consists of multiple cells, and x-cycle because the loop is based on the same digit ("x") all the way round).

This becomes more apparent if one includes a couple of nodes you omitted:

r6c1<>1 -> r6c7=1 -> r7c78<>1 -> r7c124=1

So either r6c1=1 or r7c124=1 (or both), allowing us to eliminate 1 from any common peer of both of these end nodes (only r8c1 in this case).

In Eureka notation, we would express this loop (and the associated elimination) as:

(1): r6c1=r6c7-r7c78=r7c124 => r8c1<>1

Because (being an x-cycle) the same digit is used throughout, it is only listed once at the beginning of the expression.

Prelims

a) 11(3) cage at R1C3 = {128/137/146/236/245}, no 9
b) 10(3) cage at R2C7 = {127/136/145/235}, no 8,9
c) 23(3) cage at R3C6 = {689}
d) 11(3) cage at R4C4 = {128/137/146/236/245}, no 9
e) 22(3) cage at R5C3 = {589/679}
f) 11(3) cage at R7C1 = {128/137/146/236/245}, no 9
g) 21(3) cage at R8C4 = {489/579/678}, no 1,2,3
h) 21(3) cage at R8C8 = {489/579/678}, no 1,2,3

Steps resulting from Prelims
1a. 23(3) cage at R3C6 = {689}, locked for NR1C8
1b. 22(3) cage at R5C3 = {589/679}, 9 locked for NR5C3

2. 45 rule on R123 1 outie R4C6 = 1 innie R3C9 + 7 -> R4C6 = {89}, R3C9 = {12]
2a. 23(3) cage at R3C6 = {689}, 6 locked for R3
2b. 18(3) cage at R2C1 cannot be 1{89} which clashes with R3C67, ALS block -> no 1 in R2C1
2c. 18(3) cage at R4C7 cannot be {89}1 which clashes with R4C6 -> no 1 in R5C8

3. 45 rule on C12 1 innie R6C2 = 1 outie R4C3 + 4, R6C2 = {56789} -> R4C3 = {12345}

4. 45 rule on R789 1 innie R7C9 = 1 outie R6C7 + 3, no 7,8,9 in R6C7, no 1,2,3 in R7C9

5. 45 rule on C789 3 innies R135C7 = 19 = {289/379/469/478/568}, no 1

6. 45 rule on C1234 1 outie R6C5 = 1 innie R4C4 + 2, no 8 in R4C4, no 1,2 in R6C5

7. 45 rule on C6789 1 outie R7C5 = 1 innie R9C6 + 2, no 1,2 in R7C5, no 8,9 in R9C6

8. 45 rule on R89 3 innies R8C136 = 8 = {125/134}, 1 locked for R8
8a. 1 in NR8C2 only in R9C1256, locked for R9

9. R4C6 = R3C9 + 7 (step 2) -> R3C9 + R4C6 = [18/29]
9a. Law of Leftovers (LoL) for R123 four outies 11(3) cage at R4C4 + R4C6 must exactly equal four innies 18(3) cage at R2C1 + R3C9
9b. R4C6 = {89} -> 18(3) cage must contain at least one of 8,9 so cannot be {567}
9c. If 18(3) cage contains 9 it cannot also contain 2 (because R3C9 + R4C6 = [29], LoL outies cannot contain two 2s -> LoL innies cannot contain two 2s)
9d. -> 18(3) cage at R2C1 = {189/369/378/459/468}, no 2
[Step 9c seems to be what Ed called Killer LoL in the ‘tag’ walkthrough.]

[I got stuck at this stage, so took a break and worked on some hard Assassin variants from Ruud’s site, from my backlog. When I came back to this puzzle I thought I’d found some useful LoLs but I’d got the first one wrong, so had to re-work again from here.]

10. R8C136 (step 8) = {125/134}
10a. LoL for R89 six outies 12(3) cage at R5C6 + 12(3) cage at R6C7 must exactly equal six innies R8C136 + 16(3) cage at R8C7
10b. All the outies are in NR5C6 -> all the innies are in an ‘effective hidden cage’
10c. The two 12(3) cages contain all 1,2,3 in NR5C6 -> R8C136 + 16(3) cage must contain all of 1,2,3
10d. R8C136 contains 1 and one of 2,3 -> 16(3) cage must contain one of 2,3 = {259/268/349/367} (cannot be {358} which clashes with R8C136, because R8C136 and 16(3) cage are in an ‘effective hidden cage’)

11. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7, no 1 in R1C7 + 16(3) cage -> no 1 in R56C6
11a. 1 in NR5C6 only in 12(3) cage at R6C7 = {129/138/156} (cannot be {147} which clashes with 21(3) cage at R8C8), no 4,7, clean-up: no 7 in R7C9 (step 4)

12. 12(3) cage at R5C6 = {237/246/345}, no 8,9
12a. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7 -> 16(3) cage (step 10d) = {268/349/367} (cannot be {259} because R56C6 cannot contain both of 2,5), no 5
12b. 15(3) cage at R7C5 cannot be {456} (which clashes with 16(3) cage at R8C7) -> each of the cages in NR5C4 must contain one of 1,2,3
12c. R135C7 (step 5) = {289/379/469/478/568}
12d. 2 of {289} must be in R5C7 -> no 2 in R1C7

13. LoL for C6789 four outies 14(3) cage at R5C4 + R7C5 must exactly equal four innies 16(3) cage at R1C6 + R9C6 -> whichever value is in R9C6 must be in 14(3) cage + R7C5 -> R9C6 effectively ‘sees’ all the cells of 16(3) cage at R8C7 using NR5C4 -> R789C6 ‘see’ all the cells of 16(3) cage at R8C7
[It’s likely that SudokuSolver can’t find the last parts of this step, which is also used for the following sub-steps.]
13a. 45 rule on C6789 3 innies R789C6 = 13 = {139/148/157/256} (cannot be {238/247/346} which clash with 16(3) cage at R8C7)
13b. R789C6 = 13 = {139/157/256} (cannot be {148} = [814] because 15(3) cage at R7C5 = [681] + R9C6 = 4 clash with 16(3) cage at R8C7, cannot be {148} = [841] because 15(3) cage = [384] clashes with 16(3) cage at R8C7), no 4,8, clean-up: no 6 in R7C5 (step 7)
13c. 3 of {139} must be in R9C6 (because 15(3) cage cannot be [393]) -> no 3 in R78C6
13d. 2 of {256} must be in R78C6 (R78C6 cannot be {56} because 15(3) cage at R7C5 cannot be {456}, step 12b) -> no 2 in R9C6, clean-up: no 4 in R7C5 (step 7)
13e. 16(3) cage at R8C7 (step 12a) = {268/349} (cannot be {367} which clashes with R789C6), no 7 in 16(3) cage
13f. 15(3) cage at R7C5 = {159/258/267/357} (cannot be {168} because R789C6 only contains one of 1,6)
13g. 14(3) cage at R5C4 = {149/158/167/347} (cannot be {248/356} which clash with 16(3) cage at R8C7, cannot be {257} which clashes with 15(3) cage), no 2

14. LoL for R789 seven outies 22(3) cage at R5C3 + R56C67 must exactly equal seven innies 15(3) cage at R7C5 + R7C9 + 16(3) cage at R8C7, 2 in NR5C4 only in 15(3) cage + 16(3) cage -> R56C67 must contain 2, locked for NR5C6

15. R8C136 (step 8) = {125/134}, 16(3) cage at R8C7 (step 13e) = {268/349}
15a. LoL for R89 six outies 12(3) cage at R5C6 + 12(3) cage at R6C7 must exactly equal six innies R8C136 + 16(3) cage at R8C7
15b. All the outies are in NR5C6 -> all the innies are in an ‘effective hidden cage’
15c. 15(3) cage at R7C5 (step 13f) = {159/258/267} (cannot be {357} = [375] because R8C136 = {12}5 clashes with 16(3) cage = {268} in the ‘effective hidden cage’), no 3, clean-up: no 1 in R9C6 (step 7)
15d. Killer pair 2,9 in 15(3) cage and 16(3) cage, locked for NR5C4, clean-up: no 7 in R4C4 (step 6)
15e. 15(3) cage = {159} must be {59}1 (cannot be [915] because R8C136 = {12}5 clashes with 16(3) cage = {268} in the ‘effective hidden cage’) -> no 1 in R7C6
15f. R789C6 (step 13b) = {139/157/256}
15g. 7 of {157} must be in R9C6 (cannot be [715] because 15(3) cage cannot be [771]), no 7 in R7C6

16. LoL for R789 seven outies 22(3) cage at R5C3 + R56C67 must exactly equal seven innies 15(3) cage at R7C5 + R7C9 + 16(3) cage at R8C7, 22(3) cage contains 9, 9 in NR5C4 must be in 15(3) cage + 16(3) cage, there cannot be another 9 in the innies -> no 9 in R7C9, clean-up: no 6 in R6C7 (step 4)

17. LoL for C6789 four outies 14(3) cage at R5C4 + R7C5 must exactly equal four innies 16(3) cage at R1C6 + R9C6, no 2 in 14(3) cage + R7C5 -> no 2 in 16(3) cage

18. LoL for R89 six outies 12(3) cage at R5C6 + 12(3) cage at R6C7 must exactly equal six innies R8C136 + 16(3) cage at R8C7, no 7 in innies -> no 7 in 12(3) cage at R5C6
18a. 12(3) cage at R5C6 (step 12) = {246/345}, 4 locked for NR5C6
18b. 12(3) cage at R6C7 (step 11a) = {129/138} (cannot be {156} which clashes with 12(3) cage at R5C6), no 5,6, clean-up: no 8 in R7C9 (step 4)

19. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7, no 7 in R3456C6 -> no 7 in R1C7
19a. R135C7 (step 5) = {289/469/568}, no 3

20. 12(3) cage at R5C6 (step 18a) = {246/345}
20a. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7
20b. Consider combinations for 16(3) cage (step 13e) = {268/349}
16(3) cage = {268} => R56C6 = {26} (cannot be {25} because 12(3) cage cannot contain both of 2,5)
or 16(3) cage = {349} => R56C6 = {34}
-> no 5 in R56C6, no 5 in R1C7 (from LoL)
20c. R56C6 = {26/34} -> R5C7 = {45}
20d. R135C7 (step 19a) = {469/568}, 6 locked for C7
20e. R5C7 = {45} -> no 4 in R1C7
20f. 16(3) cage = {268/349}
20g. 6 of {268} must be in R9C8 -> no 2,8 in R9C8

21. LoL for C89 five outies R24C7 + 23(3) cage at R3C6 must exactly equal five innies R79C8 + 24(3) cage at R8C8, no 2 in innies -> no 2 in R24C7
21a. 23(3) cage contains 8 -> innies must contain 8 in R7C8 + 21(3) cage, locked for NR5C6
21b. 12(3) cage at R6C7 (step 18b) = {129/138}
21c. 8 of {138} must be in R7C8 -> no 3 in R7C8

22. R135C7 (step 20d) = {469/568}, 12(3) cage at R5C6 (step 18b) = {246/345}
22a. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7
22b. Consider combinations for 16(3) cage (step 13e) = {268/349}
16(3) cage = {268} => R89C7 = {28}, locked for C7 => R13C7 = {69} => R5C7 = 4 => R56C6 = {26}, locked for C6 => R34C6 = {89}, locked for 23(3) cage at R3C6 => R3C7 = 6
or 16(3) cage = {349} => R56C6 = {34} => R5C7 = 5 => R13C7 = {68}
-> no 9 in R3C7
22c. 23(3) cage at R3C6 = {689}, 9 locked for C6
22d. 15(3) cage at R7C5 (step 15c) = {159/258/267}
22e. 7,8,9 only in R7C5 -> R7C5 = {789}, clean-up: no 3 in R9C6 (step 7)
22f. Min R9C6 = 5 -> max R89C5 = 7, no 7,8,9 in R89C5
[Other eliminations from R9C5 are done in step 24a.]

23. R789C6 (step 13b) = {157/256}, 5 locked for C6
23a. 16(3) cage at R1C6 = {178/349/367} (cannot be {169) = {16}9 which clashes with R789C6)
23b. R1C7 = {689} -> no 6,8 in R12C6
23c. R34C6 = {89} (hidden pair in C6) -> R3C7 = 6 (cage sum)
23d. 16(3) cage = {178/349}
23e. 11(3) cage at R1C3 = {128/236/245} (cannot be {137/146} which clash with 16(3) cage), no 7, 2 locked for NR1C3
23f. 18(3) cage at R1C5 = {369/459/567} (cannot be {189/378/468} which clash with 16(3) cage), no 1,8
23g. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7, 8 in R34C6 -> 8 in R1C7 + R89C7, locked for C7

24. 45 rule on C6789 3 outies R789C5 = 14 = {149/158/167/239/248} (cannot be {257} because 12(3) cage at R8C5 cannot be {25}5, cannot be {347} which clashes with 18(3) cage at R1C5, cannot be {356} because R7C5 only contains 7,8,9)
24a. 1 of {158/167} must be in R9C5 -> no 5,6 in R9C5
24b. 12(3) cage at R8C2 = {138/147/156/237/246/345} (cannot be {129} which clashes with R89C5), no 9
24c. 9 in NR8C2 only in 21(3) cage at R8C4 = {489/579}, no 6
24d. 12(3) cage at R8C2 = {138/156/237/246} (cannot be {147/345} which clash with 21(3) cage)
24e. 8 in NR8C2 must be in 12(3) cage at R8C2 = {138} or 21(3) cage = {489} -> 12(3) cage at R8C2 = {138/156/237} (cannot be {246}, locking-out cages), no 4
24f. 12(3) cage at R8C5 = {156/237/246} (cannot be {147/345} which clash with 21(3) cage)
24g. R789C5 = {158/167/239/248} (cannot be {149} because 12(3) cage at R8C5 cannot contain both of 1,4)

25. 45 rule on C1234 3 outies R456C5 = 13 = {148/157/238/247} (cannot be {256} which clashes with R789C5, cannot be {346} which clashes with 18(3) cage at R1C5), no 6, clean-up: no 4 in R4C4 (step 6)
25a. 5 of {157} must be in R6C5 (R45C5 cannot be {15} because 11(3) cage at R4C4 cannot be 5{15}), no 5 in R45C5

26. R789C5 (step 24g) = {158/167/239/248}, 16(3) cage at R8C7 (step 13e) = {268/349}
26a. Consider combinations for 12(3) cage at R8C5 (step 24f) = {156/237/246}
12(3) cage = {156/246}
or 12(3) cage = {237} => R89C5 = {23}, R7C5 = 9 => 16(3) cage = {268} => R9C8 = 6
-> 6 in 12(3) cage + R9C8, CPE no 6 in R9C12

27. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7
27a. A different LoL for R789 six outies 22(3) cage at R5C3 + 14(3) cage at R5C4 must exactly equal six innies R7C789 + 21(3) cage at R8C8
27b. 16(3) cage at R8C7 (step 13e) = {268/349}
27c. Consider combinations for 14(3) cage at R5C4 (step 13g) = {158/167/347}
14(3) cage = {158/167} => 16(3) cage = {349} (cannot be {268} which clashes with 14(3) cage) => R56C6 = {34} (LoL for C789), locked for NR5C6
or 14(3) cage = {347} => R7C7 = 3 (only position for 3 in innies of LoL for R789)
-> no 3 in R6C7, clean-up: no 6 in R7C9 (step 4)
[Taking this slightly further]
27d. 14(3) cage = {158/167} => R7C9 = 5 (no 4 in R7C9 from LoL for R789)
or 14(3) cage = {347} => 16(3) cage = {268} => 15(3) cage at R7C5 = {159} => R7C6 = 5
-> 5 in R7C69, locked for R7

28. 12(3) cage at R6C7 (step 18b) = {129/138}, 16(3) cage at R8C7 (step 13e) = {268/349}
28a. Consider combinations for R789C6 (step 23) = {157/256}
R789C6 = {157} = [517] (so R89C5 = 5, no 6 in R8C5), R7C5 = 9 (cage sum) => 16(3) cage = {268} => R9C8 = 6 => R8C2 = 6 (hidden single in NR8C2) => R9C15 = 6 = {15}, locked for R9, 12(3) cage at R6C7 = {138}, locked for NR5C6 => 21(3) cage at R8C8 = {579} => R9C9 = 9
or R789C6 = {256}, CPE no 6 in R9C8 => 16(3) cage = {349} => R56C6 = {34} (LoL for C789) => R5C7 = 5 (cage sum) => 21(3) cage at R8C8 = {678}
-> no 9 in R8C89, no 5 in R9C9
28b. 5 in R9 only in R9C12346, locked for NR8C2

29. LoL for C789 four outies R3456C6 must exactly equal four innies R1C7 + 16(3) cage at R8C7
29a. 12(3) cage at R6C7 (step 18b) = {129/138}, 16(3) cage at R8C7 (step 13e) = {268/349}
29b. Consider combinations for 12(3) cage at R5C6 (step 18a) = {246/345}
12(3) cage = {246} => R56C6 = {26} => 16(3) cage must contain {26} (LoL) = {268}, 8 locked for C7 => R1C7 = 9
or 12(3) cage = {345} => R56C6 = {34} => 16(3) cage must contain {34} (LoL) = {349}, 12(3) cage at R6C7 = {129}, X-Wing for 9 in 12(3) cage at R6C7 and 16(3) cage, no other 9 in C67
-> no 9 in R4C7

30. R789C5 (step 24g) = {167/239/248}
30a. R456C5 (step 25) = {148/157/238} (cannot be {247} which clashes with R789C5)
30b. 5 of {157} must be in R6C5 -> no 7 in R6C5, clean-up: no 5 in R4C4 (step 6)
30c. 14(3) cage at R5C4 (step 13g) = {158/347} (cannot be {167} because no 1,6,7 in R6C5), no 6
30d. 6 in NR5C4 only in R7C6 + R9C8, CPE no 6 in R9C6, clean-up: no 8 in R7C5 (step 7)
30e. Killer pair 7,9 in 18(3) cage at R1C5 and R7C5, locked for C5
30f. R456C5 = {148/238}, no 5, clean-up: no 3 in R4C4 (step 6)
30g. 14(3) cage at R5C4 = {158/347}
30h. 8 of {158} must be in R6C5 -> no 8 in R56C4
30i. 5 in C5 only in 18(3) cage at R1C5, locked for NR1C3
30j. 18(3) cage at R1C5 (step 23f) = {459/567}, no 3
30k. Killer pair 4,7 in 18(3) cage at R1C5 and 18(3) cage at R1C6, locked for NR1C3

31. 21(3) cage at R8C4 (step 24c) = {489} (cannot be {579} which clashes with R9C6), locked for NR8C2

32. 6 in NR8C2 only in R8C25, locked for R8
32a. 21(3) cage at R8C8 = {579/678}
32b. 6,9 only in R9C9 -> R9C9 = {69}
32c. 7 in NR5C6 only in R8C89, locked for R8

33. R789C6 (step 23) = {157/256} = [517/625] -> R7C6 = {56}, R8C6 = {12}
33a. 2 in R7 only in R7C1234, locked for NR5C3

34. 14(3) cage at R5C4 (step 30c) = {158/347}
34a. Consider combinations for R789C5 (step 24g) = {167/239}
R789C5 = {167} => R7C5 = 7 => 14(3) cage = {158} => R6C5 = 8
or R789C5 = {239}, 3 locked for C5
-> no 3 in R6C5, clean-up: no 1 in R4C4 (step 6)
34b. 14(3) cage at R5C4 = {158/347}
34c. R6C5 = {48} -> no 4 in R56C4

35. Consider combinations for 21(3) cage at R8C8 = {579/678}
21(3) cage = {579} => R9C9 = 9 => R8C4 = 9 (hidden single in NR8C2)
or 21(3) cage = {678}, 8 locked for R8
-> no 8 in R8C4
35a. 8 in NR8C2 only in R9C34, locked for R9
35b. 16(3) cage at R8C7 (step 13e) = {268/349}
35c. 8 of {268} must be in R8C7 -> no 2 in R8C7

36. LoL for R123 four outies 11(3) cage at R4C4 + R4C6 must exactly equal four innies 18(3) cage at R2C1 + R3C9, no 5,7 in outies -> no 5,7 in 18(3) cage -> 18(3) cage = {189/369/468}
36a. {189} must be 8{19} because R34C6 must be [89] to provide 9 for outies, 6 of {369/468} must be in R2C1 -> R2C1 = {68}
36b. Killer pair 8,9 in 18(3) cage and R3C7, locked for R3
36c. Max R3C45 = 12 -> min R2C3 = 4
36d. 9 in C4 only in R89C4, locked for 21(3) cage at R8C4, no 9 in R9C3

37. LoL for C1234 five outies R45C5 + 12(3) cage at R8C5 must exactly equal five innies 11(3) cage at R1C3 + R56C4, no 4 in innies -> no 4 in R45C5
37a. 11(3) cage at R4C4 = {128/236}, 2 locked for NR1C1

38. LoL for R12 five outies R3C58 + 23(3) cage at R3C6 must exactly equal five innies 18(3) cage at R1C1 + R2C13, no 2 in innies -> no 2 in R3C8
38a. R3C9 = 2 (hidden single in R3), placed for NR3C9 -> R4C6 = 9 (step 2), R3C6 = 8
38b. R3C9 = 2 -> R45C9 = 10 = [19]/{37/46}, no 5,8, no 1 in R5C9

39. 9 in R3 only in R3C12, locked for NR2C1
39a. 18(3) cage at R2C1 (step 36) = {189/369}, no 4

40. 11(3) cage at R1C3 (step 23e) = {128/236}, 14(3) cage at R5C4 (step 34b) = {158/347}
40a. LoL for C1234 five outies R45C5 + 12(3) cage at R8C5 must exactly equal five innies 11(3) cage at R1C3 + R56C4
40b. Consider combinations for 12(3) cage at R8C5 (step 24f) = {156/237}
12(3) cage = {156} => R89C5 = [61], R9C6 = 5, R45C5 = {23} (hidden pair in C5), R4C4 = 6 (cage sum), 5 must be in R56C4 (from LoL) = {15} => 11(3) cage at R1C3 = {236} => R1C3 = 6, R12C4 = {23}
or 12(3) cage = {237} => R89C5 = {23}, locked for C5 => R45C5 = {18}, R4C4 = 2 (cage sum) => 11(3) cage at R1C3 = {128} (from LoL) => R1C3 = 2, R12C4 = {18}, R56C4 = {37} (from LoL)
-> R1C3 = {26}, no 6 in R12C4, 2 in R12C4 + R4C4, locked for C4, 3 in R12C4 + R56C4, locked for C4
[Note. R1C3 must equal R4C4, but I can’t see how I can use this.]

41. 45 rule on R1234 3 outies R5C589 = 17 = {179/269/278/359/368} (cannot be {458} which clashes with R5C7, cannot be {467} because no 4,6,7 in R5C5), no 4, clean-up: no 6 in R4C9 (step 38b)

42. Hidden killer pair 4,5 in 14(3) cage and R4C789 for R4, 14(3) cage cannot contain both of 4,5 -> R4C789 must contain one of 4,5
42a. Killer pair 4,5 in R4C789 and R7C9, locked for NR3C9
42b. 15(3) cage at R6C8 = {159/348/357} (cannot be {168} which doesn’t contain 4 or 5, cannot be {456} because 4,5 only in R7C9), no 6

43. 16(3) cage at R8C7 (step 13e) = {268/349}
43a. Consider placement for 6 in NR3C9
6 in R45C8, locked for C8 => 16(3) cage = {349}
or 6 in R5C9 => 12(3) cage at R3C9 = [246] => R7C9 = 5 => R7C6 = 6, placed for NR5C4 => 16(3) cage = {349}
-> 16(3) cage = {349}, locked for NR5C4 -> R6C5 = 8, R56C4 = 6 = {15}, locked for C4 and NR5C4, R7C5 = 7, R78C6 = [62]
43b. Naked pair {34} in R56C6, locked for C6 and NR5C6 -> R5C7 = 5, placed for NR5C6, R56C4 = [15]
[Cracked. The rest is straightforward.]

44. 11(3) cage at R4C4 (step 37a) = {236} (only remaining combination) -> R4C4 = 6, placed for NR1C1, R45C5 = {23}, locked for C5 and NR1C1, R89C5 = [61], R9C6 = 5 (cage sum), R8C2 = 3

45. R6C7 = 2 (hidden single in C7) -> R7C78 = 10 = {19}, locked for R7 and NR5C6 -> R9C9 = 6, clean-up: no 4 in R5C9 (step 38b)

46. R7C9 = 5 (hidden single in R7)
46a. 18(3) cage at R4C7 = {468} (hidden triple in NR3C9) -> R4C7 = 4, R45C8 = [86]

47. R89C7 = [93], R9C8 = 4, R7C78 = [19], R2C7 = 7, R12C6 = [71], R1C7 = 8, placed for NR1C3
47a. R2C7 = 7 -> R23C8 = 3 = [21]
47b. R3C2 = 9, R3C1 = 3, placed for NR2C1, R2C1 = 6 (cage sum)

48. R8C4 = 4, R3C4 = 7 -> R23C3 = 9 = {45}, locked for C3 and NR1C1

49. 14(3) cage at R4C1 = {257} (only remaining combination) -> R4C3 = 2, R4C12 = {57}, locked for R4 and NR2C1, R45C5 = [32], R4C9 = 1 -> R5C9 = 9 (cage sum)

and the rest is naked singles, without using the nonets.

SumoCueV1=17J0=10J0+1J0=15J0+3J1+3J2=7J2+6J2=14J2+0J0+0J0=7J0=12J1+12J1+12J1=15J2+8J2+8J2=8J0+18J0+11J3=3J3=22J1=9J4+15J4=12J2+25J2=15J5=12J3+28J3+21J1+22J1+23J1=8J4+33J4=21J6+27J5=15J3=11J3+38J7+22J1=9J7+41J4=11J4+35J6+27J5+37J3=15J7+47J7=6J7=13J7+50J7+43J4+35J6=16J5=14J3=18J3+47J7+49J8+50J7=16J4=15J4=9J6+54J5+55J5+56J8+56J8=4J8+60J8+60J8+61J6+62J6+54J5+55J5=12J5+74J8+67J8=14J8+77J6+61J6+62J6

Prelims

a) R1C23 = {19/28/37/46}, no 5
b) R1C78 = {16/25/34}, no 7,8,9
c) R23C3 = {16/25/34}, no 7,8,9
d) R23C7 = {69/78}
e) R3C12 = {17/26/35}, no 4,8,9
f) R34C4 = {12}
g) R34C6 = {18/27/36/45}, no 9
h) R3C89 = {39/48/57}, no 1,2,6
i) R4C23 = {39/48/57}, no 1,2,6
j) R4C78 = {17/26/35}, no 4,8,9
k) R56C2 = {69/78}
l) R5C34 = {29/38/47/56}, no 1
m) R5C67 = {18/27/36/45}, no 9
n) R56C8 = {29/38/37/46}, no 1
o) R67C5 = {15/24}
p) R89C5 = {13}
q) R9C34 = {39/48/57}, no 1,2,6
r) R9C56 = {59/68}
s) 22(3) cage at R3C5 = {589/679}
t) 21(3) cage at R4C9 = {489/579/678}, no 1,2,3
u) 9(3) cage at R7C9 = {126/135/234}, no 7,8,9

1. Naked pair {13} in R89C5, locked for C5 and NR7C5, clean-up: no 5 in R67C5, no 9 in R9C3
1a. Naked pair {24} in R67C5, locked for C5
1b. 22(3) cage at R3C5 = {589/679}, 9 locked for C5 and NR1C5
1c. Naked quint {56789} in R12345C5, locked for NR1C5, clean-up: no 1,2,3,4 in R3C6
1d. Naked pair {12} in R34C4, locked for C4, clean-up: no 9 in R5C3

2. 45 rule on R12 2 innies R2C37 = 15 -> R2C3 = 6, placed for NR1C1, R2C7 = 9, placed for NR1C6, R3C3 = 1, R3C4 = 2, both placed for NR3C3, R3C7 = 6, placed for NR3C6, R4C4 = 1, placed for NR1C5, clean-up: no 4,9 in R1C2, no 4 in R1C3, no 1 in R1C8, no 7 in R3C12, no 8 in R3C6, no 3 in R3C89, no 3 in R4C6, no 2,7 in R4C78, no 5,9 in R5C4, no 3 in R5C6, no 5 in R56C8, no 5,8 in R9C6
2a. Naked pair {35} in R3C12, locked for R3 and NR1C1 -> R3C6 = 7, placed for NR3C6, R4C6 = 2, clean-up: no 7 in R1C23, no 2 in R5C7, no 4 in R56C8
2b. Naked pair {48} in R3C89, locked for R1 and NR1C6 -> R3C5 = 9, clean-up: no 3 in R1C78

3. Naked pair {35} in R4C78, locked for R4 and NR3C6, clean-up: no 7,9 in R4C23, no 4,6 in R5C6, no 8 in R56C8
3a. Naked pair {48} in R4C23, locked for R4 and NR3C3, clean-up: no 7 in R56C2, no 3,7 in R5C4
3b. Naked pair {69} in R56C2, locked for C2 and NR3C3

4. 22(3) cage at R3C5 = {679} (only remaining combination, cannot be {589} because 5,8 only in R5C5), locked for C5
4a. 12(3) cage at R2C4 = {345} (only remaining combination) -> R2C5 = 5, R2C46 = {34}, locked for R2, R1C5 = 8, R1C46 = 7 = [43], R2C46 = [34], clean-up: no 2 in R1C23, no 7 in R5C3, no 8 in R9C3
4b. R1C12 = [19], 1 placed for NR1C1, clean-up: no 6 in R1C8
4c. Naked pair {25} in R1C78, locked for R1 and NR1C6 -> R1C1 = 7, R1C9 = 6

5. 21(3) cage at R4C9 = {579} (only remaining combination, cannot be {489} which clashes with R3C9), locked for C9 and NR4C9 -> R2C9 = 1, R9C7 = 8, placed for NR4C9, R9C6 = 6, placed for NR7C5, clean-up: no 1 in R5C6, no 9 in R9C34
5a. Naked triple {234} in 9(3) cage at R7C9, locked for C9 and NR4C9 -> R9C8 = 1, R8C8 = 6, R7C8 = 8 (cage sum), R89C5 = [13]

6. R9C1 = 9 (hidden single in R9), R4C1 = 6, R45C5 = [76], R4C9 = 9, R56C2 = [96], R5C8 = 2, placed for NR3C6, R6C8 = 9, R5C4 = 8, placed for NR5C4, R5C3 = 3, placed for NR3C3, R5C6 = 5, placed for NR5C4, R5C6 = 4, R5C1 = 1, R6C9 = 8 (cage sum), placed for NR4C1, R67C6 = [19], R6C7 = 3 (cage sum), R6C4 = 7, R9C4 = 5, placed for NR7C5, R9C3 = 7, placed for NR4C1, R7C4 = 6, R6C3 = 2 (cage sum), R2C1 = 2

7. R9C1 = 9 -> R78C1 = 7 = {34}, locked for C1 and NR4C1 -> R9C2 = 2

and the rest is naked singles, without using the nonets.

SumoCueV1=11J0+0J0=15J0+2J0=7J1=7J2+5J2=11J2+7J2+0J0=13J0=12J0+11J0+4J1=15J2+14J2=9J2+7J2=18J3+10J3=10J0=17J1+21J1+21J1=21J2+16J4=14J4+18J3+18J3+20J3+20J1+21J5+24J1+24J4+26J4+26J4=7J3+36J3=18J5+38J1=9J5=18J1+41J5=14J4+43J4=7J3=15J3+38J5=7J5+40J5=5J5+41J5=15J4=14J4+45J6+46J6=18J7+48J7=17J7+50J7=15J7+52J8+53J8=19J6+46J6+56J6+56J7+58J7+60J7+60J8+52J8=13J8+63J6+63J6+56J6=14J6+75J7+75J8+60J8+71J8+71J8

I was wondering if anyone knows what this error message means:

System.NullReferenceException: Objet reference not set to an instance of an object.
at SumoCue.Cage.GetNonetNumbers()
at SumoCue.Cage.get_FullName()
at SumoCue.Solver.FoundUnplaceableDigitsInCage(Boolean Execute)
at SumoCue.Solver.SolveStep(Boolean Execute)

I got it when I got as far as I could and hit F8. Also happens if you hold down F11 on this puzzle.

After several times working around the no-hints, the puzzle became unsolvable. Haven't re-tried my logic yet, for the next reason - I've noticed in prior regular Assassin Killers that if you make an assumption and it's the wrong one, after the clearing the pop-up stating that a digit in a particular cell can't be eliminated, it will place the digit in the cell that caused the pop-up, and at this point, may or may not eliminate the markups in the R/C, but sometimes they do disappear after you click on and then off the cell. I've discovered that in most cases, you can't hit Ctrl+z and 'fix' the error, the puzzle just has to be restarted.

So if anyone can tell me if this error is a bug that makes the puzzle unsolveable, or just a nuisance message, you can clear it and hold F11 and get it again, at which there are no more steps (I found one after this point that made it unsoveable.

Maybe there's an error message topic? I guess I didn't look...

Turns out there's only a couple hard steps -- after that the whole puzzle falls quite easily.
...Or maybe not. I recall it being easier the time I wasn't writing a walkthrough.


Here's the layout for how I number the nonets:


1. Eliminate candidates from cages:
1a. 5/2 in C6 = {14|23}
1b. 7/2s in C1 and R5/N4 and C4 and C5/N2 and R1/N3 = {16|25|34}
1c. 9/2s in C5 and C8 <> 9
1d. 12/2 in R2/N1 = {39|48|57}
1e. 13/2 in C2 = {49|58|67}
1f. 14/2s in R5/N6 and C9 = {59|68}
1g. 15/2s in R1/N1 and R2/N3 = {69|78}
1h. 17/2 in C5/N8 = {89}
1i. 10/3 in R34C34 = {127|136|145|235}
1j. 11/3s in N1 and N3 <> 9
1k. 19/3 in N7 <> 1
1l. 21/3 in R34C67 = {489|579|678}
2. R78C5 = naked pair (89) on C5/N8 -> 9/2 in C5 <> 1
3. 9 of R1 locked in cage 15/2 -> no other 9 in N1, cage 15/2 in R1 = {69} naked pair in N1/R1
4. Outies of R1 = R2C159 = 6 = {123} naked triple in R2 -> 7/2 in C5/N2 = [43|52], 9/2 in C8 = [45|54|63|72|81], 12/2 in R2 = {48|57}
5. LOL -> N13=R12 -> R3C37 = 7 = [25|34] and furthermore R3C3=R2C5 and R3C7 = R1C5
6. Here's one big step.
6a. Combinations for 17/4 in R34C456: {1259|1268|1349|1358|1367|1457|2348|2357|2456}
6b. All cells in cage 17/4 in R34C456 can see either R3C3 or R2C5, these cells are either both 2 or both 3, so 17/4 in R34C456 cannot contain both 2 and 3. Remaining combinations: {1259|1268|1349|1358|1367|1457|2456}
6c. All cells in cage 17/4 in R34C456 can see either R3C7 or R1C5, these cells are either both 4 or both 5, so 17/4 in R34C456 cannot contain both 4 and 5. Remaining combinations: {1259|1268|1349|1358|1367|1457} -> there is a 1 in 17/4 in R34C456
7. 7/2 in R1 <> 1 (combinations)
8. 1s in R12 are locked in the two 11/3 cages -> those cages = {128|137}
9. 9 in R2 locked in cage 15/2 -> 15/2 in R2 = {69}
10. 4 and 5 of N1 locked in R2 -> no 4 or 5 in R2C8
11. Combinations for 9/2 in C8 = {72|81}
12. Innies for N1 = R2C2+R3C3 = 7 -> R2C2 = {45}
13. Outies for R12 = R3C28 = 10 -> R3C2 = {89}
14. LOL -> C12 = N47 -> 6 and 9 not in R489C3+R9C4
15. LOL -> C89 = N69 -> 6 and 9 not in R489C7+R9C6
16. Combinations for 21/3 in R34C67: [498|597} -> R4C6 = 9 -> 15/2 in R2/N3 = [69]
17. 9 in N5 locked in C3 -> 15/2 in R1/N1 = [69]
18. 9/2 in C5 cannot be {45} due to R1C5
19. R2C5 and 9/2 in C5 must contain both 2 and 3 -> elim from rest of C5
20. LOL -> C89 = N69 -> 1 locked in R489C7+R9C6; not in R4C7 -> 1 locked in R89C7+R9C6 -> not in rest of N9
21. R5C89+R67C9 = naked quad elims {5689} from R34C9 (R7C9 cannot be the same as one of R5C89 as if it were, then so would R6C9, thus all 4 are different and this is a valid naked quad)
22. 9 of R3 locked in N4 -> not in R6C2
23. Innies of R6789 = R6C357 = 21 = {579|678} -> no 7 in rest of N5/R6
24. Innies of N36 = R6C89 = 8 = [26|35]
25. 9 of N6 locked in 14/2 in R5 -> R6C3 = 9, 14/2 in R5/N6 = {59} naked pair -> 11 naked singles/last-digit-in-cage moves
26. Recalling step 5, R3C37 = [34] -> R3C9 = 7, R4C7 = 8, R6C7 = 5
27. 7/2 in R1/N3 = {25} naked pair -> 8 NS/LDIC moves
28. 15/3 in C8 = [249] -> 14/2 in R5 = [59] -> R9C8 = 6
29. combinations for 15/4 in R789C67 -> R8C6 = 4
30. 5/2 in C6 = [32] -> R4C6 = 8, 13/2 in C2 = [49]
31. 17/4 in R34C456 = [2681] -> naked singles/LDIC moves solve it.

Tada!

(Archive Note) Minor typos have been corrected. I’ve changed R9 to C9 in step 1f and deleted (actually just marked in red) {1457} from step 6c, because of the logic of that step which says that the cage cannot contain both 4 and 5.

Prelims

a) R1C34 = {69/78}
b) R12C5 = {16/25/34}, no 7,8,9
c) R1C67 = {16/25/34}, no 7,8,9
d) R23C2 = {49/58/67}, no 1,2,3
e) R2C34 = {39/48/57}, no 1,2,6
f) R2C67 = {69/78}
g) R23C8 = {18/27/36/45}, no 9
h) R5C12 = {16/25/34}, no 7,8,9
i) R56C5 = {18/27/36/45}, no 9
j) R5C89 = {59/68}
k) R67C1 = {16/25/34}, no 7,8,9
l) R67C4 = {16/25/34}, no 7,8,9
m) R67C6 = {14/23}
n) R67C9 = {59/68}
o) R78C5 = {89}
p) 11(3) cage at R1C1 = {128/137/146/236/245}, no 9
q) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9
r) 10(3) cage at R3C3 = {127/136/145/235}, no 8,9
s) 21(3) cage at R3C7 = {489/579/678}, no 1,2,3
t) 19(3) cage at R8C1 = {289/379/469/478/568}, no 1

1. Naked pair {89} in R78C5, locked for C5 and NR7C3, clean-up: no 1 in R56C5

2. 45 rule on R1 3 outies R2C159 = 6 = {123}, locked for R2, clean-up: no 1,2,3 in R1C5, no 9 in R2C34, no 6,7,8 in R3C8

3. R2C67 = {69} (cannot be {78} which clashes with R2C34), locked for R2 and NR1C6, clean-up: no 1 in R1C67, no 4,7 in R3C2, no 3 in R3C8
3a. R1C34 = {69} (cannot be {78} which clashes with R2C34), locked for R1 and NR1C1, clean-up: no 1 in R2C5
3b. Killer pair 4,5 in R1C45 and R1C67, locked for R1

4. 11(3) cage at R1C1 = {128/137}, 1 locked for NR1C1
4a. Killer pair 7,8 in 11(3) cage and R2C34, locked for NR1C1, clean-up: no 5,6 in R3C2
4b. Killer pair 4,5 in R2C2 and R2C34, locked for R2 and NR1C1, clean-up: no 4,5 in R3C8
4c. 10(3) cage at R3C3 = {127/136/235} (cannot be {145} because R3C3 only contains 2,3), no 4 in R4C34

5. 11(3) cage at R1C8 = {128/137}
5a. Killer pair 7,8 in 11(3) cage and R2C8, locked for NR1C6
5b. 21(3) cage at R3C7 = {489/579} (cannot be {678} because R3C7 only contains 4,5), no 6, 9 locked for R4
5c. R3C7 = {45} -> no 4,5 in R4C67

6. R56C5 = {27/36} (cannot be {45} which clashes with R1C5), no 4,5
6a. Killer pair 2,3 in R2C5 and R56C5, locked for C5

7. 45 rule on C1234 2 innies R39C4 = 10 = {19/28/37/46}, no 5

8. 45 rule on C6789 2 innies R39C6 = 9 = {18/27/36/45}, no 9

9. 45 rule on R5 3 outies R6C357 = 21 = {579/678} (cannot be {489} because no 4,8,9 in R6C5), no 1,2,3,4, 7 locked for R6 and NR4C5, clean-up: no 6 in R5C5

10. 7 in R5 only in R5C46, locked for NR1C5, clean-up: no 3 in R9C4 (step 7), no 2 in R9C6 (step 8)

[Just spotted.]
11. X-Wing for 9 in R2C67 and 21(3) cage at R3C7, no other 9 in C67
11a. 9 in NR4C5 only in R56C3, locked for C3 -> R1C34 = [69], clean-up: no 1 in R39C4 (step 7)

12. R4C6 = 9 (hidden single in NR1C5), R2C67 = [69], clean-up: no 3 in R39C4 (step 8)

13. 17(4) cage at R3C4 = {1268} (cannot be {2456} which clashes with R1C5, cannot be {2348} which clashes with R2C5, cannot be {1358} which clashes with R12C5) -> R34C5 = {16}, locked for C5, R3C46 = {28}, locked for R3 and NR1C5 -> R2C5 = 3, R1C5 = 4, both placed for NR1C5, R3C2 = 9, R2C2 = 4, R3C3 = 3, placed for NR1C1, R3C8 = 1, placed for R3C8, R2C8 = 8, placed for NR1C6, R3C5 = 6, placed for NR1C5, R4C5 = 1, placed for NR4C5, R4C4 = 5, placed for NR1C5, R4C3 = 2 (cage sum), placed for NR3C1, R2C4 = 7, placed for NR1C1, R2C3 = 5, R5C46 = [17], R5C5 = 2, placed for NR4C5, R6C5 = 7, R9C5 = 5, placed for NR7C3
[I’ll skip clean-ups and use “only remaining combinations” when necessary.]

14. R9C5 = 5 -> R9C46 = 9 = {18} -> R9C4 = 8, placed for NR7C1, R9C6 = 1, placed for NR7C8, R3C4 = 2

15. R67C6 = {23} (only remaining combination) -> R6C6 = 3, placed for NR4C5, R7C6 = 2, R1C6 = 5, R1C7 = 2, both placed for NR1C6, R3C7 = 4, R4C7 = 8 (cage sum), placed for NR3C8

16. R5C89 = {59} (only remaining combination), locked for R5 and NR3C8 -> R5C7 = 6, placed for NR4C5, R6C7 = 5, R6C4 = 4, placed for NR4C5, R7C4 = 3, R6C9 = 6, placed for NR3C8, R7C9 = 8, R6C1 = 1, R7C1 = 6, placed for NR7C1

17. 19(3) cage at R8C1 = {379} (only remaining combination), locked for NR7C1 -> R9C3 = 4, R8C3 = 1, placed for NR7C1, R78C2 = [52]

18. R9C89 = [62] (hidden pair in R9), R8C9 = 5 (cage sum)

and the rest is naked singles, without using the nonets.

SumoCueV1=19J0+0J0=12J0+2J0=22J1=28J2+5J2=10J2+7J2+0J0+2J0+2J0=12J1+4J1=11J1+5J2+5J2+7J2=14J0=18J3+12J1+12J1+4J1+14J1=6J4+24J4=7J2+18J0+19J3+19J3+19J5+4J1=21J5+32J4+32J4+26J2=10J3+36J3=24J5+38J5=18J5=8J5+41J5=17J4+43J4+36J6=23J3+46J3+38J5+40J7+41J5=21J4+51J4+43J8=12J6+46J3+46J3=19J7+40J7+57J7+51J7+51J4=19J8+54J6+54J6=23J6+57J7+57J7+57J7=19J8+62J8+62J8+65J6+65J6+65J6=12J6+75J7+75J8+69J8+69J8+69J8

Prelims

a) R23C6 = {29/38/47/56}, no 1
b) R34C1 = {59/68}
c) R3C78 = {15/24}
d) R34C9 = {16/25/34}, no 7,8,9
e) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
f) 10(3) cage at R1C8 = {127/136/145/235}, no 8,9
g) 21(3) cage at R4C6 = {489/579/678}, no 1,2,3
h) 10(3) cage at R5C1 = {127/136/145/235}, no 8,9
i) 24(3) cage at R5C3 = {789}
j) 8(3) cage at R5C6 = {125/134}
k) 19(3) cage at R7C9 = {289/379/469/478/568}, no 1
l) 12(4) cage at R1C3 = {1236/1245}, no 7,8,9
m) 28(4) cage at R1C6 = {4789/5689}, no 1,2,3

1. 12(4) cage at R1C3 = {1236/1245}, 2 locked for NR1C1
1a. Killer pair 5,6 in 12(4) cage and R34C1, locked for NR1C1

2. Naked triple {789} in 24(3) cage at R5C3, locked for NR4C4
2a. 8(3) cage at R5C6 = {125/134}, 1 locked for NR4C4
2b. 21(3) cage at R4C6 = {489/579/678}
2c. R4C6 = {456}, no 4,5,6 in R4C78

3. 10(3) cage at R1C8 = {127/136/235} (cannot be {145} which clashes with 28(4) cage at R1C6), no 4

4. 45 rule on NR6C1 2 innies R6C1 + R9C4 = 10 = [19/28]/{37/46}, no 5, no 1,2 in R9C4

5. 45 rule on NR6C9 2 innies R6C9 + R9C6 = 7 = {16/25/34}, no 7,8,9

6. 19(5) cage at R7C4 contains 1, locked for NR6C5
6a. 45 rule on C5 2 innies R89C5 = 5 = [14/23/32]

7. 45 rule on NR3C2 2(1+1) outies R4C4 + R6C1 = 6 = [24/33/42/51], clean-up: no 3,4 in R9C4 (step 4)

8. 45 rule on NR3C7 3(1+1+1) outies R4C6 + R6C9 + R7C7 = 20
8a. Max R4C6 + R6C9 = 12 -> min R7C7 = 8
8b. Min R4C6 + R6C9 = 11 -> R4C6 = {56}, R6C9 = {56}, clean-up: R9C6 = {12} (step 5)
8c. 21(3) cage at R4C6 = {579/678}, 7 locked for R4 and NR3C7
8d. R23C6 = {29/38/47} (cannot be {56} which clashes with R4C6), no 5,6 in R23C6

9. 17(3) cage at R5C8 = {269/359/368/458}, no 1
9a. R6C9 = {56} -> no 5,6 in R5C89
9b. Hidden killer pair 8,9 in 24(3) cage at R5C3 and 17(3) cage for R5, 17(3) cage contains one of 8,9 -> 24(3) cage must contain one of 8,9 in R5C34 -> R5C34 = {78/79}, 7 locked for R5, R6C4 = {89}

10. 19(5) cage at R7C4 = {12358/12367/12457/13456} (cannot be {12349} which clashes with R9C5), no 9
10a. Killer triple 2,3,4 in 19(5) cage and R9C5, locked for NR6C5

11. 45 rule on NR6C5 2 innies R7C7 + R9C5 = 1 innie R5C5 + 8, IOU no 8 in R7C7 -> R7C7 = 9, locked for NR6C5
11a. R7C7 = 9 -> R5C5 = R9C5 + 1, no 2,6 in R5C5

12. R4C6 = 6 (hidden single in R4C4), R4C78 = 15 = {78}, locked for R4 and NR3C7, clean-up: no 6,8 in R3C1, no 1 in R3C9

13. Naked pair {59} in R34C1, locked for C1 and NR1C1
13a. 19(3) cage at R1C1 = {478} (only remaining combination), locked for NR1C1

14. R4C6 + R6C9 + R7C7 = 20 (step 8), R4C6 = 6, R7C7 = 9 -> R6C9 = 5, R9C6 = 2 (step5), both placed for NR6C9, clean-up: no 9 in R23C6, no 2 in R34C9
14a. R6C9 = 5 -> R5C89 = 12 = {39}, locked for R5 and NR3C7
14b. R9C6 = 2 -> R9C45 = 10 = [64/73], clean-up: no 1,2 in R6C1 (step 4), no 4,5 in R4C4 (step 7)
14c. R6C4 = 9 (hidden single in NR4C4)

15. R1C6 = 9 (hidden single in C6), R2C5 = 9 (hidden single in R2), placed for NR1C5

16. 18(3) cage at R5C5 = {468/567}, 6 locked for C5 and NR6C5
16a. R5C5 = {45} -> no 5 in R7C5

17. 45 rule on C6789 2 remaining innies R78C6 = 11 = {38/47}, no 1,5
17a. Killer pair 3,4 in R78C6 and R9C5, locked for NR6C5
17b. R78C6 = 11 -> R78C4 + R8C5 = 8 = {125}, 5 locked for C4
[Alternatively R78C4 + R8C5 = {125} (hidden triple in NR6C5.]

18. 4 in C4 only in R23C4, locked for NR1C5, clean-up: no 7 in R23C6
18a. Naked pair {38} in R23C6, locked for C6 and NR1C5
18b. Naked pair {47} in R78C6, locked for C6 and NR6C5 -> R9C5 = 3, R9C4 = 7 (cage sum), R6C1 = 3 (step 4), both placed for NR6C1, R5C34 = [78], R8C5 = 2 (step 6a)
18c. Naked pair {15} in R78C4, locked for C4
18d. 12(3) cage at R2C4 = {246} (only remaining combination, or hidden triple for NR1C5)

19. R56C6 = [51], R5C7 = 2 (cage sum), R5C5 = 4
[I’d overlooked that R5C7 was placed for NR4C4, which would have made later steps a bit quicker.]

20. R7C7 = 9 -> 21(4) cage at R6C7 = {2469} (only remaining combination because 1,5 only in R7C8), locked for NR3C7, 2 also locked for C8
20a. Naked pair {15} in R3C78, locked for R3 -> R34C1 = [95], R34C5 = [71], R1C5 = 5, clean-up: no 6 in R3C9
20b. Naked pair {34} in R34C9, locked for C9 and NR1C6, R5C89 = [39]
20c. 10(3) cage at R1C8 = {127} (only remaining combination), locked for NR1C6

21. 19(3) cage at R7C9 = {478} (only remaining combination) -> R8C8 = 4, placed for NR6C9, R8C6 = 7, R8C9 = 8, placed for NR6C9
21a. Naked pair {16} in R9C79, locked for NR6C9 -> R8C7 = 3, R9C8 = 9

22. Naked pair {458} in R9C123, locked for NR6C1, R8C3 = 6 (cage sum), R3C3 = 2
22a. Naked pair {13} in R12C3, locked for C3 and NR1C1

23. R8C1 = 1, R5C1 = 6, placed for NR3C2, R5C2 = 1, R7C1 = 2, R7C8 = 6, R6C78 = [42]
23a. R6C23 = [78] = 15 -> R7C23 = 8 = [35]

24. Naked pair {46} in R23C4, locked for C4 -> R1C4 = 2

and the rest is naked singles, without using the nonets.

Jigsaw nonet design: Moonlotus by Leonid Kreysin and Rebecca Schwartz

SumoCueV1=18J0+0J0=11J1+2J1=16J1=18J1+5J1=13J2+7J2+0J0=12J0+10J0+2J1+4J1+5J1=8J2+15J2+7J2=19J3+10J0=15J0+20J0+4J1=24J2+23J2+15J2=13J4+18J3+18J3+20J0=14J5=10J5=21J5+23J2+26J4+26J4=17J3+36J3+36J3+30J5+31J5+32J5=8J4+42J4+42J4=9J3+45J3=17J6+30J5+31J5+32J5=10J7=24J4+52J4+45J3=8J6+47J6+47J6=19J8+51J7+51J7=21J7+52J4=20J6+55J6+55J6=20J8+58J8=6J8+61J7+61J7=14J7+63J6+63J6+66J8+66J8+58J8+68J8+68J8+71J7+71J7

Whoo. First try on this one fizzled, came back a few days later (that is, today.) Took a good couple hours but I've got a walkthrough.



1. cagesize/sum eliminations:
1a. 6/3 in N8 = {123} naked triple in N8
1b. digits in 8/3s in N3, N6/R5 and N7 <= 5, must have 1 -> no other 1s in N3, N6, R5, and N7
1c. digits in 9/3 in N4 <= 6
1d. digits in 10/3s in N5 and N9 <= 7
1e. digits in 11/3 in N2 <= 8
1f. digits in 19/3 in N4 >= 2
1g. digits in 20/3 in N7 >= 3
1h. digits in 21/3s in N5 and N9 >= 4
1i. 24/3s in N3 and N6 = {789} naked triples in N3 and N6

2. Innies of R5 = R5C456 = 20 -> R5C456 != 1|2
3. 6s in N36 locked in C89 -> not in rest of C89
4. 1s of N4789 locked in R6789 -> no 1s in rest of R6789
5. 1 of N5 locked in R4 -> not in rest of R4
6. 1 of N3689 locked in C6789 -> not in rest of C6789
7. 19/3 in C5 = {469|478|568} -> 10/3 in C5 != {145} -> no 4 in 10/3 in C5
8. 19/3 in C5 = {469|478|568} -> 16/3 in C5 != {268|367|457} -> 16/3 in C5 = {169|178|259|349|358}
9. LOL: R37C6 == R19C7; R3C6 != R9C7 -> R3C6 == R1C7 && R7C6 == R9C7 -> R1C7 = {789} && R7C6 = {123}
10. R789C6 = naked triple {123} on C6
11. R134C7 = naked triple {789} on C7
12. 21/3 in N9 = {489|579|678}; R8C7 = {456} -> R78C8 = {78|79|89}
13. R678C8 = naked triple {789} on C8
14. LOL: R3C19 == R4C37; R3C9!=R4C7 -> R3C9 == R4C3 && R3C1 == R4C7 -> R3C1 = {789} && R4C3 = {23456}
15. R3C167 = naked triple {789} on R3
16. 7s, 8s, and 9s of R345 locked in N345 -> not in rest of N345
17. R6C389 = hidden triple {789}
18. 10/3 in N9 = {136|145|235} -> 14/3 in N9 != {158|347} -> 14/3 in N9 = {149|239|248|257}
19. 8/3 in R5 = {125|134} -> 17/3 in R5 != {359|458} -> 17/3 in R5 = {269|278|368|467} -> no 5 in 17/3 in R5
20. 1 of N4 locked in 9/3 -> 9/3 = {126|135} -> 19/3 in N4 != {568} -> no 5 in 19/3 in N4 -> 5 of N4 locked in 9/3 -> 9/3 in N4 = {135} naked triple
21. How'd I miss this before -- outies of R12 = R3C258 = 6/3 = {123} naked triple in R3
22. combinations for 15/3 in N1 = {456} naked triple in N1
23. Innies of R6789 = R6C456 = 12/3 with max 6 = {156|246|345} but {156|345} conflict with 9/3 in N4 so R6C456 = {246} naked triple in R6/N5
24. 18/3 in N2 with min 4 has exactly one of {789}, R1C7 = {789} -> not in rest of 18/3 in N2
25. R126C6 = naked triple {456} in C6
26. innies of R5 = R5C456 = 20/3 -> even number of odd digits -> odd number of even digits -> must contain 8, elim 8 from rest of R5/N5
27. 8 of N4 locked in 19/3 -> 19/3 = {289|478} -> no 6 in 19/3 in N4
28. 6 of C12 locked in N47 -> not in rest of N47
29. 17/3 in N7 = {278|359|458} -> contains either 2 or 5 -> 8/3 in N7 != {125} -> 8/3 in N7 = {134} naked triple -> 17/3 in N7 = {278} naked triple -> 20/3 in N7 = {569}
30. 2 of N7 locked in R7 -> not in rest of R7
31. 10/3 in N9 = {136|145} -> 10/3 has either 4 or 6, both of which can only be in R7C7 -> R7C7 = {46}
32. 21/3 in C6 = {489|678} -> must contain 8 -> R5C6 = 8
33. 5 of C6 locked in N2 -> not in rest of N2
34. 4 of C1 locked in N4 -> not in rest of N4
35. 4 of C2 locked in N7 -> not in rest of N7
36. R7C3489 = naked quad {2789} in R7
37. LOL -> R37C6 == R19C7 -> R1C7 = {79} and R9C7 = {13}
38. 18/3 in N2 = {459|567} -> 18/3 has either 4 or 6 -> 11/3 in N2 != {146} -> no 4 in 11/3 in N2
39. 16/3 in N2: R3C5 = {123} -> no 1|2|3 in rest of 16/3
40. 9 of R6 locked in N6 -> not in rest of N6
41. R7C8 = 9 (hidden single)
42. R6C9 = 9 (hidden single)
43. 21/3 in N9 = [948|957] -> no 6 in 21/3 in N9
44. R7C7 = 6 (hidden single)
45. 10/3 in N9 = {136} naked triple in N9
46. 5 of R6 locked in N4 -> not in rest of N4
47. R7C16 = naked pair {13} in R7 -> R7C2 = 4 -> R7C5 = 5
48. R8C23 = naked pair {13} in R8 -> R8C6 = 2
49. R69C7 = naked pair (13) in C7
50. outies of C12 = R258C3 = 13/3 = [193|391|823]
51. combinations for 17/3 in R5 = {269} naked triple -> 19/3 in N4 = {478} && 8/3 in R5 = {134} naked triple && R5C7 = 4 -> lots of naked and hidden singles and last-digit-in-cage moves solve it.

That was fun!

Better late than never - thought I'd give one of the Texas Jigsaws from the back catalog a spin. Made a bit of a meal of it the first time through, but was able to tighten up the solving path considerably whilst working on this walkthrough. By the way, preparing the walkthrough took ages this time, mainly because of the extremely high symmetry of the puzzle, requiring an equally meticulous solution.

This puzzle was very heavy on naked/hidden subsets and involved some powerful applications of the Law of Leftovers.

Anyway - here's the walkthrough:


Walkthrough - Texas Jigsaw Killer 022

Nonet layout:

112222233
111222333
411123335
441666355
444666555
447666855
477798885
777999888
779999988

1. Preliminaries:

a) 11/3 cage at R1C3 - no 9
b) 24/3 cage at R3C6 = {789} -> no 7,8,9 anywhere else in N3
c) 8/3 cage at R2C7 = {1(25|34)} -> no 1 in 13/3 cage at R1C8
d) 6 in N3 now locked in 13/3 cage at R1C8 = {6(25|34)}
e) 19/3 cage at R3C1 - no 1
f) 10/3 cage at R4C5 - no 8,9
g) 21/3 cage at R4C6 - no 1,2,3
h) 24/3 cage at R6C8 = {789} -> no 7,8,9 anywhere else in N5
i) 8/3 cage at R5C7 = {1(25|34)} -> no 1 elsewhere in N5 and R5
j) 1 in N4 now locked in 9/3 cage at R6C1 = {1(26|35)}, no 4,7,8,9
k) 6 in N5 now locked in 13/3 cage at R3C9 = {6(25|34)}
l) 10/3 cage at R6C7 - no 8,9
m) 8/3 cage at R7C2 = {1(25|34)} -> no 1 elsewhere in N7
n) 21/3 cage at R7C8 - no 1,2,3
o) 20/3 cage at R8C1 - no 1,2
p) 6/3 cage at R8C6 = {123} -> no 1,2,3 anywhere else in N9

2. Innies/outies:

a) Innies R5: R5C456 = 20/3 -> no 2,3 in R5C4, no 2 in R5C5 -> no 7 in R46C5
b) Innies R1234: R4C456 = 13/3 -> no 9 in R4C4
c) Innies R6789: R6C456 = 12/3 -> no 8,9 in R6C4
d) Outies R12: R3C258 = 6/3 = {123} -> no 1,2,3 elsewhere in R3 and R12C5 -> no 7,8,9 in R4C3
e) Outies C89: R258C7 = 11/3 -> no 9 in R8C7
f) Outies R89: R7C258 = 18/3
g) Outies C12: R258C2 = 13/3

3. LoL C789: R7C6 = R9C7 = {123}, R1C7 = R3C6 = {789}
4. Naked triple on {789} in C7 at R134C7 -> no 7,8,9 elsewhere in C7
5. Naked triple on {123} in C6 at R789C6 -> no 1,2,3 in R12C6
6. Hidden killer triple on {456} in C6 at R12C6 and 21/3 cage at R4C6 -> no 7,8,9 in R12C6

Clarification:
i) 21/3 cage at R4C6 can only contain one of {456}
ii) Only other 2 candidate positions for {456} in C6 are R12C6, which therefore cannot contain
any other value

Note: We could have also arrived at the same result by analysing the possible permutations for
the 18/3 cage at R1C6.

7. LoL R123: R4C3 = R3C9 = {456}, R3C1 = R4C7 = {789}
8. Naked triple on {789} in R3 at R3C167 -> no 7,8,9 in R3C34 -> cage 15/3 at R3C3 = {456} ->
no 4,5,6 elsewhere in N1

9. LoL C12:

a) R89C12+R7C2 must contain {456} -> no 4,5,6 elsewhere in N7
b) Cage 8/3 at R7C2 can only contain 1 of {456} -> cage 20/3 at R8C1 must contain 2 of {456} ->
cage 20/3 at R8C1 = {569} -> cage 8/3 at R7C2 = {134} -> cage 17/3 at R6C3 = {278}
c) 4 in N7 locked in R78C2 -> no 4 elsewhere in C2
d) (other direction now) R25C3 = {(1|3)9} -> R2C3 = {139}, R5C3 = {39}
e) 9 is thus locked in R25C3 -> no 9 in R9C3
f) {1|3} in R25C3 forms hidden naked pair with R8C3 -> no 1,3 in R1C3

10a. LoL R789: R6C3 = R7C9 = {78} -> R6C7 = R7C1 -> no 4 in R6C7
10b. 9 in 24/3 cage at R6C8 locked in R6C89 -> no 9 in R6C6

11. LoL R89:

a) Innie R89C89+R8C7 = {278..} -> 2 locked in 14/3 cage at R8C9 -> no 2 elsewhere in N8
b) 14/3 cage at R8C9 can only accommodate one of {78} -> other must go in R8C8 -> R8C8 = {78}
c) 14/3 cage at R8C9 must contain one of {78} -> {2(57|48)} -> no 1,3,6,9
d) Hidden single in C9 at R5C9 = 1
e) Hidden single in N8 at R7C8 = 9
f) Split cage R8C78 = 12 -> no 6 in R8C7
g) {13} locked in 10/3 cage at R6C7 = {136}

12. Naked pair on {78} in cage 24/3 at R6C8 -> no 7,8 in R6C9 -> R6C9 = 9
13. LoL C789: R9C7 = R7C6 = {13}
14. LoL R789: R7C1 = R6C7 = {136}
15. Hidden single in R7 at R7C5 = 5 -> Split cage 14/2 ar R89C5 = {68} -> no 6,8 elsewhere in N9
16. 20/3 cage at R8C4 = {479}, 9 locked in R89C4 -> no 9 in R5C4
17. Hidden single in R7 at R7C2 = 4
18. Split 4/2 cage at R8C23 = {13} -> no 1,3 elsewhere in R8 -> R8C6 = 2
19. Cage 17/3 at R5C1 = {269|359|368} -> no 4,7 -> Hidden single in N4 at R4C1 = 4
20. 7 in N4 locked in split cage 15/2 at R3C1+R4C2 = {78}
21. 9/3 cage at R6C1 = {135} (4 no longer available)
22. Naked pair on {13} in R7 at R7C16 -> no 1,3 in R7C7 -> R7C7 = 6
23. Cage 17/3 at R5C1 = {269} -> R5C3 = 9, no 2,6,9 elsewhere in R5
24. Hidden single in N6 at R4C6 = 9
25. Split cage 7/2 at R5C78 = {34} -> no 3,4 elsewhere in N5
26. Innie 20/3 cage at R5C456 = {578} -> no 5,7,8 elsewhere in N6
27. Naked single at R5C5 = 7
28. Hidden single in C6 at R3C6 = 7 -> R34C7 = [98], R4C2 = 7, R3C1 = 8, R1C7 = 7
29. Hidden single in C1 at R2C1 = 7
30. Hidden single in C6 at R5C6 = 8 -> R5C4 = 5, R6C6 = 4
31. Hidden single in C7 at R2C7 = 2
32. Hidden single in C7 at R8C7 = 5 -> R8C8 = 7 -> R6C8 = 8, R7C9 = 7
33. Naked single at R6C3 = 7
34. Naked single at R9C3 = 4
35. Naked single at R8C4 = 9 -> R9C4 = 7

...and so on...

The rest is all singles.

I've made a few minor corrections, after Ed's feedback prompted me to work through my steps again.

Prelims

a) 11(3) cage at R1C3 = {128/137/146/236/245}, no 9
b) 8(3) cage at R2C7 = {125/134}
c) 19(3) cage at R3C1 = {289/379/469/478/568}, no 1
d) 24(3) cage at R3C6 = {789}
e) 10(3) cage at R4C5 = {127/136/145/235}, no 8,9
f) 21(3) cage at R4C6 = {489/579/678}, no 1,2,3
g) 8(3) cage at R5C7 = {125/134}
h) 9(3) cage at R6C1 = {126/135/234}, no 7,8,9
i) 10(3) cage at R6C7 = {127/136/145/235}, no 8,9
j) 24(3) cage at R6C8 = {789}
k) 8(3) cage at R7C2 = {125/134}
l) 19(3) cage at R7C5 = {289/379/469/478/568}, no 1
m) 21(3) cage at R7C8 = {489/579/678}, no 1,2,3
n) 20(3) cage at R8C1 = {389/479/569/578}, no 1,2
o) 20(3) cage at R8C4 = {389/479/569/578}, no 1,2
p) 6(3) cage at R8C6 = {123}

Steps resulting from Prelims
1a. 8(3) cage at R2C7 = {125/134}, 1 locked for NR1C8
1b. Naked triple {789} in 24(3) cage at R3C6, locked for NR1C8
1c. 8(3) cage at R5C7 = {125/134}, 1 locked for R5 and NR3C9
1d. Naked triple {789} in 24(3) cage at R6C8, locked for NR3C9
1e. 8(3) cage at R7C2 = {125/134}, 1 locked for NR6C3
1f. Naked triple {123} in 6(3) cage at R8C6, locked for NR7C5
1g. 1 in NR3C1 only in 9(3) cage at R6C1 = {126/135}, no 4

2. 45 rule on R12 3 outies R3C258 = 6 = {123}, locked for R3
2a. Max R3C5 = 3 -> min R12C5 = 13, no 1,2,3 in R12C5

3. 45 rule on C89 3 outies R258C7 = 11 = {128/137/146/236/245}, no 9

4. 45 rule on R5 3 innies R5C456 = 20 = {389/479/569/578}, no 2
4a. 3 of {389} must be in R5C5 -> no 3 in R5C4

5. 13(3) cage at R1C8 = {256/346}, 13(3) cage at R3C9 = {256/346}
5a. Caged X-Wing for 6 in 13(3) cages for C89, no other 6 in C89

6. Killer triple 7,8,9 in R3C6 and 21(3) cage at R4C6, locked for C6
6a. Max R12C6 = 11 -> min R1C7 = 7
6b. Naked triple {789} in R134C7, locked for C7

7. 21(3) cage at R7C8 = {489/579/678}
7a. R8C7 = {456} -> no 4,5 in R78C8
7b. Naked triple {789} in R678C8, locked for C8

8. 19(3) cage at R7C5 = {469/478/568}
8a. 10(3) cage at R4C5 = {127/136/235} (cannot be {145} which clashes with 19(3) cage), no 4

9. Law of Leftovers (LoL) for C789. Two outies R37C6 must be exactly the same as two innies R19C7. R1C7 and R3C6 are both {789} -> R7C6 and R9C7, which don’t contain any of 7,8,9, must be equal -> R7C6 = {123}
[It’s probably possible to get the same result by more conventional steps but it would take longer. Combinations for 18(3) cage at R1C6 don’t quite give this result since {369} is still possible.]
9a. Naked triple {123} in R789C6, locked for C6

10. LoL for R789 R6C37 must be exactly equal to R7C19. R7C9 = {789} -> R6C3 = {789} (because no 7,8,9 in R6C7), R7C1 = {12356} -> R6C7 = {12356}
10a. Naked triple {789} in R6C389, locked for R6
10b. 21(3) cage at R4C6 = {489/579/678}
10c. R6C6 = {456} -> no 4,5,6 in R45C6
[With hindsight, naked triple {456} in R126C6, locked for C6 is better than steps 10b and 10c.]

11. LoL for R123 R3C19 must be exactly equal to R4C37. R4C7 = {789} -> R3C1 = {789} (because no 7,8,9 in R3C9), R3C9 = {456} -> R4C3 = {456}
11a. Naked triple {789} in R3C167, locked for R3
11b. Naked triple {456} in 15(3) cage at R3C3, locked for NR1C1

12. 4 on R6 only in R6C46, locked for NR4C4
12a. 45 rule on R6789 3 innies R6C456 = 12 = {246/345}, no 1

13. 1 in R4 only in R4C45
13a. 45 rule on R1234 3 innies R4C456 = 13 = {139/157}, no 2,6,8
13b. R4C6 = {79} -> no 7,9 in R4C45

14. 8 in NR4C4 only in R5C46, locked for R5
14a. R5C456 (step 4) = {389/578}, no 6

15. R6C456 = {246} (hidden triple in NR4C4), locked for R6

16. 9(3) cage at R6C1 (step 1g) = {135} (only remaining combination, cannot be {126} because 2,6 only in R7C1, or R7C1 = {135} from LoL, step 10), locked for NR3C1
16a. 17(3) cage at R5C1 = {269/467}, 6 locked for NR3C1

17. 21(3) cage at R4C6 = {489/678} -> R5C6 = 8
17a. 24(3) cage at R3C6 = {789}, 8 locked for C7

18. 5 in C6 only in R12C6, locked for NR1C3
18a. 18(3) cage at R1C6 = {459/567}

19. LoL for C123, R19C3 must be exactly equal to R37C4
19a. No 1 in R37C4 -> no 1 in R1C3
19b. R3C4 = {456} -> R19C3 must contain {456}
19c. Killer triple {456} in R19C3 and R34C3, locked for C3
19d. R19C3 can only contain one of 4,5,6, R3C4 = {456} -> no 4,5,6 in R7C4

20. 6 in NR6C3 only in 20(3) cage at R8C1 = {569}, locked for NR6C3
20a. 8(3) cage at R7C2 = {134} (only remaining combination), locked for NR6C3, 4 also locked for C2
20b. Naked triple {278} in 17(3) cage at R6C3, 2 locked for R7
20c. 9 in R6 only in R6C89, locked for 24(3) cage at R6C8, no 9 in R7C9

21. Naked triple {278} in R7C349, locked for R7 -> R7C8 = 9, R6C9 = 9 (hidden single in R6)
21a. R7C8 = 9 -> R8C78 = 12 = [48/57], no 6

22. R7C7 = 6 (hidden single in C7) -> R6C7 + R7C6 = 4 = {13}, locked for NR6C7
22a. 5 in R6 only in R6C12, locked for 9(3) cage at R6C1, no 5 in R7C1

23. Naked pair {13} in R7C16, locked for R7 -> R7C2 = 4
23a. Naked pair {13} in R8C23, locked for R8 -> R8C6 = 2

24. 45 rule on C12 3 outies R258C3 = 13 = {139} (cannot be {238} which clashes with 17(3) cage at R6C3, ALS block) -> R5C3 = 9, R28C3 = {13}, locked for C3
24a. R5C3 = 9, placed for NR3C1 -> R5C12 = 8 = {26}, locked for R5 and NR3C1
24b. R4C1 = 4 (hidden single in NR3C1)

25. 8(3) cage at R5C7 = {134} (only remaining combination), locked for R5 and NR3C9

26. Naked pair {13} in R69C7, locked for C7 -> R5C7 = 4, R8C7 = 5, R8C8 = 7 (cage sum), both placed for NR6C7
26a. R2C7 = 2, R23C8 = 6 = [51], 2,5 placed for NR1C8, R5C89 = [31]

27. R4C456 = {139} (hidden triple in NR4C4) -> R4C6 = 9, R6C6 = 4 (cage sum), R3C6 = 7, R3C1 = 8, R4C2 = 7, R34C7 = [98]

28. R1C7 = 7, R2C6 = 6, both placed for NR1C3

29. 11(3) cage at R1C3 = {128} (only remaining combination), locked for NR1C3 -> R12C5 = {49}, locked for C5, R3C5 = 3, R4C5 = 1, R56C5 = 9 = [72], R456C4 = [356], R3C4 = 4
29a. Naked triple {568} in 19(3) cage at R7C5, locked for NR7C5 -> R89C4 = [97]

31. R3C2 = 2, R2C23 = 10 = [91]
and the rest is naked singles, without using the nonets.

Jigsaw nonet design: "H" by Bob & Debbie Scott

SumoCueV1=12J0+0J0+0J0=12J1=14J1=6J2=15J2+6J2+6J2=23J0+9J0=9J1+3J1+4J1+5J2=14J3=24J2+16J2+9J0+9J0+11J1=13J1+4J1=15J4+15J3+16J2+16J2=13J5=14J0+21J0+21J4=18J1+23J4+23J3=12J3=14J3+27J5+28J5=11J5+38J4+31J4=11J4+41J3+34J3+35J3+27J5+28J5=15J5+47J4+31J6=19J4+50J7+34J7+35J3=14J8+54J8=13J5+47J4=13J6+50J6=6J6=24J7+61J7+54J8+54J8+56J5=9J8+58J6=9J6+60J6+61J7+61J7=22J8+72J8+72J8+66J8+58J6+68J6=11J7+78J7+78J7

Prelims

a) R12C4 = {39/48/57}, no 1,2,6
b) R12C6 = {15/24}
c) R23C3 = {18/27/36/45}, no 9
d) R23C7 = {59/68}
e) R5C34 = {29/38/47/56}, no 1
f) R5C67 = {29/38/47/56}, no 1
g) R78C3 = {49/58/67}, no 1,2,3
h) R78C7 = {15/24}
i) R89C4 = {18/27/36/45}, no 9
j) R89C6 = {18/27/36/45}, no 9
k) 19(3) cage at R6C6 = {289/379/469/478/568}, no 1
l) 22(3) cage at R9C1 = {589/679}
m) 11(3) cage at R9C7 = {128/137/146/236/245}, no 9
n) 14(4) cage at R7C1 = {1238/1247/1256/1346/2345}, no 9

1. R89C6 = {18/27/36} (cannot be {45} which clashes with R12C6), no 4,5
1a. 45 rule on NR6C5, 2 innies R6C5 + R7C6 = 17 = {89}, locked for NR6C5, clean-up: no 1 in R89C6
1b. 19(3) cage at R6C6 = {379/469/478/568} (cannot be {289} which clashes with R6C5), no 2
1c. R7C6 = {89} -> no 8,9 in R6C67
1d. 45 rule on NR6C7 2 innies R6C78 = 10 = {37/46}, no 1,5,8,9
1e. R6C78 = 10 -> R6C67 cannot total 10 (CCC) -> no 9 in R7C6 -> R7C6 = 8, placed for NR6C5, R6C5 = 9, clean-up: no 3 in R5C7, no 5 in R8C3
1f. R7C6 = 8 -> R6C67 = 11 = [47/56/74], clean-up: no 7 in R6C8

2. 45 rule on R789 1 remaining innie R7C4 = 6, placed for NR3C6, clean-up: no 5 in R5C3, no 5 in R5C7, no 7 in R8C3, no 3 in R89C4

3. 45 rule on R123 2 innies R3C46 = 6 = {15/24}
3a. 45 rule on NR1C4 2 innies R3C4 + R4C5 = [28/46], clean-up: no 1,5 in R3C6
3b. Naked pair {24} in R3C46, locked for R3, clean-up: no 5,7 in R2C3
3c. 18(3) cage at R4C5 = {189/369} -> R5C5 = {13}

4. 9 in NR1C4 only in R12C4 = {39}, locked for C4 and NR1C4, clean-up: no 6 in R23C3, no 2,8 in R5C3

5. R12C6 = {15} (cannot be {24} which clashes with R3C6), locked for C6 and NR1C6, clean-up: no 6 in R5C7, no 6 in R6C7 (step 1f)
5a. Naked pair {47} in R6C67, locked for R6

6. 45 rule on R1 3 innies R1C456 = 18 = {189/459} (cannot be {567} because 6,7 only in R1C5, other combinations don’t contain 1 or 5 for R1C6) -> R1C4 = 9, R1C5 = {48}, R2C4 = 3

7. 45 rule on C123 3 innies R456C3 = 15
7a. Hidden killer triple 1,2,3 in R1C3, R23C3 and R456C3 for C3, R23C3 can only contains one of 1,2, R456C3 can only contain one of 1,2,3 -> R1C3 = {123}, R23C3 = {18}/[27], no 4,5 and R456C3 must contain one of 1,2,3

8. R3C4 + R4C5 (step 3a) = [46] (cannot be [28] which clashes with R23C3), 4 placed for NR1C4, R1C5 = 8, placed for NR1C4, R23C5 = 6 = {15}, locked for C5 and NR1C4, R3C6 = 2, R5C5 = 3, both placed for NR3C6, R23C3 = [27], clean-up: no 9 in R5C3, no 8 in R5C4, no 8,9 in R5C7, no 5 in R89C4, no 7 in R89C6
8a. Naked triple {479} in R456C6, locked for NR3C6 -> R5C4 = 5, R5C3 = 6, placed for NR4C1
8b. R1C456 (step 5) = {189} (only remaining combination) -> R12C6 = [15], placed for NR1C6, R23C5 = [15], R1C3 = 3, placed for NR1C1, clean-up: no 9 in R23C7
8c. Naked pair {68} in R23C7, locked for C7 and NR2C7
8d. R1C3 = 3 -> R1C12 = 9 = {27/45}, no 6

9. R89C4 = {27} (hidden pair in C4), locked for NR7C1
9a. 22(3) cage at R9C1 = {589}, only remaining combination, locked for R9 and NR7C1

10. R7C4 = 6 -> R6C34 = 9 = {18} (only remaining combination), locked for R6
10a. R3C4 = 4 -> R4C34 = 9 = {18} (only remaining combination), locked for R4
10b. R3C6 = 2 -> R4C67 = 13 = {49} (only remaining combination), locked for R4

11. R6C8 = 6 (hidden single in R8), placed for NR6C7, R45C8 = 6 = [24/51]

12. 15(3) cage at R1C7 = {267} (only remaining combination) -> R1C9 = 6, R1C89 = {27}, locked for R1 and NR1C6
12a. Naked pair {45} in R1C12, locked for NR1C1

[I’d seen this 45 after seeing the one in step 1d, but it’s only recently become useful.]
13. 45 rule on NR1 2 innies R4C23 = 10 = [28], placed for NR1C1, R4C8 = 5, placed for NR2C7, R5C8 = 1 (step 11), clean-up: no 5 in R7C3

14. Naked pair {49} in R78C3, locked for C3 and NR4C1 -> R9C3 = 5

15. R4C2 = 2 -> R56C2 = 12 = [75], R1C2 = 4, clean-up: no 4 in R5C67
15a. R5C67 = [92], R5C1 = 8, R46C1 = 5 = [32]

16. R9C7 = 3 (hidden single in C7) -> R9C89 = 8 = [71], all placed for NR6C7

17. R456C9 = [743]
17a. Naked pair {89} in R23C9, locked for C9 and 24(4) cage at R2C8 -> R23C8 = [43]

and the rest is naked singles, without using the nonets.