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Jigsaw nonet design: Worm by Leonid Kreysin

SumoCueV1=18J0+0J0+0J1=16J1=18J1+4J1=11J2+6J2+6J2=11J0+9J0=9J0+3J1+4J1=8J3=14J2=7J2+16J2=9J0+18J0+11J1+3J1+4J1+14J3+15J4=14J2+25J2=9J0+27J0=19J5=11J5=18J3+31J3=14J4=9J4+34J2=7J5+36J5+29J5+30J5+31J3=22J4+33J4=7J4+43J4=8J6+45J5+29J5+41J3+41J3+41J4+33J4=11J7+52J7=13J6+54J6=5J5=11J3=15J8=16J8=10J8=9J7+61J7=14J6+63J6+56J6+57J3+58J8+59J8+60J7=8J7+70J7=13J6+72J6+72J6+58J8+58J8+59J8=21J8+78J7+78J7

Prelims

a) R2C12 = {29/38/47/56}, no 1
b) R23C3 = {18/27/36/45}, no 9
c) R23C6 = {17/26/35}, no 4,8,9
d) R23C7 = {59/68}
e) R2C89 = {16/25/34}, no 7,8,9
f) R3C12 = {18/27/36/45}, no 9
g) R3C89 = {59/68}
h) R4C12 = {18/27/36/45}, no 9
i) R45C4 = {29/38/47/56}, no 1
j) R4C89 = {18/27/36/45}, no 9
k) R5C12 = {16/25/34}, no 7,8,9
l) R5C89 = {16/25/34}, no 7,8,9
m) R6C12 = {17/26/35}, no 4,8,9
n) R6C89 = {29/38/47/56}, no 1
o) R7C12 = {49/58/67}, no 1,2,3
p) R78C3 = {14/23}
q) R78C4 = {29/38/47/56}, no 1
r) R78C7 = {19/28/37/46}, no 5
s) R7C89 = {18/27/36/45}, no 9
t) R8C12 = {59/68}
u) R8C89 = {17/26/35}, no 4,8,9
v) 11(3) cage at R1C7 = {128/137/146/236/245}, no 9
w) 19(3) cage at R4C3 = {289/379/469/478/568}, no 1
x) 21(3) cage at R9C7 = {489/579/678}, no 1,2,3

1. 45 rule on C3 2 innies R19C3 = 12 = {39/48/57}, no 1,2,6

2. 45 rule on C7 2 innies R19C7 = 7 = [16/25/34]
2a. 21(3) cage at R9C7 = {489/579/678}
2b. R9C7 = {456} -> no 4,5,6 in R9C89

3. 45 rule on NR1 1 outie R1C3 = 1 innie R2C3 + 2, no 4,8 in R2C3, clean-up: no 1,5 in R3C3

4. 45 rule on NR1C7 2 outies R3C7 + R4C8 = 10 = [64/82/91], clean-up: no 9 in R2C7, R4C9 = {578}

5. 45 rule on NR6C1 2 outies R6C2 + R7C3 = 8 = [53/62/71], clean-up: no 5,6,7 in R6C1, no 1 in R8C3

6. 45 rule on NR6C8 1 outie R9C7 = 1 innie R8C7 + 4 -> R8C7 = {12}, R9C7 = {56}, clean-up: R1C7 = {12} (step 2), R7C7 = {89}
6a. 21(3) cage at R9C7 = {579/678}, 7 locked for R9 and NR6C8, clean-up: no 5 in R1C3 (step 1), no 4 in R6C89, no 2 in R7C89, no 1 in R8C89
6b. 4 in NR6C8 only in R7C89 = {45}, locked for R7 and NR6C8, clean-up: no 6 in R6C89, no 8,9 in R7C12, no 6,7 in R8C4, no 3 in R8C89
6c. Naked pair {26} in R8C89, locked for R8 and NR6C8 -> R8C7 = 1, R7C7 = 9, placed for NR7C5, R1C7 = 2, placed for NR1C7, R9C7 = 5 (step 2), placed for NR7C5, clean-up: no 7 in R1C3 (step 1), no 5 in R2C89, no 1 in R4C8 (step 4), no 8 in R4C9, no 9 in R6C89, no 3 in R7C3, no 8 in R8C12
6d. Naked pair {38} in R6C89, locked for R6 and NR6C8, clean-up: no 5 in R6C2

7. Killer pair 3,4 in R19C3 and R8C3, locked for C3, clean-up: no 5,6 in R23C3
7a. 5,6 in C3 only in 19(3) cage at R4C3 = {568}, locked for C3 and NR4C3 -> R6C2 = 7, placed for NR4C3, R6C1 = 1, R7C2 = 6, placed for NR6C1, R7C1 = 7, clean-up: no 4,5 in R2C1, no 4 in R2C2, no 1 in R2C3, no 2,3 in R3C1, no 2,8 in R3C2, no 2,3 in R4C1, no 2,8 in R4C2, no 2 in R5C1, no 1,2 in R5C2
7b. Naked pair {34} in R5C12, locked for R5 and NR4C3
7c. Naked pair {59} in R8C12, locked for R8 and NR6C1, clean-up: no 2 in R7C4

8. R1C3 = 9 (hidden single in C3), placed for NR1C3, R9C3 = 3 (step 1), R8C3 = 4, placed for NR6C1, R7C3 = 1
8a. 45 rule on NR1C3 1 remaining innie R3C2 = 2, placed for NR1C3, R2C3 = 7, clean-up: no 6 in R2C6, no 1 in R3C6

9. Naked pair {29} in R45C4, locked for C4
9a. Naked pair {38} in R78C4, locked for C4 and NR2C6, clean-up: no 5 in R23C6
9b. Naked pair {28} in R9C12, locked for R9

10. 16(3) cage at R1C4 = {457} (only remaining combination), locked for C4 and NR1C3 -> R6C4 = 6, placed for NR2C6, R3C6 = 7, R2C6 = 1, both placed for NR2C6, R6C3 = 5, R6C7 = 4, placed for NR3C7, R4C8 = 2, placed for NR3C7, R3C7 = 8 (step 4), placed for NR3C7, R2C7 = 6, placed for NR1C7, R4C9 = 7, placed for NR1C7, clean-up: no 5 in R2C2, no 1 in R3C2, no 5 in R5C89

11. R4C4 = 9, R4C56 = {45} = 9 -> R5C5 = 9, R6C56 = [29], R5C6 = 5 (cage sum)

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

Jigsaw nonet design: Lego man by Cyndie Smith

SumoCueV1=8J0+0J0+0J0=18J1+3J1+3J1=19J2+6J2+6J2=17J0+9J0+9J0=13J0+12J1+12J2=15J2+15J2+15J2=9J3=11J0=11J0=15J1=6J1=9J1=13J2=7J2=9J4+18J3+19J3+20J5+21J5+22J1+23J5+24J5+25J4+26J4=12J3+36J3+36J3=10J5+39J1+39J5=23J4+42J4+42J4=16J3=14J3=10J3=9J5=12J5=8J5=5J4=8J4=8J4+45J6+46J6+47J6+48J6+49J7+50J8+51J8+52J8+53J8=12J6+63J6+63J6=23J7+66J7+66J7=10J8+69J8+69J8=15J6+72J6+72J7=12J7+75J7+75J7=18J7+78J8+78J8

I've been taking an inventory of the walkthroughs on the forum and noticed that many of the earlier ones for TJK don't have walkthroughs.

This TJK #25 was very simple - I was really surprised. When I first started doing the TJKs I wouldn't attempt them unless there was a walkthrough posted that I could follow step by step. I just assumed that anything without a walkthrough was too difficult for me.

Haven't bothered with small text for this as it is an old one (as a side note when I was scouring the forum looking for walkthroughs I could follow I thought that the tiny text was some kind of special code that you needed to be a member to be able to interpret I know better know!)

Cheers
Caida


TJK 25 Walkthrough

Nonets:


111222333
111123333
411222336
445525566
444525666
444555666
777789999
777888999
778888899


Preliminaries:

a. 8(3)n1 = {125/134} (no 6..9) -> 1 locked for n1 and r1
b. 9(2)n4 and n25 and n6 and n57 = {18/27/36/45} (no 9)
c. 11(2)n14 and n15 = {29/38/47/56) (no 1)
d. 15(2)n25 = {69/78} (no 1..5)
e. 6(2)n2 = {15/24} (no 3,6..9)
f. 13(2)n35 = {49/58/67} (no 1..3)
g. 7(2)n36 = {16/25/34} (no 7..9)
h. 10(3)n52 and n9 = {127/136/145/235} (no 8,9)
i. 23(3)n6 and n8 = {689} (no 1..5,7) -> 6,8,9 locked for n6 and n8 and r5 and r8
k. 16(2)n47 = {79} (no 1..6,8) -> 7, 9 locked for c1
l. 14(2)n47 = {59/68} (no 1..4, 7)
m. 10(2)n47 = {19/28/37/46} (no 5)
n. 12(2)n58 = {39/48/57} (no 1,2,6)
o. 8(2)n59 and n69(x2) = {17/26/35} (no 4,8,9)
p. 5(2)n69 = {14/23} (no 5..9)

cleanup:
q. r3c8 no 1
r. r34c9 no 1,3
s. r6c5 no 3,4
t. r7c89 no 2
u. r34c1 no 2



1. hidden pair {89} in r9c89 -> no 8,9 elsewhere in r9
1a -> 18(3)n89 = [1]{89}

2. 12(3)n8 = {237/345}
2a. -> 3 locked in 12(3)n8 for n8 and r9
2b -> 2 in n8 locked in r9 -> no 2 elsewhere in r9
2c. cleanup: r6c5 no 9
2d. 5(2)n69 = {23} -> locked for c7

3. killer pair {45} locked in c5 for 6(2) and 12(2) -> no 4,5 elsewhere in c5

4. 1 locked in n3 in r2 -> no 1 elsewhere in r2

5. 4 locked in n9 in r8 -> no 4 elsewhere in r8
5a. 12(3)n7 = {237} -> locked for n7 r8
5b. 10(3)n9 = {145} -> locked in n9 r8
5c. cleanup: r67c1 = [79]
5d. r6c2 no 5
5e. r6c3 no 1,3,8
5f. r6c4 no 2,6
5g. r7c5 no 5
5h. r6c689no 3
5i. r3c2 no 4

6. LOL c1234: r49c3 = r27c4
6a. -> r7c4 no 1
6b. hidden single: r7c3 = 1
6c. r6c3 = 9
6d. cleanup r34c3 no 2
6e. r3c2 no 2
6f. r7c2 no 5
6g. r6c4 no 8

7. {68} locked in 14(2)n47 -> no 6,8 elsewhere in c2
7a. single: r9c1 = 6
7b. 14(2)n47 = [68]
7c. r6c4 no 1,3
7d. single: r6c7 = 3 -> r7c7 = 2
7e. 9(2)n57 = {45} -> 4,5 locked for c4
7f. r3c8 no 4


8. LOL c6789: r49c7 = r27c4
8a. -> r9c7 = 1 -> r2c6 = 1
8b. cleanup: r34c6 no 8
8c. r7c6 no 7

9. 7 locked in n9 in r7 -> no 7 elsewhere in r7
9a. single: r7c5 = 4
9b. r7c4 = 5
9c. r6c4 = 4
9d. single: r9c2 = 4
9e. cleanup: r9c3 = 5
9f. r6c5 = 8
9g.r3c4 no 7
9h. hidden single: r5c4 = 1

10. 12(3)n4 = {345} -> locked for n4 and r5
10a. single: r4c2 = 2
10b. r3c2 = 9
10c. pair: r34c1 = {18} -> locked for c1
10d. single: r1c2 = 1
10e. r3c3 no 3,7; r34c3 no 6
10f. r4c4 no 6

11. 1 in r6 locked in n6 -> no 1 elsewhere in n6
11a. r3c8 no 5,6
11b. r3c9 no 7
11c. single: r4c9 = 7; r3c9 = 2
11d. r34c8 = [34]
11e. r34c3 = [83]
11f. r34c4 = [69]
11g. r34c1 = [18]
11h. r34c5 = [51]
11i. r34c6 = [45]
11j. r34c7 = [76]
11k. r67c6 = [26]
11l. r67c8 = [17]
11m. r67c9 = [53]

12. 10(3)n9 = [451]
12a. 23(3)n8 = [869]
12b. 10(3)n52 = [127]
12c. 12(3)n8 = [273]
12d. 18(3)n2 = [738]
12e. 13(3)n1 = [391]
12f. 8(3)n1 = [512]
12g. 17(3)n1 = [476]
12h. 19(3)n3 = [964]
12i. 15(3)n3 = [528]
12j. 12(3)n4 = [354]
12k. 12(3)n7 = [237]
12l. 23(3)n6 = [896]
12m. 18(3)n89 = [189]

(Archive Note) Some typos have been corrected.

As with other TJKs with low SS scores, I didn't use Law of Leftovers.

Prelims

a) R34C1 = {18/27/36/45}, no 9
b) R34C2 = {29/38/47/56}, no 1
c) R34C3 = {29/38/47/56}, no 1
d) R34C4 = {69/78}
e) R34C5 = {15/24}
f) R34C6 = {18/27/36/45}, no 9
g) R34C7 = {49/58/67}, no 1,2,3
h) R34C8 = {17/26/35}, no 4,8,9
i) R34C9 = {18/27/36/45}, no 9
j) R67C1 = {79}
k) R67C2 = {59/68}
l) R67C3 = {19/28/37/46}, no 5
m) R67C4 = {18/27/36/45}, no 9
n) R67C5 = {39/48/57}, no 1,2,6
o) R67C6 = {17/26/35}, no 4,8,9
p) R67C7 = {14/23}
r) R67C8 = {17/26/35}, no 4,8,9
s) R67C9 = {17/26/35}, no 4,8,9
t) 8(3) cage at R1C1 = {125/134}
u) 19(3) cage at R1C7 = {289/379/469/478/568}, no 1
v) 10(3) cage at R5C4 = {127/136/145/235}, no 8,9
w) 23(3) cage at R5C7 = {689}
x) 23(3) cage at R8C4 = {689}
y) 10(3) cage at R8C7 = {127/136/145/235}, no 8,9

Steps resulting from Prelims
1a. 8(3) cage at R1C1 = {125/134}, 1 locked for R1 and NR1C1
1b. Naked triple {689} in 23(3) cage at R5C7, locked for R5 and NR3C9, clean-up: no 1 in R3C8, no 1,3 in R34C9, no 2 in R7C8, no 2 in R7C9
1c. Naked pair {79} in R67C1, locked for C1, clean-up: no 2 in R34C1
1d. Naked triple {689} in 23(3) cage at R8C4, locked for R8 and NR7C5, clean-up: no 3,4 in R6C5
1e. 1 in NR1C7 only in R2C6789, locked for R2
1f. 1,2 in NR7C5 only in R9C34567, locked for R9

2. R9C89 = {89} (hidden pair in NR7C6), locked for R9, R9C7 = 1 (cage sum), clean-up: no 4 in R67C7
2a. Naked pair {23} in R67C7, locked for C7, locked for C7
2b. 1 in C5 only in R345, locked for NR1C4, clean-up: no 8 in R4C6

3. 6 in R9 only in R9C12, locked for NR7C1, clean-up: no 8 in R6C2, no 4 in R6C3, no 3 in R6C4
3a. R6C4 = 4 (hidden single in R6), placed for NR4C3, R7C4 = 5, placed for NR7C1, clean-up: no 7 in R3C3, no 5 in R3C6, no 9 in R3C7, no 9 in R6C2, no 7 in R6C5, no 3 in R6C6, no 3 in R6C8, no 3 in R6C9

4. 5 in NR7C6 only in 10(3) cage at R8C7 = {145} (only remaining combination, cannot be {235} which clashes with R7C7), locked for R8 and NR7C6
4a. Naked triple {237} in 12(3) cage at R8C1, locked for NR7C1 -> R7C1 = 9, R7C2 = 8, R6C1 = 7, R6C2 = 6, both placed for NR3C1, R9C2 = 4, placed for NR7C1, R9C1 = 6, R9C3 = 5 (cage sum), R7C3 = 1, R6C3 = 9, placed for NR3C1
4b. R7C5 = 4 (hidden single in R7), R6C5 = 8, placed for NR4C3
4c. 12(3) cage at R5C1 = {345} (only remaining combination), locked for R5 and NR3C1 -> R4C2 = 2, R3C2 = 9, placed for NR1C1
Clean-ups: no 2,3,6 in R3C3, no 7 in R3C4, no 2 in R3C5, no 7 in R3C6, no 5 in R3C7, no 5 in R3C8, no 7 in R3C9, no 6 in R4C3, no 6 in R4C4, no 7 in R6C6, no 7 in R6C8, no 7 in R6C9, no 2 in R7C6

5. R6C7 = 3 (hidden single in R6), placed for NR3C9, R7C7 = 2, clean-up: no 4 in R3C8
5a. Naked pair {15} in R34C5, locked for C5 and NR1C4

6. Naked pair {18} in R34C1, locked for C1
6a. R2C3 = 6 (hidden single in C3), R2C12 = 11 = [47], 4 placed for NR1C1, 12(3) cage at R8C1 = [237], 12(3) cage at R5C1 = [354], R1C12 = [51], R1C3 = 2, placed for NR1C1

7. R3C3 = 8, placed for NR1C1, R2C4 = 3, R2C56 = 10 = [28/91]
7a. R3C4 = 6, placed for NR1C4, R6C4 = 9, R8C4 = 8, R1C4 = 7, placed for NR1C4, R5C5 = 2, placed for NR1C4, R2C5 = 9, R1C5 = 3, both placed for NR1C4, R1C6 = 8 (cage sum), R3C6 = 4, R4C6 = 5, R4C5 = 1, R4C8 = 4, R4C9 = 7, R7C9 = 3, R6C9 = 5

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

Jigsaw nonet design: Orig 8

SumoCueV1=15J0+0J0=13J0=29J1+3J1+3J1+3J1=11J1+7J2=6J3+0J0+2J0+3J1=22J1+13J1+3J2+7J1=24J2+9J3=30J3+2J0=15J0+21J4+13J4+17J2+17J2+17J2+9J3+19J3+19J0=16J0+21J4=16J4+32J2=11J2=6J2+19J3+19J3=14J3+30J4+32J4+32J4=21J4+34J5+35J5=18J6+45J6+38J3=18J6+48J6=8J4+42J5+42J5+35J5=15J6+45J6+38J6+48J6+48J7+50J7=8J7+60J5=14J5+54J8=24J8+64J8=13J6=20J7+67J7+67J7+62J5+62J5+54J8+64J8+64J8+66J8+66J8=7J8+77J7=11J7+79J7

Fascinating creatures, these jigsaw killers, requiring all the skills for solving a regular killer, plus the ability to "think round corners". Thought they may be becoming extinct after waiting 4 weeks since puzzle #25. To show that they are very much appreciated (despite the relative inactivity on this forum), I've decided to post this walkthrough:


Walkthrough - Texas Jigsaw Killer 026

Nonets (N1 through N9 are numbered in order of their top-left cells, reading from left to right, top to bottom, as shown below:

111222223
411222323
441155333
441155333
444555566
774775666
777788866
999788866
999999888

1. Preliminaries:

a) 11/3 at R1C8: no 9
b) 6/3 at R234C1 = {123} -> no 1,2,3 elsewhere in C1 or N4
c) 9 locked in 22/3 at R2C5 -> no 9 in R1C6
d) 16/2 at R45C4 = {79} -> no 7 or 9 in R1236789C4 and R4C56
e) 11/2 at R45C8: no 1
f) 6/3 at R456C9 = {123} -> no 1,2,3 elsewhere in C9
g) 21/3 at R5C7: no 1,2,3
h) 8/2 at R67C6: no 4,8,9
i) 15/3 at R789C1 = {456} -> no 4,5,6 elsewhere in C1
j) 8/2 at R7C78: no 4,8,9
k) 20/3 at R8C567: no 1,2
l) 7/2 at R9C67: no 7,8,9
m) 11/2 at R9C89: no 1

2. Outie N4: R47C3 = 5 -> R47C3 = {1234}

3. Outie R89: R7C19 = 14 = [59|68] -> R8C8 = {12}, R8C9 = {45}

4. Innie N8: R7C567+R9C7 = 14 -> no 9 in R7C5

5. Innie/outie difference N9: R7C1+R8C4-R9C6 = 7 -> R8C4-R9C6 = 1|2 -> no 1 in R8C4

6. Naked triple {123} at R56C9+R8C8 in N6 -> no 1,2,3 elsewhere in N6

7. Innie N1: R34C4+R4C3 = 17 -> no 1 in R4C3 (would require 7|9 in R3C4 -> clash with R45C4)

8. R4C3 is peer of all candidate positions for digit 4 in N4 -> no 4 in R4C3 -> R47C3 = {23} (see step 2)

9. Naked triple {123} at R4C139 in R4 -> no 1,2,3 elsewhere in R4

10. Innie N789: R7C367 = 9 = {1(26|35)} (no 4 available) -> R7C67 = {156}

11. From step 10: Innie N789 (R7C367) forms killer pair on {56} with R7C1 -> no 5,6 elsewhere in R7

12. Naked single R7C8 = 7 -> R7C7 = 1, R7C6 = {56}, R6C6 = {23}

13. Innie N8: Split cage 13/3 at R7C56+R9C7 -> no 8 in R7C5, no 2 in R9C7

14. R45C8 = {56} -> no 5,6 elsewhere in C8

15. LoL R89: Outie R7C567 = {1..} -> innie R8C489 = {1..} -> R8C8 = 1 -> R4C9 = 1

16. 11/3 at R1C8 = {2..} (1 no longer available) -> 2 in N2 locked in R12C8 -> no 2 elsewhere in C8 or N2

17. Hidden single N8: R7C5 = 2 -> R7C3 = 3, R4C3 = 2 (see step 8), R4C1 = 3

18. Innie N789: R7C6 = 5 (see step 10) -> R6C6 = 3 -> R6C9 = 2, R5C9 = 3

19. R7C19 = [68] -> R7C24 = [94], R89C1 = [45], R8C9 = 5 -> R45C8 = [56]

20. LoL R89: Outie R7C567 = [251] -> R8C4 = 2

...

(Archive Note) One typo corrected.

I just finished this one -- couldn't do it in one sitting but finished it in the next (which is a rarity, usually if I can't finish it at once I give up altogether).

I'm going to post my walkthrough as it's rather different -- it may be somewhat circuitous, I'm sure it isn't minimal, but it's the way I solved it. Despite what was said at the top (that the puzzle requires "lots of LoL") I only used one LoL move in the entire puzzle.
Archive Note: Because of the way that the Texas Jigsaw Killers were posted on sudocue.net, Ruud’s introductory notes weren’t available except for his latest TJK.

Nonet numbering:
111222223
411222323
441155333
441155333
444555566
774775666
777799966
888799966
888888999


0a. 7/2 in R9 does not contain 789 (sums)
0b. 8/2 in C6 does not contain 489 (sums)
0c. 8/2 in R7 does not contain 489 (sums)
0d. 11/2 in C8 does not contain 1 (sums)
0e. 11/2 in R9 does not contain 1 (sums)
0f. 16/2 in C4 = {79} naked pair (sums), elim 79 from rest of C4
0g. 6/3 in C1 = {123} naked triple (sums), elim 123 from rest of C1 & N4
0h. 6/3 in C9 = {123} naked triple (sums), elim 123 from rest of C9
0i. 11/3 in R12 does not contain 9 (sums)
0j. 20/3 in R8 does not contain 12 (sums)
0k. 21/3 in R78 does not contain 123 (sums)
0l. 22/3 in R23 does not contain 1234, must contain 9 (sums), elim 9 from R1C6
1. 15/3 in C1 without 123 must be {456} naked triple (sums), elim 456 from rest of C1
2. outies of R89 = 14/2 = [59|68]
3. 4 of c1 locked in N8 -> elim from rest of N8
4. outies of N4 = R47C3 = 5/2 = {14|23}
5. combinations for 14/3 in N6 = [914|815|824]
6. R56C9+R8C8 = {123} naked triple, elim 123 from rest of N6
7. combinations for 8/2 in R7 = [17|26|35]
8. 24/4 in N8 must have 5 or 6, forms killer triple {456} with R89C1, elim 456 from rest of N8
9. Outies-innies of N14 = R7C3-R34C4 = -12 -> R34C4 >= 13 -> R3C4 >= 4; additionally, since R34C4 cannot be 16 (sums) R7C3 is not 4
10. R5C569 = hidden triple {123}
11. 8/2 in C6 must have 1, 2 or 3; forms killer triple {123} with R59C6; elim 123 from rest of C6
12. Outies-innies of N8 = R7C1+R8C4-R9C6 = 7 -> 8<=R7C1+R8C4<=10 -> 2<=R8C4<=5
13. sums for 7/2 in R9 = [16|25|34]
14. 1 of N9 locked in R7 -> elim from rest of R7
15. 1 of R9 locked in N8 -> elim from rest of N8
16. R8C8 = hidden single 1 in R8
17. R4C9 = hidden single 1 in C9
18. 1 of R5 locked in N5 -> elim from rest of N5
19. combinations for 11/3 in R12 = {236|245} (elim 78 from 11/3), 2 locked in R12C8 -> elim 2 from rest of N2 and C8
20. combinations for 11/2 in R9 = [38|47|56|65|74]
21. 9 of R9 locked in N8 -> elim from rest of N8
22. 9 of R8 locked in N9 -> elim from rest of N9
23. Repeat step 4, hidden cage 5/2 in R47C3 = {23} pair (sums) -> elim 23 from rest of C3
24. R4C13 = {23} naked pair -> elim 23 from rest of R4
25. combinations for 11/2 in C8 = {47|56}
26. R36C8 = hidden pair {89}
27. Outies-innies of C89 = R37C7-R6C8 = -5 -> 3<=R37C7<=4 -> R7C7 = 1 && R3C7 = {23} -> R7C8 = 7 (last digit in cage)
28. combinations for 11/2 in C8 = {56} pair -> elim 56 from rest of C8
29. Combinations for 11/2 in R9 = [38|47]
30. Combinations for 11/3 in R12 = [263|254|362|452] -> elim 4 from R1C9
31. LOL on R6789 -> R6C3 = {56}, R6C6 = {23}
32. Combinations for 14/3 in C3 = [563|653|752], elim 5 from rest of C3 and N4
33. only possibility for 13/3 in C3 is {148} -> elim 148 from rest of N1 and C3
34. R1C1+R4C4 = naked pair {79} in N1, elim 79 from rest of N1
35. R6C69 = naked pair {23} in R6, elim 23 from rest of R6
36. combinations for 8/2 in C6 = [26|35]
37. R7C16 = naked pair {56} in R7, elim 56 from rest of R7
38. R9C3 = hidden single 9 in C3
39. 4 of N4 locked in C2 -> elim 4 from rest of C2
40. 4 of R7 locked in 18/4 in C45, elim 4 from R6C45
41. rest is naked singles, hidden singles, and last-digit-in-cage moves

(Archive Note) One typo corrected.

As with other TJKs with low SS scores, I didn't use Law of Leftovers.

Prelims

a) R45C4 = {79}
b) R45C8 = {29/38/47/56}, no 1
c) R67C6 = {17/26/35}, no 4,8,9
d) R7C78 = {17/26/35}, no 4,8,9
e) R9C67 = {16/25/34}, no 7,8,9
f) R9C89 = {29/38/47/56}, no 1
g) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9
h) 6(3) cage at R2C1 = {123}
i) 22(3) cage at R2C5 = {589/679}
j) 6(3) cage at R4C9 = {123}
k) 21(3) cage at R5C7 = {489/579/678}, no 1,2,3
l) 20(3) cage at R8C5 = {389/479/569/578}, no 1,2

Steps resulting from Prelims
1a. Naked triple {123} in 6(3) cage at R2C1, locked for C1 and NR2C1
1b. Naked pair {79} in R45C4, locked for C4
1c. Naked triple {123} in 6(3) cage at R4C9, locked for C9, clean-up: no 8,9 in R9C8
[There’s also CPEs from the 22(3) cage at R2C5 and the naked pair {79} using a jigsaw nonet but, since this puzzle has a low score I didn’t use them.]

2. 15(3) cage at R7C1 = {456} (only remaining combination), locked for C1
2a. 15(3) cage at R1C1 cannot contain more than one of 7,8,9, R1C1 = {789} -> no 7,8,9 in R12C2

3. 45 rule on R89 2 outies R7C19 = 14 = [59/68]
3a. R7C9 = {89} -> R8C89 = 5,6 = [14/15/24]
3b. Naked triple {123} in R56C9 + R8C8, locked for NR5C8, 3 also locked for C9, clean-up: no 8,9 in R4C8, no 5,6,7 in R7C7
3c. 4 in C1 only in R89C1, locked for NR8C1, clean-up: no 3 in R9C7

4. R5C569 = {123} (hidden triple in R5)
4a. 16(4) cage at R4C6 cannot contain more than two of 1,2,3 -> no 1,2,3 in R4C67

5. 14(3) cage at R5C3 must contain at least one of 1,2,3 -> R7C3 = {123}

6. 1 in NR7C5 only in R7C567 + R9C7
6a. 45 rule on NR7C5 4 innies R7C567 + R9C7 = 14 = {1238/1247/1256/1346}, no 9 in R7C5

7. 45 rule on C89 1 innies R6C8 = 2 outie R37C7 + 5
7a. Min R37C7 = 3 -> min R6C8 = 8
7b. R6C8 = {89} -> R37C7 = 3,4 = {12/13}, 1 locked for C7, clean-up: no 6 in R9C6
7c. Naked pair {89} in R6C8 + R7C9, locked for NR5C8, clean-up: no 2,3 in R4C8
7d. 1 in NR7C5 only in R7C567, locked for R7

8. 14(3) cage at R5C3 = {248/257/347/356} (cannot be {239} because 2,3 only in R7C3), no 9

9. 9 in NR2C1 only in 30(5) cage at R3C2 = {15789/24789/25689/34689/35679} -> R4C3 = {123}
9a. Naked triple {123} in R4C139, locked for R4

10. 13(3) cage at R1C3 = {148/157/238/247/256/346} (cannot be {139} which clashes with R47C3, ALS block), no 9
10a. Killer triple 1,2,3 in 13(3) cage, R4C3 and R7C3, locked for C3
10b. 1 in C3 only in R1234C3, locked for NR1C1
10c. 9 in C3 only in R89C3, locked for NR8C1

11. R1C1 = {789} -> 15(3) cage at R1C1 must contain one of 2,3, 13(3) cage at R1C3 (step 10) = {148/157/238/247/256/346}
11a. Killer triple 1,2,3 in 15(3) cage, 13(3) cage and R4C3, locked for NR1C1
11b. Min R3C4 + R4C5 = 9 -> max R3C5 = 6

12. Max R23C9 + R3C7 = 20 -> min R3C8 = 4
12a. R23C7 + R4C9 = {123} (hidden triple in NR1C9)
12b. Naked triple {123} in R237C7, locked for C7, clean-up: no 5 in R9C6
12c. Killer triple 1,2,3 in R5C6, R67C6 and R9C6, locked for C6
12d. 1 in R9 only in R9C2456, locked for NR8C1

13. R7C567 + R9C7 (step 6a) = {1247/1256/1346} (cannot be {1238} because R9C7 only contains 4,5,6), no 8
13a. Killer triple 1,2,3 in R7C3 and R7C567, locked for R7

14. 45 rule on NR8C1 3(1+1+1) outies R7C1 + R8C4 + R9C7 = 14
14a. Max R7C1 + R9C7 = 12 -> min R8C4 = 2
14b. Min R7C1 + R9C7 = 9 -> max R8C4 = 5
14c. 1 in NR6C1 only in R6C245, locked for R6, clean-up: no 7 in R7C6

15. R8C8 = 1 (hidden single in R8), placed for NR5C8
15a. R4C9 = 1 (hidden single in C9), placed for NR1C9
15b. R7C7 = 1 (hidden single in C7), R7C8 = 7, placed for NR5C8, clean-up: no 4 in R45C8, no 7 in R6C6, no 4 in R9C9
15c. Naked pair {56} in R45C8, locked for C8, clean-up: no 5,6 in R9C9

16. 11(3) cage at R1C8 = {236/245} -> R12C8 = {23/24}, 2 locked for C8 and NR1C4, R1C9 = {56}, clean-up: no 9 in R9C9
16a. Naked triple {234} in R129C8, locked for C8
16b. R9C67 = [16/25] (cannot be [34] which clashes with R9C8)
16c. R9C3 = 9 (hidden single in R9)

17. 21(3) cage at R5C7 = {489/579/678}
17a. R6C7 = {456} -> no 4,5,6 in R5C7
17b. Naked triple {789} in R5C147, locked for R5
17c. 4 in R5 only in R5C23, locked for NR3C1

18. 1 in C3 only in 13(3) cage at R1C3 = {148/157}, no 2,3,6
18a. 14(3) cage at R5C3 (step 8) = {248/257/356} (cannot be {347} which clashes with 13(3) cage)
18b. Killer pair 4,5 in 13(3) cage and 14(3) cage, locked for C3
18c. Killer triple 7,8,9 in R1C1, 13(3) cage and R4C4, locked for NR1C1

19. R7C567 + R9C7 (step 13) = {1256/1346}, 6 locked for NR7C5
19a. 20(3) cage at R8C5 = {389/479} (cannot be {578} which clashes with R9C9), no 5
[Alternatively 9 in R8 only in 20(3) cage …]
19b. Killer pair 3,4 in 20(3) cage and R9C8, locked for NR7C5, clean-up: no 5 in R6C6
19c. Naked triple {256} in R7C156, locked for R7 -> R7C3 = 3, placed for NR6C1, R4C3 = 2, placed for NR1C1, R4C1 = 3
19d. R7C3 = 3 -> 14(3) cage at R5C3 (step 18a) = {356} (only remaining combination), locked for C3 and NR2C1 -> R5C2 = 4
19e. Naked triple {789} in R347C2, locked for C2, 7 also locked for NR2C1

20. R7C4 = 4 (hidden single in R7)
20a. 18(3) cage at R7C1 = {189/279}, 9 locked for NR7C1
20b. 1,2 only in R7C2 -> R7C2 = {12}

21. 13(3) cage at R1C3 (step 18) = {148} (only remaining combination), locked for C3 and NR1C1 -> R8C3 = 7, placed for NR8C1

22. R9C9 = 7 (hidden single in R9), R9C8 = 4, placed for NR7C5, R12C8 = {23}, locked for NR1C3, R1C9 = 6 (cage sum), placed for NR1C9, R4C8 = 5, placed for NR1C9, R4C7 = 7 (hidden single in NR1C9), R4C4 = 9, placed for NR1C1, R4C2 = 8, R5C4 = 7, placed for NR3C5

23. 4 in NR1C9 only in R23C9, locked for C9 -> R8C9 = 5, R7C9 = 8 (cage sum), both placed for NR5C8, R6C8 = 9, R5C7 = 8, R6C7 = 4 (cage sum), 20(3) cage at R8C5 = [389]

24. R3C7 = 3 (hidden single in R3), R2C7 = 2, R12C8 = [23]

25. R1C1 = 7, R1C2 = 3 (hidden single in R1), R2C2 = 5 (cage sum), placed for NR1C1, R3C4 = 6, R4C5 = 4, R3C5 = 5 (cage sum), R4C6 = 6, clean-up: no 2 in R67C6

26. R67C6 = [35]

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

Jigsaw nonet design: ZigZag by Gérard Coteau

SumoCueV1=12J0=16J0+1J0=17J0+3J1=12J2+5J2=5J2+7J2+0J0+1J0=15J3+3J0=20J1+13J1+5J2=20J2+16J2+0J0+11J3+11J3=16J4+13J1=8J1+23J1=16J2+16J5=15J0+27J3+21J3+21J4=9J4+23J1=13J1+25J2=8J5=10J3=19J3=15J6+38J4+31J4+33J4+33J1+25J5+35J5+36J3+37J7+38J6=14J6+31J4=20J4+50J5=13J5+52J8=12J3+37J7+48J6+48J6=12J6+50J4=20J5+60J5=10J8+54J7+54J7=13J7+58J6+58J6=17J8+60J5=18J8+62J8=10J7+72J7+65J7+65J7+68J6+68J8+70J8+70J8+62J8

This was another very enjoyable jigsaw killer to do on paper. Thanks, Ruud.

In contrast to Ruud's warning that the solving path was "long, very long", the puzzle could actually be solved pretty quickly.

Here's the walkthrough:


Texas Jigsaw Killer 027

Nonet layout:

111123333
114122333
144522236
144552236
447555266
487755669
487775669
888779699
888879999

1. Preliminaries:

a) 5/2 cage at R1C8 - no 5,6,7,8,9
b) 20/3 cage at R2C5 - no 1,2
c) 20/3 cage at R2C8 - no 1,2
d) 8/3 cage at R3C6 = {1(25|34)} -> no 1 elsewhere in N2
e) 15/2 cage at R4C1 = {69|87}
f) 9/3 cage at R4C5 - no 7,8,9
g) 8/2 cage at R4C9 - no 4,8,9
h) 10/2 cage at R5C1 - no 5
i) 19/3 cage at R5C2 - no 1
j) 20/3 cage at R6C6 - no 1,2
k) 13/2 cage at R6C8 - no 1,2,3
l) 20/3 cage at R7C7 - no 1,2
m) 10/3 cage at R7C9 - no 8,9
n) 10/2 cage at R9C1 - no 5

2. Innies/outies:

a) Outies N1: R1C5+R4C2 = 15/2 = {69|87} -> no 2,3,4,5 in R1C5
b) Outies N3: R3C9+R5C8 = 8/2 -> no 4,8,9
c) Outies N8: R5C2+R7C1 = 9/2 -> no 9
d) Outies N9: R6C8+R9C5 = 13/2 -> no 1,2,3 in R9C5
e) Innies N48: R4C23 = 11/2 -> no 1 in R4C3
f) Outies N12: R4C2+R5C6 = 11/2 -> no 1 in R5C6 -> no 9 in 13/3 cage at R4C7
g) From steps 2e) and 2f) above, we can conclude that R4C3 = R5C6 (common cell, matching cage sums)
-> no 9 in R4C3
h) Innies N36: R6C78 = 9/2 -> {36|45} (1,2 not available)
i) Outies N79: R5C4+R6C8 = 9/2 -> no 9 in R5C4
j) Outies C9: R126C8 = 11/3 -> no 9 in R2C8
k) Innies N7: R56C3+R9C5 = 19/3 -> no 1 in R56C3
l) Outies C12: R123C3 = 19/3 -> no 1
m) Outies C89: R789C7 = 20/3 -> no 1,2 in R9C7
n) Innies C9: R1236C9 = 27/4 = {(37|46)89} -> no 1,2,5
o) Outies N3679: R5C4+R67C6 = 20/3 -> no 1,2 in R5C4
p) Innies R1234: R4C579 = 9/3 (no eliminations currently possible)
q) Outies C1: R489C2 = 14/3 (no eliminations currently possible)
r) Innies C6789: R289C6 = 17/3 (no eliminations currently possible)
s) Innies C89: R789C8 = 18/3 (no eliminations currently possible)
t) Innie/outie difference C89: R9C7 - R7C8 = 0 -> R9C7 = R7C8

3. {89} in N6 locked in 20/3 cage at R7C7 = {389} -> no 3,8,9 elsewhere in N6
4. No 3 in 20/3 cage at R2C8
5. {12} in N6 locked in 8/2 cage at R4C9 and hidden 8/2 cage R3C9+R5C8 (see step 2b) = {{6,2},{7,1}}
-> R3C9 = {67} and R5C8 = {12}, no 6 in R6C78 = {45}
-> no 4,5 elsewhere in R6 or N6, R6C9 = {89}, R67C6 = {6789}, R9C5 = {89} (see step 2d), no 6 in R5C1

6. 5 in C9 now locked in 10/3 cage at R7C9 = {(14|23)5} -> no 6,7
7. R34C8 (14/2 or 15/2) = {59|68|69|78} -> no 1,2,3,4

8. 5/2 cage R1C89 = [14|23] -> no 3,4 in R1C8
9. 20/3 cage R2C89+R3C9 = [596|587|497] -> R2C8 = {45}, R2C9 = {89}

10. Naked pair on {12} in C8 at R15C8 -> no 1,2 in R89C8
11. Naked pair on {45} in C8 at R26C8 -> no 4,5 in R89C8

12. R26C8 = {45} = 9/2 -> R1C8 = 2 (see step 2j) -> R1C9 = 3, R5C8 = 1 -> R3C9 = 7 (see step 2b)
-> 8/2 cage at R45C9 = {26} -> 10/3 cage at R7C9 = {145} -> no 1,4,5 elsewhere in N9

13. 1 in N3 locked in 12/3 cage at R1C6 = {1(47|56)} -> no 8,9
14. 2 in N9 locked in C6 -> 17/3 cage at R8C6 = {2(69|78)} -> no 3, no 2 elsewhere in C6

15. R4C9 = {26} -> innies R1234 (R4C579) = {126} (see step 2p) -> no 1,2,6 elsewhere in R4
-> 15/2 cage at R4C1 = {78} -> no 7,8 elsewhere in R4, R1C5 = {78} (see step 2a)

16. Naked single R4C8 = 9 -> R3C8 = 6
17. Naked single R2C9 = 8 -> R2C8 = 5
18. Naked single R6C8 = 4 -> R6C9 = 9, R9C5 = 9 (see step 2d)
19. Naked single R6C7 = 5
20. Hidden single in R5 at R5C1 = 9 -> R6C1 = 1
21. Hidden single in R3 at R3C4 = 9
22. Hidden single in C6 at R2C6 = 9
23. Hidden single in R4 at R4C5 = 1

24a. R5C6 = R4C3 = {34} (see steps 2e and 2g)
24b. 16/3 cage at R3C4 = {349} -> no 5 in R4C4 -> {34} in R4 locked in R4C34
-> no 3,4 elsewhere in R4 -> Naked single R4C6 = 5 -> R3C6 = 1, R3C7 = 2

25. Naked single R4C7 = 6 -> R4C9 = 2 -> R5C9 = 6
26. Hidden single in C6 at R5C6 = 3 -> R5C7 = 4
27. Naked single R4C4 = 4 -> R4C3 = 3 -> R4C2 = 8 (see step 2e) -> R4C1 = 7, R1C5 = 7 (see step 2e)

28. Naked single R1C6 = 4 -> R1C7 = 1 -> R2C7 = 7
29. Naked single R2C5 = 3 -> R3C5 = 8
30. Hidden single in R3 at R3C1 = 3
31. Hidden single in N4 at R5C2 = 7 -> R7C1 = 2 (see step 2c)
32. Hidden single (in 15/3 cage at R2C3) R2C3 = 6
33. Naked single R2C1 = 4 -> R1C1 = 5

Now we're on the way home...


Please note that there was a logic error in the above walkthrough at step 24, which I have now corrected.

(Archive Note) Some typos have been corrected.

Thanks Ed for pointing out a few things I'd missed, which would have simplified some of my steps.

Prelims

a) R1C89 = {14/23}
b) R4C12 = {69/78}
c) R45C9 = {17/26/35}, no 4,8,9
d) R56C1 = {19/28/37/46}, no 5
e) R6C89 = {49/58/67}, no 1,2,3
f) R9C12 = {19/28/37/46}, no 5
g) 20(3) cage at R2C5 = {389/479/569/578}, no 1,2
h) 20(3) cage at R2C8 = {389/479/569/578}, no 1,2
i) 8(3) cage at R3C6 = {125/134}
j) 9(3) cage at R4C5 = {126/135/234}, no 7,8,9
k) 19(3) cage at R5C2 = {289/379/469/478/568}, no 1
l) 20(3) cage at R6C6 = {389/479/569/578}, no 1,2
m) 20(3) cage at R7C7 = {389/479/569/578}, no 1,2
n) 10(3) cage at R7C9 = {127/136/145/235}, no 8,9

1. 8(3) cage at R3C6 = {125/134}, 1 locked for NR1C5

2. 45 rule on C1 3 outies R489C2 = 14
2a. Min R4C2 = 6 -> max R89C2 = 8, no 8,9 in R89C2, clean-up: no 1,2 in R9C1

3. 45 rule on C9 3 outies R126C8 = 11 = {137/146/236/245} (cannot be {128} because 1,2 only in R1C8), no 8,9 in R26C8, clean-up: no 4,5 in R6C9
3a. 20(3) cage at R2C8 = {389/479/569/578}
3b. 3 of {389} must be in R2C8 -> no 3 in R23C9

4. 45 rule on C12 1 outie R1C3 = 1 innie R3C2 + 4, R1C3 = {56789}, R3C2 = {12345}

5. 45 rule on C1234 1 outie R1C5 = 1 innie R8C4 + 4, R1C5 = {56789}, R8C4 = {12345}

6. 45 rule on NR1C6 2 outies R3C9 + R5C8 = 8 = [53/62/71]
6a. R26C9 = {89} (hidden pair in C9), clean-up: no 6,7 in R6C8
6b. 20(3) cage at R2C8 = {479/569/578} (cannot be {389} because R3C9 only contains 5,6,7), no 3

7. 45 rule on NR6C9 1 outie R9C5 = 1 innie R6C9 -> R9C5 = {89}
7a. 17(3) cage at R8C6 cannot contain both of 8,9 -> no 8,9 in R89C6

8. 45 rule on NR1C6 + NR3C9 1 outie R6C9 = 1 innie R6C7 + 4, R6C9 = {89} -> R6C7 = {45}
8a. Naked pair {45} in R6C78, locked for R6 and NR3C9, clean-up: no 3 in R45C9, no 6 in R5C1, no 3 in R5C8 (step 6)
8b. Killer pair 6,7 in R3C9 and R45C9, locked for C9 and NR3C9
8c. 20(3) cage at R2C8 (step 6b) = {479/569/578}
8d. 4,5 only in R2C8 -> R2C8 = {45}
8e. Naked pair {45} in R26C8, locked for C8, clean-up: no 1 in R1C9
[Ed pointed out that I missed -> R1C8 = 2 (step 3), which would have simplified the rest of this step and given an extra elimination in step 9. I only got that placement in step 16c.]
8f. 10(3) cage at R7C9 = {145/235}, 5 locked for NR6C9
8g. Killer pair 1,2 in R45C9 and 10(3) cage, locked for C9, clean-up: no 3 in R1C8
8h. Naked pair 1,2 in R15C8, locked for C8

9. 16(3) cage at R3C8 = {169/178/259/268} (other combinations don’t contain 1 or 2 for R5C8), no 3
9a. Killer pair 8,9 in R2C9 and 16(3) cage, locked for NR1C6

10. 45 rule on C6789 1 outie R9C5 = 1 innie R2C6 -> R2C6 = {89}
10a. Naked pair {89} in R2C69, locked for R2
10b. 20(3) cage at R2C5 = {389/479/569/578}
10c. 3 of {389} must be in R2C5 -> no 3 in R3C5

11. 20(3) cage at R6C6 = {479/569/578} (cannot be {389} because R6C7 only contains 4,5), no 3
11a. R6C7 = {45} -> no 4,5 in R7C6
11b. Killer pair 8,9 in R2C6 and 20(3) cage, locked for C6

12. 45 rule on NR5C3 1 innie R9C5 = 1 outie R5C4 + 4, R9C5 = {89} -> R5C4 = {45}

13. 45 rule on NR1C1 1 outie R1C5 = 1 innie R4C1, no 5 in R1C5, clean-up: no 1 in R8C4 (step 5)

14. 45 rule on NR6C2 2 outies R5C2 + R7C1 = 9 = {27/36/45}/[81], no 9, no 8 in R7C1

15. 45 rule on NR2C3 + NR6C2 1 outie R4C1 = 1 innie R4C3 + 4 -> R4C3 = {2345}
15a. Max R4C3 = 5 -> min R34C4 = 11, no 1 in R34C4

[I’d seen this 45 earlier but it wasn’t helpful then]
16. 45 rule on C89 1 outie R9C7 = 1 innie R7C8 -> R9C7 = {389}
16a. Naked triple {389} in R789C7, locked for C7
16b. 18(3) cage at R8C8 = {369/378}, 3 locked for NR6C9
16c. 10(3) cage at R7C9 (step 8f) = {145} (only remaining combination), locked for C9 and NR6C9 -> R1C9 = 3, R1C8 = 2, placed for NR1C6, R5C8 = 1, R3C9 = 7 (step 6), clean-up: no 9 in R6C1
16d. 2 in NR6C9 only in R89C6, locked for C6

17. 1 in NR3C4 only in 9(3) cage at R4C5 = {126/135}, no 4, 1 locked for C5
17a. 12(3) cage at R7C5 = {237/246/345}, no 8,9

[I think the SS score is just high enough to allow me to use Law of Leftovers (LoL). This cage pattern has plenty of scope for using LoL for the columns.]
18. LoL for C1234 three innies R345C4 must exactly equal three outies R789C5, R5C4 = {45} -> R78C5 must contain at least one of 4,5 -> 12(3) cage at R7C5 (step 17a) = {246/345}, no 7, 4 locked for NR5C3
18a. 7 in C5 only in R12C5, locked for NR1C5
[Ed pointed out that I missed CPE no 7 in R2C4.]
18b. 7 in C7 only in R12C7, locked for NR1C6
18c. 12(3) cage at R1C6 = {147} (only remaining combination), locked for NR1C6 -> R2C8 = 5, R2C9 = 8 (cage sum), placed for NR1C6, R6C78 = [54], R6C9 = 9, placed for NR6C9, R9C5 = 9 (step 7), placed for NR5C3, R2C6 = 9, clean-up: no 9 in R4C1 (step 13), no 6 in R4C2, no 5 in R8C4 (step 5)
18d. R9C5 = 9 -> R89C6 = 8 = {26}, locked for NR6C9
18e. Naked pair {78} in R67C6, locked for C6 and NR3C4

19. R4C12 = {78} (hidden pair in R4), clean-up: no 6 in R1C5 (step 13)
[With hindsight I ought to have spotted R4C12 = {78}, CPE no 7,8 in R567C1 using NR2C3.
Ed pointed out that I’d also missed CPE no 7,8 in R12C2 using NR1C1.]

20. 6 in C7 only in R45C7, locked for NR1C5
20a. R2C6 = 9 -> R23C5 = 11 = [38/74]
20b. 5 in NR1C5 only in 8(3) cage at R3C6 = {125} (only remaining combination) -> R3C7 = 2, R45C6 = {15}, locked for C6 -> R1C6 = 4, R5C6 = 3, placed for NR3C4, clean-up: no 6 in R1C3 (step 4), no 7 in R6C1
[Alternatively 8(3) cage at R3C6 = {125} (only remaining combination, cannot be {134} which clashes with R23C5) …]

21. 9(3) cage at R4C5 (step 17) = {126} (only remaining combination), locked for C5 and NR3C4
21a. Naked triple {459} in R345C4, locked for C4
21b. Naked pair {26} in R5C59, locked for R5 -> R45C7 = [64], R34C8 = [69], R45C9 = [26], R5C4 = 5, 9(3) cage at R4C5 = [126], R34C6 = [15] (order of these placements changed), 16(3) cage at R3C4 = [934], 3 placed for NR2C3
21c. Naked pair {78} in R45C2, locked for C2 and NR2C3 -> R5C1 = 9, R6C1 = 1, placed for NR2C3
[I’ve omitted routine clean-ups from this stage.]

22. Naked pair {45} in R3C23, locked for R3 -> R3C5 = 8
22a. R3C23 = {45} = 9 -> R2C3 = 6 (cage sum)
22b. R7C1 = 2 (hidden single in NR2C3), R5C2 = 7 (step 14), R67C2 = 12 = [39], both placed for NR6C2, R5C3 = 8, R6C3 = 2 (cage sum), both placed for NR5C3, R6C4 = 7, R7C34 = 7 = [16]

23. R1C5 = 7 -> R12C4 = 10 = [82], R9C4 = 1, placed for NR6C2, R89C3 = 12 = {57}, locked for C3 and NR6C2

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

Jigsaw nonet design: Orig 8

SumoCueV1=7J0+0J0=12J1+2J1=19J1=16J2+5J2=11J2+7J2=21J3=11J0+10J0+2J1+4J1+4J1+5J2+7J2=21J2+9J3+9J3+10J0=27J0=16J1+22J1=8J1+24J2+17J2=10J3+27J3+27J3+21J0+21J0+22J4=17J4+33J4+17J4=16J5=14J3+37J3=24J6+21J0=11J7+41J7+33J4=15J4+36J5+36J5+37J3+39J6+39J6+41J7=14J7+44J4+44J4=16J5+54J5+54J8=22J8+39J6=16J6+51J7+51J7+44J4=9J5=16J5+64J8+57J8=11J8+59J6+59J6=20J7+70J7+63J5+64J5+57J8+57J8+67J8+67J8=5J6+78J6+70J7

It was a great challenge - took ages to find the key. On the 3rd attempt finally found the way without chains. I here-by declare my love for LoL!!

Please let me know of any improvements/short-cuts to this walk-through.
Cheers. Ed
[edit: thanks to mike and Andrew for some corrections]

Texas Jigsaw Killer 28
1. "45" r1234: r5c58 = 16 = {79}: locked for r5
1a. 17(3)r4c7: must have 7/9 -> {368/458} blocked

2. Law of Leftovers (LOL) on r1234: 2 X 6 cell Leftovers. Below: {79} locked -> {79} locked above the line in 21(3)r2c1 or 10(3)r4c1
2a. 7 and 9 locked for n4

3. 14(3)r5c2 cannot have 7 or 9 = {158/248/356} = [4/5,2/5..]

4. 10(3)r4c1 = {127/136}(no 4,5)({145/235} blocked by 14(3))
4a. = 1{27/36} = [6/7..]
4b. 1 locked r4, n4

5. "45"n5(r4c6): r4c69 = 13 = h13(2)r4c6 = {49/58}({67} blocked by 10(3)r4c1step 4a)
5a.h13(2) = [5/9..]

6. 17(3)r4c7 = {179/269/278/467}(no 3) ({359} blocked by h13(2))

7. 3 in N5 only in 15(4)r5c9 = 3{129/147/156/246}(no 8)

8. r1234: 2 X 6 cell Leftovers. Below: must have 3 and no 8 -> above: must have 3 and no 8
8a. ->21(3)r2c1: no 8 = {579}: locked n4

9. 10(3) = {136}:locked for n4 & r4

10. LoL on r1234: {579/136}:locked above the line -> must be below the line
10a. 7 and 9 locked in r5c58
10b. -> no 7 or 9 in 15(4)(r5c9)
10c. 15(4) must have the 1,3,5,6 for below the line
10d. 15(3) = {1356}
10e. {1356} locked n5

11. 17(3)r4c7 = {278}
11a. r5c8 = 7
11b. r4c78 = {28}: locked for r4

12. h13(2) = {49}: locked r4

13. deleted

14. r5c5 = 9
14a. r4c45 = {57} Locked n1

15. r3c4 = 6 (addition in 27(4)r3c4)

16. 7(2)r1c1 = {34}: locked r1, n1

17. "45" n3: r4c9 - 3 = r3c8
17a. r4c9 = 4
17b. r3c8 = 1
17c. r3c7 = 7
17d. 1 in r1 only in 12(3)n2 -> no 1 r2c4

18. r4c6 = 9

19. r3c12 = {59}: locked r3, n4
19a. r2c1 = 7

20. r23c9 = [98]

21. r3c3 = 2
21a. r2c23 = {18}:locked for r2

22. r5c2 = 2 (hsingle n4)
22a. r56c3 = {48}: locked for c3

23. r2c23 = [81]

24. r1c4 = 1 (hsingle r1)

25. r3c56 = {34}: locked for n2

26. r1c5 = 8 (hsingle n2)
26a. r2c56 = 11 = {56}: locked for r2, n2

27. r1c3 = 9, r2c4 = 2

28. 16(3)(r1c7) = {367/457} = 7{36/45}
28a.r1c6 = 7
28. r12c7 = [63/54]

29. "45" c123: r9c3 = 6

30. r4c3 = 3

31. r78c3 = {57}: Locked for n9(r7c3)

32. r789c4 = 16 = {349}: locked for c4, n9

33. r9c6 = 8 (hsingle n9)
33a. r89c5 = {12}:locked for c5

34. 24(4)r5c4 = 78{36/45} = [3/4..]:
34a. 8 locked n7(r5c4)
34a. 5(2)r9c7 = {14/23} = [3/4]
34b. Killer pair 3/4 for n7(r5c4)
34c. r8c7 = 9 (hsingle n7)

35. X-wing 8 r56c34: 8 locked r56

36. Killer pair 1/2 r9c5 and 5(2)r9c7: locked r9
36a. r89c1 = [63/54/45]

37. r7c1 = 8 (hsingle c1), r8c8 = 8 (hsingle r8)

38. r4c78 = [82]

39. r1c9 = 2 (h single r1)

40. r6c1 = 2 (h single c1)

41. r7c7 = 2 (h single n8(r5c6))

42. r8c6 = 2 (h single c6)
and the rest falls out with cage sums and singles

Prelims

a) R1C12 = {16/25/34}, no 7,8,9
b) R3C78 = {17/26/35}, no 4,8,9
c) R89C1 = {18/27/36/45}, no 9
d) R9C78 = {14/23}
e) 19(3) cage at R1C5 = {289/379/469/478/568}, no 1
f) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9
g) 21(3) cage at R2C1 = {489/579/678}, no 1,2,3
h) 11(3) cage at R2C2 = {128/137/146/236/245}, no 9
i) 21(3) cage at R2C9 = {489/579/678}, no 1,2,3
j) 10(3) cage at R4C1 = {127/136/145/235}, no 8,9
k) 11(3) cage at R5C6 = {128/137/146/236/245}, no 9
l) 11(3) cage at R8C5 = {128/137/146/236/245}, no 9
m) 20(3) cage at R8C8 = {389/479/569/578}, no 1,2
n) 27(4) cage at R3C4 = {3789/4689/5679}, no 1,2

1. 45 rule on NR4C6 2 innies R4C69 = 13 = {49/58/67}, no 1,2,3
1a. 45 rule on R123 2 outies R4C69 = 1 innie R3C4 + 7, R4C69 = 13 -> R3C4 = 6, placed for NR1C1, clean-up: no 1 in R1C12, no 2 in R3C78
1b. R3C4 = 6 -> R4C45 + R5C5 = 21 = {489/579}, no 3
1c. Killer pair 4,5 in R1C12 and R4C45 + R5C5, locked for NR1C1

2. 45 rule on NR7C3 2 innies R78C3 = 12 = {39/48/57}, no 1,2,6

[While archiving I noticed that Jean-Christophe posted a nice shortcut which works after this step.
Law of Leftovers for C123 4 outies 27(4) cage at R3C4 must exactly equal 4 innies R1789C3, R78C3 = 12 -> R19C3 = 15, 27(4) cage contains 6 in R3C4 -> R19C3 must contain 6 = {69}.]

3. 45 rule on NR1C3 2(1+1) outies R3C8 + R4C6 = 10 = [19/37/55], clean-up: no 1 in R3C7, no 5,7,9 in R4C9 (step 1)

4. 21(3) cage at R2C9 = {489/678} (cannot be {579} because R4C9 only contains 4,6,8), no 5, 8 locked for C9

5. 45 rule on R1234 2 outies R5C58 = 16 = {79}, locked for R5

6. Hidden killer triple 1,2,3 in 10(3) cage at R4C1 and 17(3) cage at R4C7 for R4, 17(3) cage cannot contain more than one of 1,2,3 -> 10(3) cage must contain two of 1,2,3 -> 10(3) cage = {127/136/235} (cannot be {145} which only contains one of 1,2,3), no 4 in 10(3) cage

7. Hidden killer triple 1,2,3 in R3C3, 16(3) cage at R3C5 and R3C78 for R3, 16(3) cage cannot contain more than one of 1,2,3, R3C78 contains one of 1,3 -> R3C3 = {123}

8. Law of Leftovers (LoL) for C12 three outies R456C3 must exactly equal three innies R1C12 + R2C2 -> no 6,9 in R456C3
8a. 1,2,3 in NR1C1 only in R1C12 + R2C2 + R23C3 -> R23456C3 must contain 1,2,3 (using the LoL), locked for C3, clean-up: no 9 in R78C3 (step 2)

[With hindsight I ought to have spotted R19C3 = {69} (hidden pair in C3).]

9. 9 in NR2C1 only in 21(3) cage at R2C1 = {489/579}, no 6

10. R456C3 must exactly equal three innies R1C12 + R2C2 (LoL, step 8), R1C12 = {25/34} -> R456C3 must contain {25/34}
10a. Killer pair 4,5 in R456C3 and R78C3, locked for C3
10b. Min R1C3 = 6 -> max R12C4 = 6, no 7,8,9 in R12C4

11. R4C45 + R5C5 (step 1b) = {489/579} -> R4C69 = 13 (step 1) = [76/94] (cannot be [58] which clashes with R4C45), no 5 in R4C6, no 8 in R4C9
11a. 8 in C9 only in R23C9, locked for NR1C6
11b. Naked pair {79} in R4C6 + R5C8, locked for NR4C6
11c. 17(3) cage at R4C7 = {269/278/359/467} (cannot be {179} because 7,9 only in R5C8, cannot be {368/458} because R5C8 only contains 7,9), no 1
11d. Hidden killer triple 1,2,3 in 10(3) cage at R4C1 and 17(3) cage for R4, 10(3) cage contains two of 1,2,3 -> 17(3) cage must contain one of 1,2,3 = {269/278/359} (cannot be {467} which doesn’t contain 1,2 or 3), no 4

12. 1 in R4 only in 10(3) cage at R4C1, locked for NR2C1
12a. 10(3) cage at R4C1 (step 6) = {127/136}, no 5

13. R4C69 (step 11) = [94] (cannot be [76] which clashes with 10(3) cage at R4C1), both placed for NR4C6, R5C8 = 7, R5C5 = 9
13a. R4C9 = 4 -> R23C9 = 17 = {89}, locked for C9 and NR1C6
13b. R3C4 + R5C5 = [69] = 15 -> R4C45 = 12 = {57}, locked for R4 and NR1C1, clean-up: no 2 in R1C12
13c. R5C8 = 7 -> R4C78 = 10 = {28}, locked for R4 and NR4C6
13d. Naked pair {34} in R1C12, locked for R1 and NR1C1
13e. Naked triple {128} in 11(3) cage at R2C2, 8 locked for R2 -> R23C9 = [98]
13f. Naked triple {136} in 10(3) cage at R4C1, locked for NR2C1
13g. 21(3) cage at R2C1 = {579} (only remaining combination), locked for NR2C1
13h. R4C3 = 3 (hidden single in C3)
13i. 1 in C3 only in R23C3, locked for 11(3) cage at R2C2, no 1 in R2C2

14. R1C3 = 9 (hidden single in R1), R12C4 = 3 = {12}, locked for C4 and NR1C3
14a. R4C6 = 9 -> R3C56 = 7 = {34}, locked for R3 and NR1C3, clean-up: no 5 in R3C78
14b. R3C7 = 7, placed for NR1C3, R3C8 = 1, placed for NR1C6, clean-up: no 4 in R9C7
14c. Naked pair {56} in R2C56, locked for R2 and 19(3) cage at R1C5 -> R1C5 = 8, R2C1 = 7, clean-up: no 2 in R89C1

15. R1C9 = 2 (hidden single in C9), R12C4 = [12]
15a. R1C6 = 7 (hidden single in R1)

16. R2C2 = 8, R23C3 = [12]
16a. Naked pair {48} in R56C3, locked for C3 and 14(3) cage at R5C2 -> R5C2 = 2
16a. Naked pair {57} in R78C3, locked for NR7C3 -> R9C3 = 6, placed for NR7C3, clean-up: no 3 in R8C1

17. 11(3) cage at R8C5 = {128} (only remaining combination) -> R9C6 = 8, placed for NR7C3, R89C5 = {12}, locked for C5, clean-up: no 1 in R8C1
17a. Naked triple {349} in R789C4, locked for C4

18. 20(3) cage at R8C8 = {569/578} (cannot be {389/479} because 4,8,9 only in R8C8), no 3,4 -> R8C8 = {89}, 5 locked for C9 and NR5C6
18a. Naked triple {136} in R567C9, locked for C9 and NR4C6 -> R6C8 = 5, R1C8 = 6, R2C8 = 3 (cage sum), R12C7 = [54], clean-up: no 2 in R9C7
18b. R89C9 = {57} -> R8C8 = 8, R4C78 = [82], R9C8 = 4, R9C7 = 1, both placed for NR5C4, R89C5 = [12], clean-up: no 5 in R8C1

19. R7C8 = 9 -> R67C7 = 5 = {23}, locked for C7 and NR5C6 -> R5C7 = 6, R56C6 = {14}, locked for C6

20. R8C7 = 9 -> R78C6 = 7 = {25}, locked for C6 and NR5C4 -> R5C4 = 8

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

Jigsaw nonet design: Cross by Andrew Smith

SumoCueV1=22J0+0J1=14J1+2J1=13J1=17J1+5J2=14J2+7J3+0J0+0J0+2J1=17J1+4J1+12J2+5J2+7J3+7J3=17J4+18J0=18J0+12J1+12J2+12J2=15J3=24J3+25J3+18J4+20J4+20J0=45J0+30J2+30J2+24J3+24J3+25J5=16J4+36J4+36J4+30J0+30J0+30J2=8J3+42J5+42J5=12J4=17J4+46J6+30J6+30J7+30J7=9J5+51J5=17J5+45J4+45J6+46J6=28J6+57J8+57J7+51J7+53J5+53J5=20J6+63J6=16J6+57J8=6J8+57J8=18J7=22J7+70J5+63J6+63J8+65J8+65J8+67J8+69J8+69J7+70J7+70J7

Fwah Ruud: you're torturing me. That is a fantastic puzzle. Had to use a chain move to unlock it. But perhaps someone can find a better way. Found some really interesting ways to use LoL - hope they are valid. Couldn't find a way to use Jean-Christophe's really great use of LoL and cage sums from TJK28 though. Perhaps that's why I found this puzzle so hard.

Won't put the walk-through into tiny text this time. Please let me know of any shortcuts missed, or improvements/corrections. [Thanks to Glyn for some improvements/corrections and naming ] [some more typos fixed: thanks Mike][And more improvements and corrections from Andrew. Thanks!]
Enjoy! Ed

Texas Jigsaw Killer 29

1. "45" r12: 3 outies r3c456 = 7 = {124}
1a. {124} locked for r3 and not elsewhere in 17(5) cage

2. "45" r12: 2 innies r2c46 = 10 = {37}
2a. {37} locked for r2
2b. Common Peer Elimination (CPE) -> no 3 or 7 r1c6

3. 13(2)r1c5 = {49/58}/[76](no 6 r1c5)

4. 24(3)r3c8 = {789}
4a. -> no 789 in r12c9 or r4c78

5. 14(4)r1c8: no 9

6. 8(3)r5c7 = 1{25/34}
6a. 1 locked for r5
6b. CPE ->no 1 r6c7

7. 15(3)r3c7. All combinations have 7/8/9 which are only in r3c7. The exception is {456}
7a. -> r3c7 = 5..9

8. 7 and 9 now only in r3 in n4(r1c9)
8a. 7 & 9 locked for r3
8b. 17(3)r3c1 - no 1,2,5,7 r4c1
8c. 17(3) = {359/368/458}
8d. deleted
8e. 18(3)r3c3: {279} combo blocked by r3c3 -> no 2 r4c23

9. 3 in r3 only in c123
9a. CPE: no 3 r1c1 or r4c2

10. 6(2)r8c5 = {15/24}
10a. -> 1 in c5 locked in 6(2) or r3c5 (6(2) = {24} -> r3c5 = 1)

11. "45" c123: r19c4 = 17 = {89}: locked for c4

12. 14(3)r1c3 must have 8/9
12a. = {149/158/239/248}
12b. cannot have both 8 and 9 -> r12c3 = {14/15/23/24} = 1..5

13. 16(3)r8c3 must have 8/9
13a. {367/457} blocked
13b. cannot have both 8 & 9 -> no 8,9 r89c3

14. "45" c123: 4 innies: r1289c3 = 13 = h13(4)c3
14a. must have 1: 1 locked for c3
14b. -> {259} combo blocked from 16(3)r8c3: (because 13(4) cannot contain both of 2&5)
14c. -> 16(3)r8c3 = {169/178/268/349/358}
14d. -> r12-89c3 = {14-26/14-35/15-34/23-17/24-16}

15. deleted

16. "45" r1234: 3 innies = 9 = h9(3)r4
16a. = {126/135/234}(no 789) = [2/5..]
16b. ->{258} combo blocked from 15(3)r3c7

17. 15(3)r3c7: all combo's without 1 have 2/3/4 in r4
17a. -> 1 in r4 in h9(3) or r4c78

18. 1 in n5(r3c1) only in r67c1 or r6c2
18a. CPE: no 1 r7c2

19. LoL r6789: 3 innies r67c1+r6c2 = 3 outies r4c9+r5c89
19a.innies must have 1 (step 18) -> outies must have 1
19b. 1 in r5c89
19c. 1 locked for r5 (no 1 r5c7) & 1 locked for n6(r4c9)
19d. deleted, faulty [Thanks Andrew]

20. 9(3)r6c7 = {126/135/234} -> r7c7 = 1..4

21. 16(3)r5c1 = {259/268/349/367} ({358/457} blocked by 8(3)r5c7)
21a. 16(3) = [2/3..], 8(3)r5c7 = [2/3]: Killer pair 2/3: 2 & 3 locked for r5
[Andrew noticed that “45” on r5 3 innies r5c456 = 21 is a simpler way to do the above and I use that "45" in the next step. Thanks]
21b. 2 in n5(r3c1) is either in the innies of LoLr6789 (step 19) -> must be in r5c89 for outies OR 2 in n5 is in 16(3)r5c1.
21c. Either way, no 2 r5c7. Is there a name for this move? ( LoLR - "Law of Left(&Right)-overs")
21d. -> no 5 r5c89 ({125} combo must have 5 in r5c7)
21e. -> no 5 in innies of LoL r6789 (step 19) in r6c12+r7c1 (rest of this step deleted)
21f. no 6 in the outies -> no 6 in innies-> no {156} combo in 12(3)r7c1

22. "45" r5: r5c456 = 21 = h21(3)r5
22a. = {489/579/678}

23. "45" r6789: 3 innies r6c456 = 15 = h15(3)r6
23a. = {159/168/249/267/348/357} ({258/456} blocked by h9(3)r4 in 45(9) see step 16a.)

24. "45" r89: 2 innies r8c46 = 8 = h8(2):
24a. = {17/26/35}(no 4,8,9)
24b.-> 2 locked for n9(r7c5) in h8(2) or 6(2) ({15} in 6(2) -> {26} in h8(2))

25. "45" r89: 3 outies r7c456 = 20 = h20(3)
25a. = {389/479/569/578}(no 1,2, no 3 in r7c56)
25b. 28(5)r7c4: all combinations must work for both the h8(2) and h30(3)
25c. {14689/24589/34678} all blocked

26. "45" c89: 1 outie r5c7 + 5 = 2 innies r46c8
26a. min r5c7 = 3 -> min r46c8 = 8 -> no 1 r4c8

27. "45" c89: 5 outies = 19
27a. -> must have 1:1 locked for c7

28. "45" c789: 2 outies r19c6 = 9.
28a. no 9, no 6 r9c6

29. "45" n4(r1c9): r1c8 + 16 = 3 innies
29a. Max. 3 innies = {589} = 22
29b. max, r1c8 = 6
29c. CPE: no 8 r1c6 since it sees all 8's in n3(r1c7)
29d. no 1 r9c6 (h9(2)c7)

30. 17(3)r1c7 must have 8/9 for n3(r1c6): r5c6 only other place with 8/9
30a. 17(3) = {179/269/278/359/368/458} = [8/9..] not both
30b. only other place for 8 or 9 is in r5c6 = {89}

31. [18] blocked from h9(2)c6.
31a. 1 in r1c6 -> r23c7 = [79] -> r5c6 = 8: clash with r9c6
31b. no 1 r1c6, no 8 r9c6

32. LoL c1234: 4 outies r1c56+r25c5 = 4 innies r8c4+r9c234
32a. no 1,3 in 4 outies -> no 1 or 3 in 4 innies
32b. 16(3)r8c3 = [1/2/3] only in r8c3 = {123}

33. LoL c6789: 3 outies r346c5 = 3 innies r189c6
33a. no 8 or 9 in innies -> no 8 or 9 in r6c5

Now - a contradiction chain. Is there another way?
34. no 2 r1c6 because of LoL c6789 (step 33)
34a. 2 in r1c6 -> r9c6 = 7 (h9(2)c6) -> {27} in innies
34b. -> {27} in outies r346c5 -> 6(2)r8c5 = {15} -> 13(2)r1c5 = {49}
34c. but {249} in n2 forces 1 into both r3c4 and {158} in 14(3)r1c3
34d. no 2 r1c6
34e. no 7 r9c6 (h9(2)c6)
34e. no 2 r8c4 (LoL c1234 step 32)
34f. oops should have done this earlier too: h8(2)r8 = [53/62/71]

35. 17(3)r1c6 must have 4,5,6
35a. = {269/359/368/458}

36. LoL c1234: Outies r125c5+r1c6 must have 4/5. Here's how.
36a. since 13(2)r1c5 = [4/5] or {67}
36b. if {67} -> r1c6 = 4/5
36c. -> LoLc1234: innies r8c4 + r9c234 must have 4/5
36d. -> Killer pair 4/5 with 6(2)r8c5: 4 and 5 locked for n9(r7c5)

37. r9c6 = 3 finally!
37a. r89c7 = {69/78} = [8/9..]
37b. no 5 r8c4 (h8(2)r8)

38. r2c46 = [37]

39. r1c6 = 6 (h9(2)c6)
39a. r12c7 = {29}/[38]=[8/9..]

40. Killer Pair 8/9 in r1289c7:8 and 9 locked for c7

41. 9 in r3 only in 24(3)r3c7 -> no 9 r4c9
41a. -> no 9 in r7c12 + r7c1 (innies LoL r6789)

42. 13(2)r1c5 = {49/58}(no 7) = [4/5..]

43. Killer Pair 4/5 in 13(2) & 6(2)c5: 4 and 5 locked c5

44. r1c2 = 7 (hsingle n2)

45. 15(3)r3c7 = {267/357/456}(no 1)

46. r7c7 = 1 (hsingle c7)
46a. r6c78 = {26/35}(no 4)

47. 1 in r4 in h9(3) = 1{26/35}(no 4)
47a. 1 locked for 45(9)

48. h15(3)r6 = {249/267/348/357} ({258/456}blocked by r6c78)

49. h13(4)c3: r12-89c3
49a. from step 14d ={14-35/14-26/15-34/24-16}(no 7 r9c3)
49b. = 14{35/26}
49c. 4 locked c3
49b. CPE for 7's n9(r7c5): no 7 r7c4

50. Killer pair 1/2 in c5 in r3c5 and 6(2)
50a. 2 locked c5
50b. [edit:this step added]. LoLc6789: no 7 in 3 innies -> no 7 r6c5. (or 3 and 6 placed in innies -> outies r347c5 must have {36} for c5)
50c.{36} pair r46c5: 3 and 6 locked c5 and 45(9)

51. 28(5)r7c4 = 67{159/249/258}
51a. deleted

52. r4c5 = 6 (hsingle n3) How long has that been there?
52a. r6c5 = 3
52b. h9(3)r4: r4c46 = {12}:
52c. 1 and 2 locked r4 & 45(9)

53. naked pair {12}:locked for n3
53a. r3c6 = 4
53b. r12c7 = [38]

the rest is cage sums, hidden/naked singles.

Thanks for the great walkthrough, Ed!


Yes, there is. See steps 35d, 36, 38 and 40 in the modified walkthrough below.

Also, for another way of progressing, see steps 22 and 23, which build nicely on your fascinating "LoLR" move to lead to an early placement.

Here is my modified walkthrough, taking as much as possible from Ed's excellent opening and early mid-game:

Edit: Simplified step 46 in response to Ed's post below - thanks, Ed

Texas Jigsaw Killer 29

1 - 21f. As for Ed's walkthrough

22. "45" n6(r4c9): 2 outies (r57c7) + 11 = 2 innies (r48c9)
22a. Min. 2 outies = {13} = 4
22b. -> Min. r48c9 = 15
22c. -> Min. r8c9 = 6 = {6789}

23. r48c9 contain 2 of {56789}
23a. 17(3)r6c9 must contain exactly 2 of {56789}
(if only 1 of {56789}, highest possible sum = 16 {349} - too low;
if 3 of {56789}, lowest possible sum = 18 {567} - too high)
23b. -> one of {56789} in n6(r4c9) unaccounted for -> must go in r6c78
23c. -> 9(3)r6c7 = {1(26|35)} (no 4)
23d. 1 in n6(r4c9) already locked in r5c89 (step 19b) -> r7c7 = 1

24. 1 in n5(r3c1) now locked in r6 -> no 1 elsewhere in r6 (r6c4)

25. 1 in 45(9)r4c4 now locked in r4 -> no 1 elsewhere in r4 (r4c8)
25a. h9(3)r4 (step 16) = {1(26|35)} (no 4)

26. 1 in n4(r1c9) locked in 14(4)r1c8 = {1..}
26a. no 1 in r1c8

27. no 1,5,6 in r7c1 -> 12(3)r6c1 <> {156}
27a. only other combo with 5 ({345}) already blocked (step 19d)
27b. -> no 5 in r7c2

28. "45" r789: 4 outies (r6c1239) = h22(4)r6
28a. 1 in r6 locked in h22(4)r6 = {1(489|579|678)} -> no 2,3 in r6c1239

29. "45" r5: r5c456 = 21 = h21(3)r5
29a. = {489/579/678}
29b. {89} only in r5c56 -> no 4 in r5c56

30. "45" r6789: 3 innies r6c456 = 15 = h15(3)r6
30a. = {159/168/249/267/348/357} ({258/456} blocked by h9(3)r4 in 45(9)split-cage see step 16a.)

31. "45" c789: 2 outies r19c6 = 9.
31a. no 9 r1c6, no 2,6 r9c6

32. "45" n4(r1c9): r1c8 + 16 = 3 innies
32a. Max. 3 innies = {589} = 22
32b. max, r1c8 = 6
32c. CPE: no 8 r1c6 since it sees all 8's in n3(r1c7)
32d. no 1 r9c6 (h9(2)c7)

33. 17(3)r1c6 must have 8/9 for n3(r1c6): r5c6 only other place with 8/9
33a. 17(3) = {179/269/278/359/368/458} = [8/9..] not both
33b. only other place for 8 or 9 is in r5c6 = {89}

34. "45" r89: 2 innies r8c46 = 8 = h8(2):
34a. = {17/26/35}(no 4,8,9)
34b.-> 2 locked for n9(r7c5) in h8(2) or 6(2) ({15} in 6(2) -> {26} in h8(2))

35. "45" r89: 3 outies r7c456 = 20 = h20(3)
35a. = {389/479/569/578}(no 1,2)
35b. 28(5)r7c4: all combinations must work for both the h8(2) and h30(3)
35c. {14689/24589/34678} all blocked
35d. {89} only in r7c56 -> no 3 in r7c56

36. 3 in c5 locked in 45(9)r4c4 -> not elsewhere in 45(9)r4c4 (r46c46)

37. CPE: no 9 r6c5 since it sees all 9's in c6

38. LoL c1234: 4 outies r1c56+r25c5 = 4 innies r8c4+r9c234
38a. no 3 in 4 outies -> no 3 in 4 innies

39. 3 in n9(r7c5) locked in c6 -> not elsewhere in c6 (r2c6)
39a. CPE: no 3 r8c7 since it sees all 3's in c6

40. r2c6 = 7 (naked single)
40a. -> r2c4 = 3
40b. no 2 in r1c6 (step 31)
40c. 7 no longer available in 17(3)r1c6 -> no 1 in r1c6
40d. no 8 in r9c6 (step 31)

41. CPE: no 8 r6c5 since it sees all 8's in c6

42. 7 in n2(r1c2) locked in r1 -> not elsewhere in r1 (r1c1)

43. CPE: no 7 r5c3 since it sees all 7's in n1(r1c1)

44. LoL c789: 3 outies r6c56+r7c6 = 3 innies r1c78+r2c7
44a. no 7 in 3 innies -> no 7 in 3 outies

45. LoL c1234 (revisited): 4 outies r1c56+r25c5 = 4 innies r8c4+r9c234
45a. no 1,2 in 4 outies -> no 1,2 in 4 innies
45b. 16(3)r8c3 = [1/2/3] only in r8c3 = {123}
45c. h8(2)r8 (step 34): no 5,6 in r8c6

46. Hidden killer pair on {12} in n9(r7c5), as follows:
46a. 6(2)r8c5 must contain exactly one of {12}
46b. only other place for {12} in n9 is r8c6 -> r8c6 = {12}
46c. r9c6 = 3 (hsingle n9)
46d. r1c6 = 6 (step 31)
46e. no 7 in 13(2) r12c5 = {49|58} = {4/5..}
46f. split 11(2)r12c7 = {29}|[38]
46g. split 15(2)r89c7: no 2,4,5

47. r1c2 = 7 (hsingle r1)

48. r4c5 = 6 (hsingle n3)
48a. split h3(2) r1234 innie cage at r4c46 = {12}, locked for r4 and 45(9)r4c4

49. r6c5 = 3 (hsingle c5)
49a. split h12(2) r6789 innie cage at r6c46 = [48|75]
49b. split 8(2) cage at r6c78 = {26}, locked for r6 and n6

50. r1c8 = 5 (hsingle n3)

51. r1c7 = 3 (hsingle n3) -> r2c7 = 8

52. r5c6 = 9 (naked single)

53. r6c7 = 2 (hsingle c7) -> r6c8 = 6

54. r4c8 = 3 (hsingle n4)
54a. r34c7 = [75] (only permutation possible)

55. r5c7 = 4 (naked single) -> r5c89 = [13]

The rest is now all naked and hidden singles.

(Archive Note). A few typos have been corrected in steps 31, 31a and 45b.

Prelims

a) R12C5 = {49/58/67}, no 1,2,3
b) R89C5 = {15/24}
c) 24(3) cage at R3C8 = {789}
d) 8(3) cage at R5C7 = {125/134}
e) 9(3) cage at R6C7 = {126/135/234}, no 7,8,9
f) 14(4) cage at R1C8 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R12 3 outies R3C456 = 7 = {124}, locked for R3 and 17(5) cage at R2C4
1a. R3C456 = 7 -> R2C46 = 10 = {37}, locked for R2, clean-up: no 6 in R1C5
1b. R2C46 = {37}, CPE no 3,7 in R1C6 using NR1C2

2. 45 rule on R89 3 outies R7C456 = 20 = {389/479/569/578}, no 1,2
2a. R7C456 = 20 -> R8C46 = 8 = {17/26/35}, no 4,8,9

3. 8(3) cage at R5C7 = {125/134}, 1 locked for R5, CPE no 1 in R6C7 using NR4C9

4. 45 rule on R5 3 innies R5C456 = 21 = {489/579/678}, no 2,3
4a. 45 rule on R1234 3 innies R4C456 = 9 = {126/135/234}, no 7,8,9
4b. 6 of {126} must be in R4C56 (R4C56 cannot be {12} which clashes with R3C56, ALS block), no 6 in R4C4

5. Naked triple {789} in 24(3) cage at R3C8, CPE no 7,8 in R12C9
[I missed no 7,8,9 in R4C78 as in Ed's walkthrough.]
5a. 17(3) cage at R3C1 = {269/359/368/458/467} (cannot be {179/278} which clash with 24(3) cage, ALS block), no 1
5b. 4 of {467} must be in R4C1 -> no 7 in R4C1
5c. 17(3) cage at R6C9 = {179/269/278/359/368/458/467}
5d. 1 of {179} must be in R67C9 (R67C9 cannot be {79} which clashes with 24(3) cage, no 1 in R7C8

6. 45 rule on C123 2 outies R19C4 = 17 = {89}, locked for C4
6a. Min R1C4 = 8 -> max R12C3 = 6, no 6,7,8,9 in R12C3
6b. Min R9C4 = 8 -> max R89C3 = 8, no 8,9 in R89C3
6c. R5C456 (step 4) = {489/579/678}
6d. 4 of {489} must be in R5C4 -> no 4 in R5C56

7. 45 rule on C789 2 outies R19C6 = 9 = {18/45}/[27/63], no 9, no 2,6 in R9C6

8. R7C456 (step 2) = {389/479/569/578}
8a. 3 of {389} must be in R7C4 -> no 3 in R7C56
8b. 3 in C5 only in R46C5, locked for 45(9) cage at R4C4
8c. R4C456 (step 4a) = {126/135/234}
8d. 3 of {135/234} must be in R4C5 -> no 4,5 in R4C5

9. 45 rule on C123 4 innies R1289C3 = 13 = {1237/1246/1345}, 1 locked for C3

10. 45 rule on C789 4 innies R1289C7 = 26 = {2789/3689/4589/4679/5678}, no 1

11. 45 rule on C12 2 innies R46C2 = 1 outie R5C3 + 3, IOU no 3 in R46C2
[With hindsight it’s interesting to note that 45 rule on NR3C1 2 outies R37C2 = 2 innies R46C2 but R37C2 can contain 3 since neither of these cells “sees” R5C3.]

12. 45 rule on C89 4(3+1) innies R4C8 + R5C89 + R6C8 = 13
12a. Min R5C89 + R6C8 = 6 -> max R4C8 = 7

13. Law of Leftovers (LoL) for R6789 three outies R4C9 + R5C89 must exactly equal three innies R6C12 + R7C1, no 6 in R4C9 + R5C89 -> no 6 in R6C12 + R7C1

14. LoL for C6789 three outies R346C5 must exactly equal three innies R189C6, no 9 in R189C6 -> no 9 in R6C5

15. 45 rule on NR4C9 2 innies R48C9 = 2 outies R57C7 + 11
15a. Max R48C9 = 17 -> max R57C7 = 6, no 6 in R7C7
15b. Min R57C7 = 3 -> min R48C9 = 14, no 1,2,3,4 in R8C9

[Here I spotted that 8(3) cage at R5C7 and 9(3) cage at R6C7 can’t simultaneously be {134} and {234} because 17(3) cage at R6C9 must contain one of {1234}.]

16. R3C5 = {124}, R89C5 = {15/24} -> variable combined cage R389C5 = [1]{24}/{24}{15} must contain 1, locked for C5

17. Naked triple {124} in R3C456, R4C456 (step 4a) = {126/135/234}
17a. R34C56 must contain 1 (R3C456 must have 1 in R3C56 when R4C4 contains 1 and when R4C456 = {234}), locked for NR1C7, no 1 in R1C8
[Step 17a rewritten for clarity.]

18. LoL for C789 three outies R6C56 + R7C6 must exactly equal three innies R1C78 + R2C7, no 1 in R1C78 + R2C7 -> no 1 in R6C6
18a. 45 rule on R6789 3 innies R6C456 = 15 = {159/168/249/267/348/357} (cannot be {258/456} which clash with R4C456)
18b. 9 of {249} must be in R6C6, 4 of {348} must be in R6C4 -> no 4 in R6C6

19. LoL for C1234 four outies R125C5 + R1C6 must exactly equal four innies R8C4 + R9C234, no 3 in R125C5 + R1C6 -> no 3 in R8C4 + R9C23, clean-up: no 5 in R8C6 (step 2a)

20. 3 in NR7C5 only in R89C6, locked for C6 -> R2C6 = 7, placed for NR1C7, R2C4 = 3, placed for NR1C2, clean-up: no 2 in R1C6 (step 7), no 1 in R8C4 (step 2a)
20a. 17(3) cage at R1C6 = {269/359/368/458}, no 1, clean-up: no 8 in R9C6 (step 7)

21. LoL for C1234 four outies R125C5 + R1C6 must exactly equal four innies R8C4 + R9C234, no 1,2 in R1C6 -> no 1,2 in R8C4 + R9C23, clean-up: no 6 in R8C6 (step 2a)
21a. 16(3) cage at R8C3 must contain one of 1,2,3 (cannot be 4[48]) -> R8C3 = {123}

22. R1289C3 (step 9) = {1246/1345} (cannot be {1237} because 16(3) cage at R8C3 cannot contain both of 3,7), no 7 in R9C3, 4 locked for C3

23. Hidden killer triple 1,2,3 in R89C5 and R89C6 for NR7C5, R89C5 contains one of 1,2 -> R8C6 = {123}, R9C6 = {13}, clean-up: no 4,5 in R1C6 (step 7)

24. 17(3) cage at R1C6 (step 20a) = {269/368/458} (cannot be {359} because R1C6 only contains 6,8)
24a. 3 of {368} must be in R1C7, 6,8 of {269/458} must be in R1C6 -> no 6,8 in R1C7

25. Hidden killer pair 6,7 in R1C26 and R12C5 for NR1C2, R12C5 contains both or neither of 6,7 -> R1C26 must contain both or neither of 6,7
25a. 45 rule on NR1C2 3 remaining innies R1C26 + R3C4 = 15 = {258/267} (cannot be {159/249} because R1C6 only contains 6,8, cannot be {168/456} which only contain one of 6,7) -> R3C4 = 2, placed for NR1C2, R1C26 = [58/76]
25b. 1 in NR1C2 only in R12C3, locked for C3
25c. Naked pair {14} in R3C56, locked for NR1C7
25d. 7 in NR1C2 only in R1C25, locked for R1
25e. R4C456 (step 4a) = {126/135/234}
25f. 1,4 only in R4C4 -> R4C4 = {14}

26. Killer pair {14} in R3C5 and R89C5, locked for C5, clean-up: no 9 in R12C5
26a. 14(3) cage at R1C3 = {149} (hidden triple in NR1C2) -> R1C4 = 9, R12C3 = {14}, 4 locked for C3, R9C4 = 8, placed for NR7C5

27. 17(3) cage at R1C6 (step 24) = {269/368}, no 5
27a. 17(3) cage = {269/368}, CPE no 6 in R1C8 + R45C6 using NR1C7
27b. 6 in NR1C7 only in R2C7 + R4C5, CPE no 6 in R2C5 + R4C7, clean-up: no 7 in R1C5

28. Naked pair {58} in R12C5, locked for C5 and NR1C2 -> R1C2 = 7, R1C6 = 6, clean-up: no 1 in R89C5
28a. Naked pair {24} in R89C5, locked for C5 and NR7C5 -> R4C56 = [14], clean-up: no 6 in R8C4 (step 2a)

29. R4C5 = 6 (hidden single in NR1C7)
29a. R4C456 = 9 (step 4a) -> R4C4 = 1, placed for NR1C1, R4C6 = 2, placed for NR1C7 -> R1C7 = 3, R2C7 = 8 (cage sum), placed for NR1C7, R1C8 = 5, placed for NR1C7, R5C6 = 9, R5C5 = 7, placed for NR1C1
29b. R5C456 (step 4) = {579} (only remaining combination) -> R5C4 = 5, placed for NR1C1

30. R8C4 = 7, R8C6 = 1 (step 2a), R67C4 = [46], both placed for NR6C3, R6C56 = [38], R7C6 = 5, all placed for NR6C5, R7C5 = 9, placed for NR7C5
30a. Naked pair {56} in R9C23, locked for R9
30b. R9C6 = 3 -> R89C7 = 15 = [69], 9 placed for NR6C5

31. 9(3) cage at R6C7 = {126} (only remaining combination) -> R6C8 = 6, R67C7 = [21], 2 placed for NR4C9, 1 placed for NR6C5, R5C7 = 4, placed for NR1C9
31a. Naked pair {57} in R34C7, locked for NR1C9 -> R4C8 = 3
31b. Naked pair {89} in R3C89, locked for R3 and 24(3) cage at R3C8 -> R4C9 = 7, placed for NR4C9, R34C7 = [75]
31c. R5C89 = [13], R2C2 = 2, R12C9 = [16]

32. Naked pair {48} in R7C89, locked for R7 and NR4C9
32a. Naked pair {59} in R68C9, locked for C9 -> R3C89 = [98], R7C89 = [84], R9C9 = 2, R89C8 = [47], R9C1 = 1, R89C5 = [24], R8C3 = 3, placed for NR6C3, R9C3 = 5 (cage sum), R9C2 = 6

33. R7C2 = 2, R67C3 = [97], R34C3 = [68], R4C2 = 4 (cage sum)

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

Jigsaw nonet design: ZigZag by Gérard Coteau

Cages with cells in 3 jigsaw nonets:  lilac
Cages with cells in 2 jigsaw nonets:  green, yellow and brown
Cages with cells in 1 jigsaw nonet:   red and blue

See the end of Børge's post for the explanation of this window's title.

SumoCueV1=19J0+0J0+0J0=24J0=28J1+4J2+4J2+4J2=21J2+0J0=24J0+0J3+3J0+4J1=13J1+14J2+8J2+8J2+10J0+10J3=22J3+3J4+3J1+14J1+14J1=15J2+25J5+10J0+20J3+20J3=10J4+30J4=16J1+32J1+32J2+25J5=18J3+20J3=27J6=10J4+39J4+39J4+32J1=23J5+25J5+36J3+38J7+38J6+38J6=10J4+49J4+43J5+43J5=28J8+36J3+36J7=17J6+56J6=24J6+58J4+43J5+53J5+53J8=12J7+63J7+56J7+56J6=28J6+58J8=16J5+53J8+69J8+63J7+67J7+67J7+67J7+67J6+58J8+69J8+69J8+69J8

What a beautiful puzzle. This one will surely be up in the top 5 of my all time favs. Ruud, you continue to astound me with your puzzle-making skill. Thanks.

No chains this time. . Please let me know of any improvements/corrections. Let us all know of any shortcuts.

Cheers
Ed

Texas Jigsaw Killer 030

1...23...
..4......
...5....6
.........
..7......
.8......9
.........
.........
.........
n2(r1c5)
n3(r1c6)
n4(r2c3)
n5(r3c4)
n6(r3c9)
n7(r5c3)
n8(r6c2)
n9(r6c9)

1. "45" r6789: 3 outies r5c138 = 23 = h23(3)r5
1a. = {689}:locked for r5

2. "45" n5(r3c4): 2 innies r3c4 + r7c6 = 15 = h15(2)n5
2a. = {69/78}

3. 10(2) cages n5(r3c4): no 5

4. 5 in n5 only in 10(3) = 5{14/23}(no 7)
4a. 5 locked for r5

5. "45" r1234: 3 outies r5c279 = 12 = h12(3)r5
5a. = {147/237}

6.19(5)r1c1 must have 1
6a. -> no 1 r2c24

7. 13(4)r2c6 must have 1: no 8,9
7a. -> no 1 r2c5 or r45c7

8. (not essential: but couldn't resist putting it in)
Triangular connection between 1 required in the 19(5) and 13(4) cages -> no 1 r3c1.Here's how.
8a. 1 in r2c3 -> 1 in 13(4) in r3 -> no 1 r3c1
8b. 1 required in 19(5) in n1 -> no 1 r3c1

9. 16(5)r8c7 = {12346}
9a. -> no {12346} in r8c68

10. 28(4)r6c9 = 89{47/56}

Now, time to get serious
11. LoL r789: 5 outies(r5c3 + r6c2349) = 5 innies(r7c1678 + r8c7)
11a. 27(4)r5c3 = 9{378/468/567}(no 1,2)
11b. -> no 1 or 2 in outies
11c. -> no 1 or 2 in innies

12. -> 1 and 2 required in 16(4)r8c7 are no only in n9
12a. 1 & 2 locked n9

13. 27(4)r5c3 must have 9
13a. from LoL r789 (step 11): outies must have at least one 9
13b. -> innies must have at least one 9
13c. -> 9 in innies in r7c678
13d. -> 9 locked for r7

14. but only one 9 is possible in these innies (all in same row)
14a. -> only one 9 is possible in outies (must be in 27(4))
14b. -> no 9 r6c9

15. 28(4)r6c9 must have 9: only in c8
15a.9 locked c8

16. 9 in r5 only in r5c13
16a. CPE on 9's in r5 -> no 9 r234c3

17. 3 in c9 only in r345 or r89
17a. CPE on 3's in c9 -> no 3 r8c7
17b. 3 required in 16(5)r8c7 only in n9
17c. 3 locked n9

18. r8c7 = {46} in n6(r3c9) -> the same digit 4/6 is in r67c9. Here's how.
I'll explain this move by looking at the 4 first.
18a.-> when r8c7 = 4 the 4 for c8 and c9 must come from 2 nonets apart from n6(r3c9)
18b. -> 4 must be in c89 in n3(r1c6) and n9(r6c9) in r67c9 (can't be in r89c9 or r9c8 because same cage as r8c7)
18c. same logic applies when r8c7 = 6

19. -> 4/6 required in 28(4)r6c9 in r67c9
19a. -> no 4/6 in r7c8 (since 28(4) cannot be both 4 and 6)
19b. -> hidden killer pair 4/6 in n9 in r67c9 and 4 cells from 16(5) that are in n9
19c. -> no 4/6 in r9c6

20. "45" n6(r3c9) + n9(r6c9): r3c8 +8 = r89c6
20a. min. r89c6 = {57} = 12 -> min r3c8 = 4

21. 15(4)r3c8: {1239} combo blocked by r3c8
21a. = {1248/1257/1347/1356/2346}(no 9)

22. 9 in c9 only in r12c9 in n3:
22a. 9 locked for n3
22b. 21(3)r1c9 = {189/579}(no 1..4,6)

23. 15(4)r3c8 must have exactly two of 1,2,3 in r345c9 (step 21a)
23a. the only other place for 1,2 or 3 in c9 is in r89c9
23b. r89c9 cannot have more than 1 of 1,2,3 because of r345c9
23c. -> r89c9 must have 4/6
23d. -> {46} Very Hidden Killer Pair (VHKP) in r6789c9 (remembering step 19b).
23e. 4 and 6 locked in r6789c9 for c9 and n9
23f. 4 in 21(3)r1c9 only in r2c8: no 8 r2c8
23g. r9c78 = {123}

24. "45" n23: 2 innies r3c58 = 12 = h12(2)r3
24a. = {48/57}(no 1,2,3,6,9)

25. 15(4)r3c8 now = {1248/1257/1347} = 1{248/257/347}
25a. 1 locked for c9 and n6

26. 1 in n9 now only in r9: 1 locked for r9

27. 4 in 15(4) only in r3c8 -> no 8 r3c8
27a. -> no 4 r3c5 (h12(2)r3)
27b. and no 8 r3c9 (since 8 already in r3c5 in h12(2) for r3)

28. 4 locked in r23c8. Here's how.
28a. 21(3)r1c9 = {579} -> r3c8 = 4
28b. 21(3) = {489} -> 4 in r2c8
28c. 4 locked in r23c8 for c8 and n3

29. 4 in n6(r3c9) only in c7
29a. 4 locked for c7
29b. no 9 r4c6 (since 4 in 16(4)r4c6 only in r4c6

30. "45" n6(r3c9): r3c8 + 7 = 2 innies r7c8 + r8c7
30a. -> 2 innies = 11, 12 or 14
30b. -> no 9 possible r7c8 (means 2 innies = 13/15}

31. r8c8 = 9 (hsingle c8)

32. 9 in n6(r3c9) only in c7
32a. 9 locked for c7
32b. 23(4)r5c8 must have 9 and 6/8
32c. 23(4) = 9{248/356}(no 7)
32d. 23(4) = [6/8] not both -> only in r5c8 in this cage

33. 9 in n2(r1c5) only in c5: locked for c5
33a. no 1 r4c4 or r6c6

34. 9 in r9 on in n8(r6c2)
34a. no 9 r6c2

35. Generalised X-wing on 9's in n23
35a. 9 locked for r12 in c59

36. 9 in n1 only in c1 in r34c1
36a. 9 locked c1
36b. no 9 r3c2 (same cage)

37. r4c2 = 9, r5c3 = 9, r9c4 = 9, r3c1 = 9 (hsingles)

38. 28(4)r6c9 now = 8{47/56}
38a. CPE on 8's in this cage -> no 8 r4c9

39. "45" n6(r3c9): r3c8 + 7 = 2 innies r7c8 + r8c7
39a. max. 2 innies = [76] = 13 ([86] blocked by r5c8)
39b. -> max r3c8 = 6 (no 7)
39c. no 5 in r3c5 (h12(2)r3)

40. Killer pair {45} in 21(3)r1c9 and r3c8.
40a. 4 & 5 locked n3

41. 15(4)r3c8 = {1257/1347} = 17{25/34}
41a. 7 locked for c9 and n6
41b. no 5 r2c8 (7 only in 21(3) in r2c8)

42. 28(4)r6c9 now = split 19(3) = {568}(no 4)
42a. 6 only in r67c9: 6 locked for n9

43. 6 in 16(5)r8c7 only in r8c7
43a. r8c7 = 6

44. r57c8 = [85]
44a. r67c9 = {68}:locked for c9 and n9

Now, the final move, back to where things started.
45. LoL r789: 5 outies(r5c3 + r6c2349) = 5 innies(r7c1678 + r8c7)
45a. 5 innies must have 5 (r7c8) -> outies must have 5
45b. only 5 available is in r6c234 in 27(4)
45c. r6c234 = {567}: locked for r6
All the rest is on the back straight.

sudokuEd wrote "What a beautiful puzzle... Ruud, you continue to astound me with your puzzle-making skill. Thanks."
I second that emotion. Some of these TJK's (including this one) really are gems.

In particular, they allow a lot of room for creativity. Just take look at Ed's last two walkthroughs. Two new technique variants already - "Law of Left & Rightovers" (which I would prefer to call something like "LoL Pincer") and "Very Hidden Killer Pair" (VHKP))!

I attacked this puzzle from a different direction that Ed did. In particular, there was one piece of composite logic I used (see step 23), which serves as a good example of something that is easy for a human to spot (less than a minute), but which would be very difficult to build into a computer program. Without this shortcut, the puzzle would have been significantly more difficult to solve, with many more steps required.

Anyhow, that's enough for the time being - here's the walkthrough, which I'm deliberately not posting as tiny text, since I don't think it's necessary on this forum:


Walkthrough - Texas Jigsaw Killer 30
(Archive Note). Mike's link to Ruud's original puzzle diagram has been omitted; this diagram is immediately after the SudokuSolver diagram in this archive entry.

Nonet Layout:

111123333
114122333
144522236
144552236
447555266
487755669
487775669
888779699
888879999


1. 19/5 at R1C1 = {1..}
1a. CPE: no 1 in R2C24

2. 21/3 at R1C9: no 1,2,3

3. 13/4 at R2C6 = {1(237|246|345)} (no 8,9)
3a. CPE: no 1 in R2C5 and R45C7

4. 10/2 at R4C4: no 5

5. 27/4 at R5C3 = {(378|468|567)9}: no 1,2

6. 10/3 at R5C4: no 8,9

7. 10/2 at R6C5: no 5

8. 28/4 = {(47|56)89}: no 1,2,3
8a. CPE: no 8,9 in R9C8

9. 16/5 at R8C7 = {12346}
9a. CPE: no 1,2,3,4,6 in R8C68

10. Innies N5: R3C4+R7C6 = 15/2 = {69|78}

11. 5 in N5 locked in 10/3 at R5C4 = {(14|23)5) (no 6,7), 5 locked for R5

12. Innies N23: R3C58 = 12/2 -> no 1,2,6

13. Innies N78: R7C25 = 6/2 = {15|24}

14. Outies R6789: R5C138 = 23/3 = {689}, locked for R5

15. CPE: R8C7 sees all candidate positions for 1,2,3 in C9 -> no 1,2,3 in R8C7
15a. {123} in 16/5 at R8C7 now locked in N9 -> no 1,2,3 in R9C6

16. CPE: R4C5 sees all 8's and 9's in N2 -> no 8,9 in R4C5
16a. cleanup: no 1,2 in R4C4

17. LoL R789: R5C3+R6C2349 (outies) = R7C1678+R8C7 (innies)
17a. No 1,2 in outies -> no 1,2 in innies
17b. 27/4 at R5C3 = {9..} (step 5) -> outies = {9..} -> innies must also contain a 9
17c. candidate digit 9 only available in innies in R7 -> no 9 elsewhere in R7 (R7C349)
17d. CPE: R6C8 sees all 9's in 28/4 at R6C9 -> no 9 in R6C8

18. innie/outie difference R89: R89C6 + R8C8 = R7C34 + 17
18a. max. innies = 24 -> max. outies = 7 -> no 7,8,9 in R7C34

19. LoL C89: R1C6+R12678C7 (outies) = R89C8+R6789C9 (innies)
19a. -> R8C7 must correspond to either of R67C9 (only non-peers in innies)
19b. -> 4,6 in N9 locked in R9C789+R678C9 -> no 4,6 in R9C6

20. innie/outie difference N1234: R7C2 + R3C4 = R3C8 + 5
20a. -> no 5 in R7C2 (since peers R3C48 cannot contain the same digit)
20b. no 1 in R7C5 (step 13)

21. 24/4 at R7C5 requires a digit < 5 ({5678} = 26 - too high)
21a. Only cell with candidate digit < 5 is R7C5 -> R7C5 = {24}
21b. -> R7C25 = {24} (step 13), locked for R7

22. 1 in R7 now locked in 17/4 at R7C3 within R7C34 -> not elsewhere in N7
22a. no 1 in R8C3

23. R7C34 = {1(356)}
23a. {15} blocked because it would require R89C6+R8C8 = 23 (step 18) = {689},
impossible as 6 is unavailable in R89C6+R8C8
23b. {16} blocked because it would require R89C6+R8C8 = 24 (step 18) = {789},
and would simultaneously force R7C6 to {789}, implying a naked triple on {789}
in R789C6 = 24 -> cage sum exceeded for 24/4 at R7C5
23c. Therefore R7C34 must be {13}, 3 locked for R7 and N7, no 3 in R8C3
23d. no 2 in R8C34 (otherwise 17/4 cage sum unreachable)

24. R89C6+R8C8 must now sum to 21 (step 18), with 6 unavailable -> R89C6+R8C8 = {579} (no 8),
locked for N9
24a. R89C6+R7C5 must sum to an even number -> R7C6 must be even -> R7C6 = {68}
24b. R3C4 = {79} (step 10)

25. 2 in N7 locked in C5 -> not elsewhere in C5
25a. cleanup: no 8 in R4C4 and R6C6

26. 8 in N9 locked in R67C9 -> not elsewhere in C9
26a. no 8 in R7C8 (same cage)
26b. 9 in 28/4 at R6C9 locked in R78C8 -> not elsewhere in C8

27. LoL R89: R5C3+R6C234+R7C2345 (outies) = R89C6789 (innies)
27a. no 8 in innies -> no 8 in outies
27b. -> 27/4 = {5679} (no 3,4)
27c. 5,7 locked in R6C234 for R6
27d. cleanup: no 3 in R6C56

28. LoL C123: R567C3 (innies) = R129C4 (outies)
28a. no 2,4,8 in innies -> no 2,4,8 in outies

29. LoL C6789: R123C5 (outies) = R567C6 (innies)
29a. no 7 in innies -> no 7 in outies
29b. no 2 in outies -> no 2 in innies
29c. cleanup: no 8 in R6C5

30. HS in C4 at R5C4 = 2
30a. Split 8/2 at R5C56 = {35} (only combo possible), locked for R5 and N5

The rest is hidden and naked singles only.

Prelims

a) R4C45 = {19/28/37/46}, no 5
b) R6C56 = {19/28/37/46}, no 5
c) 21(3) cage at R1C9 = {489/579/678}, no 1,2,3
d) 10(3) cage at R5C4 = {127/136/145/235}, no 8,9
e) 13(4) cage at R2C6 = {1237/1246/1345}, no 8,9
f) 27(4) cage at R5C3 = {3789/4689/5679}, no 1,2
g) 28(4) cage at R6C9 = {4789/5689}, no 1,2,3
h) 16(5) cage at R8C7 = {12346}

1. 45 rule on NR3C4 2 innies R3C4 + R7C6 = 15 = {69/78}
1a. 5 in NR3C4 only in 10(3) cage at R5C4, 5 locked for R5
1b. 5 in NR3C4 only in 10(3) cage at R5C4 = {145/235}, no 6,7

2. 45 rule on NR5C3 + NR6C2 2 innies R7C25 = 6 = {15/24}
2a. Max R7C5 = 5 -> min R789C6 = 19, no 1 in R89C6

3. 45 rule on NR1C5 + NR1C6 2 innies R3C58 = 12 = {39/48/57}, no 1,2,6

4. 16(5) cage at R8C7 = {12346}, CPE no 2,3,4,6 in R8C68
4a. 1 in NR6C9 only in R8C9 + R9C789, locked for 16(5) cage, no 1 in R8C7

5. 19(5) cage at R1C1 contains 1, CPE no 1 in R2C24

6. 45 rule on R89 5(4+1) innies R8C3468 + R9C6 = 34
6a. Max R8C3468 = 30 -> min R9C6 = 4
6b. 45 rule on R89 3 innies R8C68 + R9C6 = 2 outies R7C34 + 17
6c. Max R8C68 + R9C6 = 24 -> max R7C34 = 7, no 7,8,9 in R7C34
6d. 2,3 in NR6C9 only in R8C9 + R9C789, locked for 16(5) cage at R8C7, no 2,3 in R8C7

7. 45 rule on N2C3 2 innies R2C3 + R3C2 = 1 outie R7C2 + 5, IOU no 5 in R2C3

8. 45 rule on NR3C9 2 innies R7C8 + R8C7 = 1 outie R3C8 + 7
8a. Max R7C8 + R8C7 = 15 -> max R3C8 = 8, clean-up: no 3 in R3C5 (step 3)

9. Hidden killer triple 1,2,3 in 15(4) cage at R3C8 and 23(4) cage at R5C8 for NR3C9, 15(4) cage cannot contain more than two of 1,2,3 in R345C9 (because there’s no 9 in R3C8), 23(4) cage cannot contain more than one of 1,2,3 -> 15(4) cage must contain two of 1,2,3 in R345C9 and 23(4) cage must contain one of 1,2,3
9a. Hidden killer triple 1,2,3 in R345C9 and R89C9 for C9, R345C9 contains two of 1,2,3 -> R89C9 must contain one of 1,2,3
9b. 16(5) cage at R8C7 = {12346}, R89C9 contains one of 1,2,3 -> R9C78 = {123}

10. 28(4) cage at R6C9 contains one of 4,6 in R6C9 + R7C89, R89C9 contains one of 4,6 -> killer pair 4,6 using C9 and NR3C9, no 4,6 in R345C9

11. 15(4) cage at R3C8 = {1239/1248/1257/1347}, 1 locked for C9 and NR3C9
11a. 4 of {1248} must be in R3C8 -> no 8 in R3C8, clean-up: no 4 in R3C5 (step 3)
11b. 16(5) cage at R8C7 = {12346}, 1 locked for R9

12. Law of Leftovers (LoL) for R89 four outies R67C29 must exactly equal four innies R8C457 + R9C5, R7C25 contains one of 1,2 -> R7C5 + R8C45 + R9C5 (no 1,2 in R8C7) must contain one of 1,2 (because R7C25 contains one of 1,2)
12a. Hidden killer pair 1,2 in R7C34 and R7C5 + R8C45 + R9C5 for NR5C3, R7C5 + R8C45 + R9C5 contain one of 1,2 -> R7C34 must contain one of 1,2
12b. Killer pair 1,2 in R7C25 and R7C34, locked for R7
[Ed and Mike got this result more directly using LoL for R789 five outies R5C3 + R6C2349 must exactly equal five innies R7C1678 + R8C7, no 1,2 in R5C3 + R6C2349 -> no 1,2 in R7C17.]

13. R7C8 + R8C7 = R3C8 + 7 (step 8)
13a. R3C8 = {3457} -> R7C8 + R8C7 = 10,11,12,14 = {46/47/56/48/68}, no 9 in R7C8
13b. 28(4) cage at R6C9 = {4789/5689}, 9 locked for NR6C9

14. 24(4) cage at R7C5 = {1689/2589/2679/4569/4578}
14a. 9 of {1689/2679/4569} must be in R7C6 -> no 6 in R7C6, clean-up: no 9 in R3C4 (step 1)

15. 45 rule on R89 4 outies R7C3456 = 1 innie R8C8 + 7
15a. Min R7C3456 = 13 -> no 5 in R8C8

16. R8C3468 + R9C6 = 34 (step 6)
16a. Max R89C6 = 15 -> min R8C348 = 19, no 1 in R8C34

17. R7C3456 = R8C8 + 7 (step 15)
17a. R7C6 = {789}, R8C8 = {789} -> R7C345 = 6,7,8,9 = {123/124/125/134/126} (cannot be {135/234} because R7C25 = {15/24} and no other 1,2 in R7), 1 locked for R7 and NR5C3, clean-up: no 5 in R7C5 (step 2)

18. 24(4) cage at R7C5 = {1689/2589/2679/4569/4578}
18a. R7C5 = {124} -> no 4 in R9C6
18b. 4 in NR6C9 only in R6789C9, locked for C9

[I was struggling to make progress so I used a couple of forcing chains.]
19. 15(4) cage at R3C8 (step 11) = {1239/1248/1257/1347}, R7C8 + R8C7 = R3C8 + 7 (step 8)
19a. Consider placements for R8C7
R8C7 = 4 => max R7C8 + R8C7 = 12 => max R3C8 = 5
or R8C7 = 6, R7C8 = 4, R7C8 + R8C7 = 10 => R3C8 = 3
or R8C7 = 6 with no 4 in R7C8, 4 in 16(5) cage at R8C7 only in R89C9, locked for C9 => 28(4) cage at R6C9 = {5689}, CPE, no 5 in R34C9 => 15(4) cage cannot be 7{125} (note that {1347} can only be 4{137})
-> no 7 in R3C8, clean-up: no 5 in R3C5 (step 3)
19b. 15(4) cage = {1239/1248/1257/1347}
19c. 5 of {1257} must be in R3C8 -> no 5 in R34C9

20. R7C8 + R8C7 = R3C8 + 7 (step 8)
20a. Consider placements for R3C8
R3C8 = {34} => max R7C8 + R8C7 = 11, no 8 in R7C8
or R3C8 = 5, placed for NR1C9 => 5 in C9 must be in R67C9 => 28(4) cage at R6C9 = {5689} => R7C8 = 6 (R67C9 cannot be {56} in this chain because R7C8 + R8C7 cannot be [66] and there’s no other place for 6 in 16(5) cage at R8C7)
-> no 8 in R7C8
20b. R7C8 + R8C7 cannot total 12 -> no 5 in R3C8, clean-up: no 7 in R3C5 (step 3)

21. 28(4) cage at R6C9 = {4789/5689}, 8 locked for NR6C9
21a. 24(4) cage at R7C5 (step 14) = {2679/4569/4578} (cannot be {1689/2589} because 8,9 only in R7C6), no 1, clean-up: no 5 in R7C2 (step 2)
21b. 8,9 only in R7C6 -> R7C6 = {89}, clean-up: no 8 in R3C4 (step 1)
21c. Naked pair {24} in R7C25, locked for R7
21d. 1 in R7 only in R7C34, locked for NR5C3

22. LoL for C1234 three outies R789C5 must exactly equal three innies R345C4, no 1 in R789C5 -> no 1 in R45C4, clean-up: no 9 in R4C5

23. R7C3456 = R8C8 + 7 (step 15)
23a. Max R89C6 = 13 -> min R7C56 = 11 -> min R7C3456 = 15 -> min R8C8 = 8

24. 28(4) cage at R6C9 = {4789/5689} must have one of 8,9 in R67C9
24a. 21(3) cage at R1C9 = {579/678} (cannot be {489} which clashes with R67C9), no 4, 7 locked for NR1C6

25. 4 in C9 only in 28(4) cage at R6C9 = {4789} (no 6 in R7C8) or in R89C9 (28(4) cage = {5689}, no 4 in R8C7) -> R7C8 + R8C7 = [56/74], no 6 in R7C8
25a. R7C8 + R8C7 = [56/74] = 11 -> R3C8 (step 8) = 4, placed for NR1C6, R3C5 = 8 (step 3), placed for NR1C5, clean-up: no 2 in R4C4, no 2 in R6C6
25b. R3C8 = 4 -> R345C9 = 11 = {128/137}, no 9
25c. 4 in NR3C9 only in R68C7, locked for C7

26. 28(4) cage at R6C9 = {4789/5689}
26a. R7C8 = {57} -> no 5,7 in R67C9

27. 5 in C9 only in R12C9, locked for NR1C6
27a. 21(3) cage at R1C9 = {579} (only remaining combination), 9 locked for NR1C6

28. LoL for C1234 three outies R789C5 must exactly equal three innies R345C4, no 8 in R789C5 -> no 8 in R4C4, clean-up: no 2 in R4C5
28a. 8 in NR3C4 only in R67C6, locked for C6

29. R89C6 = {57} (hidden pair in NR6C9), locked for C6, clean-up: no 3 in R6C5
29a. 24(4) cage at R7C5 (step 21a) = {4578} (only remaining combination) -> R7C5 = 4, placed for NR5C3, R7C6 = 8, R7C2 = 2, placed for NR6C2, R3C4 = 7 (step 1), placed for NR3C4, clean-up: no 3,6 in R4C4, no 3 in R4C5, no 2 in R6C5, no 3,6 in R6C6
29b. R3C45 = [78] = 15 -> R12C4 = 9 = {36/45}, no 1,2,9

30. 10(3) cage at R5C4 = {235} (hidden triple in NR3C4), locked for R5
30a. R7C4 = 1 (hidden single in C4)

31. R7C3456 = R8C8 + 7 (step 15)
31a. Min R7C3456 = 16 -> R8C8 = 9
31b. Max R7C3456 = 16 -> R7C3 = 3, placed for NR5C3

32. R2C8 = 7, R7C8 = 5, placed for NR3C9, R7C9 = 6, R6C9 = 8 (cage sum)

33. R345C9 (step 25b) = {137} (only remaining combination), locked for C9 and NR3C9 -> R7C7 = 9
33a. Naked pair {24} in R89C9, locked for 16(5) cage at R8C7 -> R8C7 = 6, placed for NR3C9, R56C8 = [82], R6C7 = 4, clean-up: no 6 in R6C5
33b. Naked pair {19} in R6C56, locked for R6 and NR3C4 -> R4C45 = [46], clean-up: no 5 in R12C4 (step 29b)
33c. Naked pair {36} in R12C4, locked for C4 and NR1C1 -> R6C4 = 5, placed for NR5C3, 10(3) cage at R5C4 = [253], R8C4 = 8, R8C3 = 5 (cage sum), R89C6 = [75]

34. R8C5 = 2, R9C4 = 9, R9C5 = 7, placed for NR5C3 -> R56C3 = [96]
34a. R8C5 + R9C45 = [297] = 18 -> R9C23 = 10 = [64]

35. R67C1 = [37], both placed for NR2C3, R5C1 = 6 (cage sum)

36. Naked triple {128} in R234C3, locked for C3 and NR2C3 -> R5C2 = 4

37. R5C2 = 4 -> 22(4) cage at R3C3 = {1489} (only remaining combination) -> R4C2 = 9, R34C3 = [18]

38. R12C3 = [72] = 9 -> R1C12 + R2C1 = 10 = {145} (only remaining combination), locked for NR1C1 -> R4C1 = 2

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

Mike(mhparker): In particular, they allow a lot of room for creativity. Just take look at Ed's last two walkthroughs. Two new technique variants already - "Law of Left & Rightovers" (which I would prefer to call something like "LoL Pincer") and "Very Hidden Killer Pair" (VHKP))!

Glyn: LoL Pincer sounds a good name Mike, blame me for the LoLR pun. Gets my vote if Ed approves.

Ed: Mike (mhparker) wrote " new technique variants already - "LoL Pincer" and "Very Hidden Killer Pair" (VHKP)"
LoL Pincer sounds great - I like names that describe. BTW: the VHKP is still only theoretical - 2 implied (ie hidden) sets of pairs. My move only actually had 1 hidden. Was still fun though.

Original diagram no longer available
Jigsaw nonet design: ZigZag by Gérard Coteau

SumoCueV1=19J0+0J0+0J0=24J0=28J1+4J2+4J2+4J2=21J2+0J0=24J0+0J3+3J0+4J1=13J1+14J2+8J2+8J2+10J0+10J3=22J3+3J4+3J1+14J1+14J1=15J2+25J5+10J0+20J3+20J3=10J4+30J4=16J1+32J1+32J2+25J5=18J3+20J3=27J6=10J4+39J4+39J4+32J1=23J5+25J5+36J3+38J7+38J6+38J6=34J4+49J4+43J5+43J5=28J8+36J3+36J7=17J6+56J6+49J6+49J4+43J5+53J5+53J8=12J7+63J7+56J7+56J6=28J6+49J8=16J5+53J8+69J8+63J7+67J7+67J7+67J7+67J6+49J8+69J8+69J8+69J8

I had to start making changes from my TJK 30 walkthrough right at the beginning because my previous step 1 no longer worked. Instead I’ve started with a step which I missed for TJK 30.

I’ve also learned enough from going through Ed’s and Mike’s posted walkthroughs for TJK 30 to make this V2 easier and more enjoyable than the way I solved TJK 30.

Prelims

a) R4C45 = {19/28/37/46}, no 5
b) 21(3) cage at R1C9 = {489/579/678}, no 1,2,3
c) 10(3) cage at R5C4 = {127/136/145/235}, no 8,9
d) 13(4) cage at R2C6 = {1237/1246/1345}, no 8,9
e) 27(4) cage at R5C3 = {3789/4689/5679}, no 1,2
f) 28(4) cage at R6C9 = {4789/5689}, no 1,2,3
g) 16(5) cage at R8C7 = {12346}

1. 45 rule for R6789 3 outies R5C138 = 23 = {689}, locked for R5

2. 45 rule on NR5C3 + NR6C2 2 innies R7C25 = 6 = {15/24}

3. 45 rule on NR1C5 + NR1C6 2 innies R3C58 = 12 = {39/48/57}, no 1,2,6

4. 16(5) cage at R8C7 = {12346}, CPE no 1,2,3,4,6 in R8C68

5. 19(5) cage at R1C1 contains 1, CPE no 1 in R2C24

6. 13(4) cage at R2C6 = {1237/1246/1345}, CPE no 1 in R2C5 + R45C7

7. 45 rule on N2C3 2 innies R2C3 + R3C2 = 1 outie R7C2 + 5, IOU no 5 in R2C3

8. 45 rule on R89 5(4+1) innies R8C3468 + R9C6 = 34
8a. Max R8C3468 = 30 -> min R9C6 = 4
8b. 45 rule on R89 3 innies R8C68 + R9C6 = 2 outies R7C34 + 17
8c. Max R8C68 + R9C6 = 24 -> max R7C34 = 7, no 7,8,9 in R7C34
8d. 1,2,3 in NR6C9 only in R8C9 + R9C789, locked for 16(5) cage at R8C7, no 1,2,3 in R8C7

9. Law of Leftovers (LoL) for R789 five outies R5C3 + R6C2349 must exactly equal five innies R7C1678 + R8C7, no 1,2 in outies -> no 1,2 in R7C167
9a. 27(4) cage at R5C3 contains 9 -> five innies must contain 9, which can only be in R7C1678, locked for R7
9b. Five innies only contain one 9 -> five outies can only contain one 9, which must be in 27(4) cage at R5C3 -> no 9 in R6C9
9c. 28(4) cage at R6C9 contains 9, locked for C8, clean-up: no 3 in R3C5 (step 3)

10. Hidden killer triple 1,2,3 in 15(4) cage at R3C8 and 23(4) cage at R5C8 for NR3C9, 15(4) cage cannot contain more than two of 1,2,3 in R345C9 (because there’s no 9 in R3C8), 23(4) cage cannot contain more than one of 1,2,3 -> 15(4) cage must contain two of 1,2,3 in R345C9 and 23(4) cage must contain one of 1,2,3
10a. Hidden killer triple 1,2,3 in R345C9 and R89C9 for C9, R345C9 contain two of 1,2,3 -> R89C9 must contain one of 1,2,3
10b. 16(5) cage at R8C7 = {12346}, R89C9 contains one of 1,2,3 -> R9C78 = {123}

[At this stage for TJK 30 I originally used
28(4) cage at R6C9 contains one of 4,6 in R6C9 + R7C89, R89C9 contains one of 4,6 -> killer pair 4,6 using C9 and NR3C9, no 4,6 in R345C9
However having learned from Ed’s and Mike’s walkthroughs for TJK 30 there’s the more powerful …]
11. LoL for C89 six outies R1C6 + R12678C7 must exactly equal six innies R6789C9 + R89C8, R8C7 = {46} -> R67C9 (only cells of the innies which aren’t “seen” by R8C7) must contain one of 4,6
11a. R67C9 contain one of 4,6, R89C9 contain one of 4,6 (from 16(5) cage at R8C7), killer pair 4,6 locked for R9 and NR6C9
11b. 28(4) cage at R6C9 = {4789/5689}, R67C9 contain one of 4,6 -> no 4,6 in R7C8

12. 45 rule on NR3C9 2 innies R7C8 + R8C7 = 1 outie R3C8 + 7
12a. R7C8 + R8C7 cannot total 10 -> no 3 in R3C8, clean-up: no 9 in R3C5 (step 3)

13. 15(4) cage at R3C8 = {1248/1257/1347} (cannot be {1239} because R3C8 only contains 4,5,7,8), no 9, 1 locked for C9 and NR3C9
13a. 4 of {1248} must be in R3C8 -> no 8 in R3C8, clean-up: no 4 in R3C5 (step 3)
13b. 8 of {1248} must be in R4C9 (R3C89 cannot be [48] which clashes with R3C58 = [84], step 3), no 8 in R3C9
13c. 16(5) cage at R8C6 = {12346}, 1 locked for R9

14. 9 in C9 only in R12C9, locked for NR1C6
14a. 21(3) cage at R1C9 contains 9 = {489/579}, no 6

15. R7C8 + R8C7 = R3C8 + 7 (step 12)
15a. R3C8 = {457} -> R7C8 + R8C7 = 11,12,14 = [56/74/84/86], no 9
15b. R8C8 = 9 (hidden single in C8), placed for NR6C9
15c. R7C8 + R8C7 = 11,12 = [56/74/84] (cannot be [86] which clashes with R5C8) -> R3C8 = {45}, clean-up: no 5 in R3C5 (step 3)

16. 15(4) cage at R3C8 (step 13) = {1248/1257/1347}
16a. R3C8 = {45} -> no 5 in R345C9

17. 21(3) cage at R1C9 (step 14a) = {489/579}, R3C8 = {45}, killer pair 4,5 locked for NR1C6
17a. 4 in NR1C6 only in R23C8, locked for C8
17b. 4 in NR3C9 only in R678C7, locked for C7

18. 9 in NR3C9 only in R67C7, locked for C7
18a. 34(6) cage at R6C5 must contain 9, locked for NR3C4, clean-up: no 1 in R4C45
18b. 9 in R5 only in R5C13, CPE no 9 in R234C3

19. 9 in C7 only in 23(4) cage at R5C8 = {2489/3569} (cannot be {2579/3479} because R5C8 only contains 6,8), no 7
19a. R5C8 = {68} -> no 6,8 in R6C78 + R7C8
19b. 2 of {2489} must be in R6C8 -> no 2 in R6C7
19c. 7 in NR3C9 only in R345C9 + R7C8, CPE no 7 in R67C9
19d. 7 in NR6C9 only in R89C6, locked for C6 and 34(6) cage at R6C5, no 7 in R6C5

20. 16(4) cage at R4C6 cannot be {1249} (because 4,9 only in R4C6), no 9
20a. 9 in R4 only in R4C12, CPE no 9 in R123C2 + R567C1
20b. R5C3 = 9 (hidden single in R5)
20c. R4C2 = 9 (hidden single in NR2C3)

21. 9 in NR1C5 only in R12C5, locked for C5
21a. Caged X-Wing for 9 in 28(5) cage at R1C5 and 21(3) cage at R1C9, no other 9 in R12
21b. 24(4) cage at R1C4 = {3678/4578}, no 1,2
21c. R9C4 = 9 (hidden single in C4)
21d. R3C1 = 9 (hidden single in C1)

22. 7 in C6 only in R89C6
22a. LoL for C789 three outies R189C6 must exactly equal three innies R345C7, R89C6 contains 7 -> R345C7 must contain 7, locked for C7 and NR1C5 -> R3C5 = 8, placed for NR1C5, R3C8 = 4 (step 3), clean-up: no 2 in R4C4
22b. 21(3) cage at R1C9 (step 14a) = {579} (only remaining combination), 7 locked for NR1C6
22c. 24(4) cage at R1C4 (step 21b) = {3678/4578}, 7 locked for C4, clean-up: no 3 in R4C5
[Cracked. The rest is fairly straightforward.]

23. R1C7 = 8 (hidden single in C7), placed for NR1C6

24. LoL for C1234 three outies R789C5 must exactly equal three innies R345C4, no 8 in R789C5 -> no 8 in R4C4, clean-up: no 2 in R4C5

25. R67C6 = {89} (hidden pair in NR3C4), locked for C6
25a. Naked pair {57} in R89C6, locked for C6, NR6C9 and 34(6) cage at R6C5, no 5 in R67C5, clean-up: no 1 in R7C2 (step 2)
25b. 1 in R7 only in R7C345, locked for NR5C3
25c. 8 in NR6C9 only in R67C9, locked for C9 and 28(4) cage at R6C9, no 8 in R7C8

26. 15(4) cage at R3C8 (step 13) = {1347} (only remaining combination), 3,7 locked for C9 and NR3C9 -> R7C8 = 5, placed for NR3C9, R26C8 = [72], clean-up: no 1 in R7C5 (step 2)
26a. Naked pair {24} in R7C25, locked for R7 -> R67C7 = [49], R67C6 = [98], R67C9 = [86], R8C7 = 6, placed for NR3C9, R5C8 = 8, R5C1 = 6, placed for NR2C3
26b. Naked pair {13} in R7C34, locked for R7, NR5C3 and 17(4) cage at R7C3, no 1,3 in R8C3
26c. R7C34 = {13} = 4 -> R8C34 = 13 = {58}, locked for R8 -> R89C6 = [75]
26d. Naked pair {58} in R8C34, CPE no 5 in R6C3
26e. Naked pair {24} in R78C5, locked for C5, clean-up: no 6 in R4C4

27. R7C1 = 7, placed for NR2C3
27a. R57C1 = [67] = 13 -> R6C1 + R7C2 = 5 = [14/32]

28. 27(4) cage at R5C3 = {5679} (only remaining combination), no 3, 6 locked for R6

29. 8 in NR2C3 only in R24C3, locked for C3 -> R8C3 = 5, placed for NR6C2, R8C4 = 8
29a. R6C4 = 5 (hidden single in R6)
29b. Naked pair {67} in R6C23, CPE no 6,7 in R9C3

30. R8C12 = {13} (hidden pair in R8), locked for NR6C2, R9C1 = 8 (cage sum)
30a. Naked pair {13} in R68C1, locked for C1
30b. Naked triple {245} in R124C1, locked for NR1C1
30c. Naked triple {367} in R123C4, locked for C4 -> R4C4 = 4, R4C5 = 6, both placed for NR3C4, R7C34 = [31], R5C4 = 2
30d. Naked pair {13} in R5C6 + R6C5, locked for NR3C4 -> R3C4 = 7, placed for NR3C4, R5C5 = 5, R6C5 = 3 (cage sum), placed for NR3C4, R6C5 = 1, R7C5 = 4 (cage sum)

31. R5C7 = 7 -> R4C678 = 9 = {135} (only remaining combination) -> R4C6 = 1, placed for NR1C5, R4C78 = [53]

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

Jigsaw nonet design: Orig 8

Cages with cells in 3 jigsaw nonets:  pink and orange
Cages with cells in 2 jigsaw nonets:  green and yellow 
Cages with cells in 1 jigsaw nonet:   red and blue

See the end of Børge's post for the explanation of this window's title.

SumoCueV1=24J0+0J0=16J0+2J1=18J1+4J1=15J1+6J1=17J2=14J0+0J0+0J0+2J1+4J1=12J1+6J1+8J2+8J2+9J0+9J0=12J0+20J3+20J3+14J3+6J2=27J2+8J2=25J4+27J4=12J4+29J4=22J3+14J2+25J2+25J2=26J5+27J4=19J4+29J4+31J4+31J3+31J5=13J5+25J5+35J5+27J4+37J6+37J6=14J6+31J3+42J5+42J5+35J5+35J5=15J6+37J6=23J6+48J3=20J3+58J3+58J7=6J7+61J7+54J6+54J6+56J8+48J8=12J8=19J8=24J7+69J7+61J7+54J6+56J8+56J8+67J8+67J8+68J8+68J7+69J7+69J7

This one was a breeze after that work on TJK 18. Hardly used any Jigsaw techniques though. Only 2 LOL moves.
Here's the walkthrough. As it just came out i'll keep it in tiny print.

Walkthrough TJK 31

N1 at R1C1
N2 at R1C4
N3 at R1C9
N4 at R3C4
N5 at R4C1
N6 at R4C9
N7 at R6C2
N8 at R7C7
N9 at R8C3

1. 6(3) at R7C8 = {123} -->> locked for N8

2. 27(4) at R3C8 = {3789/4689/5679}: no 1,2

3. 45 N1: 2 innies: R13C3 = 7 = {16/25/34}

4. 45 C123(including hidden cage at R13C3): 1 outie: R4C4 = 4
4a. R45C3 in 12(3) at R4C3 = {17/26/35}: no 8,9

5. 45 N8: 2 innies: R79C7 = 15 = {69/78}

6. 45 C789(including hidden cage at R79C7): 1 outie: R6C6 = 8
6a. 26(4) at R4C9 = {4679}(last possible combo) -->> locked for N6
6b. R56C7 in 13(3) at R5C7 = {23}(last possible combo) -->> locked for C7 and N6
6c. R5C8 = 5; R5C6 = 1
6d. Clean up: R4C3: no 3,7

7. R3C8 + R4C78 in 27(4) at R3C8 = {679}(last possible combo) -->> locked for N3

8. LOL C789: R456C6 = R1C78 + R2C7
8a. Outies: no 4, 6,7, 9 -->> Innies: no 4, 6, 7, 9
8b. Outies: R56C6 = [18], so innies needs {18} -->> locked for N2 and 15(4) cage at R1C7
8c. 15(4) at R1C7 = {1248} (needs {18}) -->> R1C8 = 2; R3C7 = 4(only place in cage); R12C7 = {18} -->> locked for C7
8d. Innies: R1C8 = 2, so innies need 2 -->> R4C6 = 2

9. R79C7 = {69}(last combo) -->> locked for C7 and N9
9a. R4C7 = 7; R8C7 = 5
9b. Naked Pair {69} at R34C8 -->> locked for C8
9c. Naked Pair {69} at R4C89 -->> locked for R4

10. 45 on N5: 2 innies: R5C24 = 8 = {26} -->> locked for R5 and N5
10a. R56C7 = [32]; R45C3 = [17]

11. LOL on R123: R3C456 = R4C678
11a. Outie: no 1, 3, 5, 8 -->> Innies: no 1, 3, 5, 8
11b. Outies: R4C67 = [27], so innies needs {27} -->> locked for R3 and N4
11c. R7C4 = 1(hidden); R7C8 = 3; R78C9 = [21]
11d. R6C2 = 1(hidden); R9C5 = 1(hidden); R2C8 = 1(hidden); R12C7 = [18]; R3C1 = 1(hidden)

12. 12(3) at R3C3 = {237} (last possible combo) -->> R3C3 = 3; R3C45 = {27}
12a. R1C3 = 4(Step 3)

13. 45 on N2: 1 innie: R2C8 = 4
13a. R3C6 = 6; R34C8 = [96]; R45C9 = [94]; R6C89 = [76]
13b. R9C9 = 7(hidden); R6C5 = 4(hidden); R9C2 = 4(hidden); R89C8 = [48]; R7C1 = 4(hidden)

14. 22(5) at R4C5 needs one of {358} in R4C5 and one of {26} in R5C4 -->> 22(5) = 14{368} -->>R4C5 = 3; R5C45 = [68]
14a. R5C1 = 9; R5C2 = 2; R6C1 = 3(hidden); R8C2 = 3(hidden); R7C2 = 7(hidden)
14b. R6C34 = [95]; R8C4 = 8; R7C3 = 8(hidden)

15. R12C4 = {39} (last possible combo to fill 16(3) at R1C3) -->> locked for C4 and N2
15a. R9C4 = 2; R89C1 = [26]; R89C3 = [65]; R9C67 = [39]; R8C56 = [97]
15b. R6C567 = [596]; R3C45 = [72]; R1C6 = 5; R2C3 = 2

16. 14(3) at R2C1 = {158} (last possible combo) -->> R2C1 = 5; R3C2 = 8
16a. R4C12 = [85]; R123C9 = [835]; R1C1 = 7; R12C5 = [67]; R12C4 = [39]; R12C2 = [96]

And we are done.
(Archive note) Typos corrected.

Prelims

a) 20(3) cage at R7C5 = {389/479/569/578}, no 1,2
b) 6(3) cage at R7C8 = {123}
c) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
d) 27(4) cage at R3C8 = {3789/4689/5679}, no 1,2
e) 26(4) cage at R4C9 = {2789/3689/4589/4679/5678}, no 1

1. Naked triple {123} in 6(3) cage at R7C8, locked for NR7C7

2. 45 rule on NR7C7 2 innies R79C7 = 15 = {69/78}

3. 45 rule on NR1C1 2 innies R13C3 = 7 = {16/25/34}, no 7,8,9

4. 45 rule on NR4C1 2 innies R5C24 = 8 = {17/26/35}, no 4,8,9

5. 45 rule on NR4C9 2 innies R5C68 = 6 = [15/24]
5a. 27(4) cage at R3C8 = {4689/5679} (cannot be {3789} because R5C8 only contains 4,5), no 3, 6,9 locked for NR1C9
5b. R5C8 = {45} -> no 4,5 in R3C8 + R4C78

6. 45 rule on C123 2 innies R13C3 = 1 outie R4C4 + 3
6a. R13C3 = 7 (step 3) -> R4C4 = 4, placed for NR4C1
6b. R4C4 = 4 -> R45C3 = 8 = {17/26/35}, no 8,9
[It was only after going through Para’s walkthrough that I realised that I missed a similar step for R79C7 and C789 which would give a placement for R6C6 and shorten my solving path.]

7. 45 rule on NR1C9 2 innies R3C7 + R4C6 = 1 outie R5C8 + 1
7a. Max R5C8 = 5 -> max R3C7 + R4C6 = 6, no 7,8 in R3C7 + R4C6

8. 45 rule on NR6C2 2 innies R6C4 + R7C3 = 1 outie R5C2 + 11
8a. Max R6C4 + R7C3 = 17 -> max R5C2 = 6, clean-up: no 1 in R5C4 (step 4)
8b. Min R6C4 + R7C3 = 12, no 1,2 in R6C4 + R7C3

9. 45 rule on C89 2 outies R48C7 = 1 innie R1C8 + 10
9a. Max R48C7 = 17 -> max R1C8 = 7

10. 45 rule on R123 1 innie R3C8 = 1 outie R4C6 + 7, R3C8 = {89}, R4C6 = {12}
10a. Naked pair {12} in R45C6, locked for C6
10b. 12(3) cage at R2C6 = {138/147/156/237/246} (cannot be {129} because 1,2 is only in R4C6, cannot be {345} because R4C6 only contains 1,2), no 9 in R23C6
10c. R3C7 + R4C6 = R5C8 + 1 (step 7)
10d. R5C8 = {45} -> R3C7 + R4C6 = 5,6 = [32/41/42/51], no 1,2 in R3C7
10e. 27(4) cage at R3C8 (step 5a) = {4689/5679}, 6 locked for R4, clean-up: no 2 in R5C3 (step 6b)

11. 45 rule on R789 1 innie R7C2 = 1 outie R6C4 + 2 -> R6C4 = {3567}, R7C2 = {5789}
11a. R6C4 + R7C3 = R5C2 + 11 (step 8)
11b. Max R6C4 + R7C3 = 16 -> max R5C2 = 5, clean-up: no 2 in R5C4 (step 4)
11c. Min R6C4 + R7C3 = 12, max R6C4 = 7 -> min R7C3 = 5

12. 45 rule on NR1C9 + NR4C9 3 innies R3C7 + R45C6 = 7, R45C6 = {12} = 3 -> R3C7 = 4, placed for NR1C9, clean-up: no 3 in R1C3 (step 3)
12a. R3C7 = 4 -> R1C78 + R2C7 = 11 = {128/137/236}, no 5,9

13. 45 rule on NR1C4 1 remaining outie R1C3 = 1 innie R2C6, no 1,2 in R1C3, no 3,7,8 in R2C6, clean-up: no 5,6 in R3C3 (step 3)
13a. 12(3) cage at R2C6 (step 10b) = {147/156/246} (cannot be {138/237} because 3,7,8 only in R3C6), no 3,8

14. 3 in NR1C9 only in 17(4) cage at R1C9
14a. Caged X-Wing for 17(4) cage and 6(3) cage at R7C8, no other 3 in C89
14b. 3 in NR4C9 only in 13(3) cage at R5C7 = {139/238/346}, no 5,7
14c. 4 of {346} must be in R6C6 -> no 6 in R6C6

15. R8C7 = 5 (hidden single in C7)
15a. 5 in NR8C3 only in R9C23456, locked for R9

16. 45 rule on NR8C3 2 outies R7C3 + R9C7 = 1 innie R8C4 + 9
16a. Min R7C3 + R9C7 = 11 -> min R8C4 = 2
16b. R68C4 cannot total 6 -> no 8 in R7C4

[Probably a step not consistent with the SS score, but I’ll use it anyway. It’s not going quite as far as Law of Leftovers, which would have given more eliminations.]
17. 1 in C6789 only in 15(4) cage at R1C7, NR1C9, NR4C9 and NR7C7 -> 15(4) cage must contain 1, locked for NR1C4
17a. R1C78 + R2C7 (step 12a) = {128/137}, no 6
17b. 18(3) cage at R1C5 = {369/459/468/567} (cannot be {279/378} which clash with R1C78 + R2C7), no 2
17c. 16(3) cage at R1C3 = {259/268/349/367/457} (cannot be {358} which clashes with R1C78 + R2C7)
17d. 6 of {268/367} must be in R1C3 -> no 6 in R12C4

18. 1 in NR1C1, NR4C2 and NR6C2 only in C123, locked for C123

[Missed this at the time; anyway it’s not really consistent with the SS score]
19. R7C2 = R6C4 + 2 (step 11), R6C4 + R7C3 = R5C2 + 11 (step 8)
19a. Max R6C4 + R7C3 = 15 (cannot be [79] = 16, which clashes with R6C4 + R7C2 = [79], killer combo clash) -> no 5 in R5C2, clean-up: no 3 in R5C4 (step 4)

20. 45 rule on NR4C1 + NR6C2 3(2+1) innies R56C4 + R7C3 = 19
20a. Max R56C4 = 13 -> min R7C3 = 6

[I should have remembered this 45 after I did step 15)
21. R48C7 = R1C8 + 10 (step 9), R8C7 = 5 -> R4C7 = R1C8 + 5, no 7 in R1C8, no 9 in R4C7
21a. R79C7 (step 2) = {69} (only remaining combination, cannot be {78} which clashes with R1C78, step 17a), locked for C7 and NR7C7
21b. Killer pair 7,8 in R1C78 and R4C7, locked for C7

22. 13(3) cage at R5C7 (step 14b) = {139/238} -> R6C6 = {89}
22a. Naked triple {123} in R5C67 + R6C7, locked for NR4C9, 3 also locked for C7
22b. Naked triple {123} in R5C267, locked for R5, clean-up: no 5,7 in R4C3 (step 6b)

23. 27(4) cage at R3C8 (step 5a) = {4689/5679} -> R4C8 = 6, R3C8 = 9
23a. Naked quad {4578} in R5689C8, locked for C8, 5 also locked for NR4C9

24. 20(3) cage at R7C5 = {389/479/569} (cannot be {578} because R7C7 only contains 6,9), 9 locked for R7, clean-up: no 7 in R6C4 (step 11)
24a. R6C4 + R7C3 = R5C2 + 11 (step 8), R7C2 = R6C4 + 2 (step 11)
24b. Min R6C4 + R7C3 = 12 -> no 3 in R6C4, clean-up: no 5 in R7C2 (step 11)
24c. R6C4 + R7C3 = [58/67] (cannot be [57] which clashes with R6C4 + R7C2 = [57], cannot be [68] which clashes with R6C4 + R7C2 = [68], killer combo clashes)
24d. R6C4 + R7C3 = [58/67] = 13 -> R5C2 = 2, R5C4 = 6 (step 4), both placed for NR4C1, R6C4 = 5, R7C3 = 8, R7C2 = 7, all placed for NR6C2

25. R5C6 = 1, placed for NR4C9, R56C7 = [32], R6C6 = 8 (cage sum), placed for NR4C9

26. R6C4 = 5 -> R78C4 = 9 = [18/27]
26a. Naked triple {123} in R7C489, locked for R7, 3 also locked for NR7C7

27. 20(3) cage at R7C5 (step 24) = {569} (only remaining combination), locked for R7, 5 also locked for NR3C4 -> R7C1 = 4, placed for NR6C2

28. R6C9 = 6, R5C8 = 5 (hidden pair in NR4C9), R4C7 = 7 (cage sum), placed for NR1C9, R5C3 = 7, R4C3 = 1 (cage sum), both placed for NR4C1, R45C9 = [94], R6C8 = 7
[Routine clean-ups omitted from here]

29. R4C6 = 2, placed for NR1C9, R23C6 = 10 = [46], 4 placed for NR1C4, 6 placed for NR3C4

30. Naked pair {59} in R7C56, locked for R7 and NR3C4 -> R7C79 = [69], R456C5 = [384], 3,8 placed for NR3C4, R56C1 = [93], R6C23 = [19], both placed for NR6C2
30a. Naked pair {26} in R89C1, locked for C1 and NR6C2 -> R8C2 = 3, R8C6 = 7, placed for NR8C3, R9C6 = 3 (cage sum), R8C4 = 8, R7C4 = 1 (cage sum), placed for NR3C4, R9C4 = 2, placed for NR8C3, R3C45 = [72], R3C3 = 3 (cage sum), R1C3 = 4 (step 3)

31. R89C3 = [65], R12C2 = {69} (hidden pair in C2), R2C3 = 2, R1C1 = 7 (cage sum)

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

Jigsaw nonet design: Orig 8

SumoCueV1=24J0+0J0=16J0+2J1=18J1+4J1=15J1+6J1=17J2=14J0+0J0+0J0+2J1+4J1=12J1+6J1+8J2+8J2+9J0+9J0=12J0+20J3+20J3+14J3+6J2=22J2+8J2=25J4+27J4=20J4+29J4=15J3+14J2+25J2+25J2=26J5+27J4+29J4+29J4+29J4+31J3=19J5+41J5+41J5+35J5+27J4=17J6+46J6=14J6+31J3+41J5+41J5+35J5+35J5=15J6+46J6=23J6+48J3=20J3+58J3+58J7=6J7+61J7+54J6+54J6+56J8+48J8=12J8=19J8=24J7+69J7+61J7+54J6+56J8+56J8+67J8+67J8+68J8+68J7+69J7+69J7

Prelims

a) 22(3) cage at R3C8 = {589/679}
b) 20(3) cage at R7C5 = {389/479/569/578}, no 1,2
c) 6(3) cage at R7C8 = {123}
d) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
e) 26(4) cage at R4C9 = {2789/3689/4589/4679/5678}, no 1

Steps resulting from Prelims
1a. 22(3) cage at R3C8 = {589/679}, 9 locked for NR1C9
1b. Naked triple {123} in 6(3) cage at R7C8, locked for NR7C7

2. 45 rule on NR7C7 2 innies R79C7 = 15 = {69/78}

3. 45 rule on NR1C1 2 innies R13C3 = 7 = {16/25/34}, no 7,8,9

4. 45 rule on NR1C9 2 innies R3C7 + R4C6 = 6 = {15/24}

5. 45 rule on NR6C2 2 innies R6C4 + R7C3 = 13 = {49/58/67}, no 1,2,3

6. 45 rule on R123 1 innie R3C8 = 1 outie R4C6 + 7, R3C8 = {89}, R4C6 = {12}, clean-up: no 1,2 in R3C7 (step 4)

7. 45 rule on R789 1 innie R7C2 = 1 outie R6C4 + 2, R6C4 = {4567}, R7C2 = {6789}, clean-up: no 4,5 in R7C3 (step 5)
7a. Killer quad {6789} in R7C23 and 20(3) cage at R7C5, locked for R7

8. 45 rule on C6789 1 innie R1C6 = 1 outie R7C5, no 1,2 in R1C7

9. Caged X-Wing for 3 in 17(4) cage at R1C9 and 6(3) cage at R7C8, no other 3 in C89

10. 15(4) cage at R1C7 = {1248/1257/1347/1356/2346} (cannot be {1239} because R3C7 only contains 4,5), no 9
10a. R3C7 = {45} -> no 4,5 in R1C78 + R2C7

11. 45 rule on C123 2 outies R45C4 = 2 innies R13C3 + 3
11a. R13C3 (step 3) = 7 -> R45C4 = 10 = {19/28/37/46}, no 5
11b. R45C4 = 10 -> R4C3 + R5C23 = 10 = {127/136/145/235}, no 8,9

12. 45 rule on C789 2 innies R79C7 = 2 outies R56C6 + 6
12a. R79C7 = 15 (step 2) -> R56C6 = 9 = {18/27/36/45}, no 9
12b. R56C6 = 9 -> R5C78 + R6C7 = 10 = {127/136/145/235}, no 8,9

13. R6C4 + R7C3 = 13 (step 5), R7C2 = 1 outie R6C4 + 2 (step 7) -> R7C23 = 15 = {69/78}
13a. 20(3) cage at R7C5 = {569/578} (cannot be {389/479} which clash with R7C23), no 3,4, 5 locked for R7 and NR3C4, clean-up: no 3,4 in R1C6 (step 8)

14. 45 rule on C1234 1 innie R9C4 = 1 outie R3C5, no 5 in R9C4

15. 4 in R7 only in R7C14, CPE no 4 in R6C4, clean-up: no 6 in R7C2 (step 7), no 9 in R7C3 (step 5)

16. 45 rule on NR1C4 2(1+1) outies R1C3 + R3C7 = 1 innie R2C6 + 4
16a. Max R1C3 + R3C7 = 11 -> max R2C6 = 7

17. 45 rule on NR8C3 2(1+1) outies R7C3 + R9C7 = 1 innie R8C4 + 9
17a. Max R7C3 + R9C7 = 17 -> max R8C4 = 8
17b. Min R7C3 + R9C7 = 12 -> min R8C4 = 3

18. 20(3) cage at R7C5 (step 13a) = {569/578} with 5 in R7C56, R1C6 = R7C5 (step 8) -> 5 in R16C6, locked for C6, clean-up: no 4 in R56C6 (step 12a)

19. Hidden killer quad 1,2,3,4 in 17(3) cage at R6C2 and 15(4) cage at R7C1 for NR6C2, 17(3) cage contains one of 1,2,3,4 -> 15(4) cage must contain three of 1,2,3,4 = {1239/1248/1347/2346}, no 5
19a. 5 in NR6C2 only in R6C234, locked for R6

20. 45 rule on C89 2 outies R48C7 = 2 innies R17C8 + 5
20a. Min R48C7 = 10 (cannot be [54] which clashes with R3C7) -> min R17C8 = 5, no 1 in R1C8

21. 12(3) cage at R2C6 = {129/138/147/237/246}
21a. 9 of {129} must be in R3C6 -> no 1,2 in R3C6
21b. 6 of {246} must be in R3C6 (cannot be [642] which clashes with R3C7 + R4C6 = [42], killer combo clash), no 6 in R2C6

[Maybe this puzzle needs Law of Leftovers (LoL), even though the SS score is only 1.20?]
22. LoL for R123 three outies R4C678 must exactly equal three innies R3C456, no 3,4 in R4C678 -> no 3,4 in R3C456
22a. Same LoL, no 5 in R3C456 -> no 5 in R4C78
22b. 22(3) cage at R3C8 = {679} (only remaining combination) -> R3C8 = 9, R4C78 = {67}, locked for R4 and NR1C9, R4C6 = 2 (step 6), R3C7 = 4 (step 4), both placed for NR1C9, clean-up: no 3 in R1C3 (step 3), no 3,4,8 in R5C4 (step 11a)
22c. Same LoL, R4C678 = 2{67} -> R3C456 = {267}, locked for R3 and NR3C4, clean-up: no 1,5 in R1C3 (step 3), no 6,7 in R1C6 (step 8)

23. R1C3 + R3C7 = R2C6 + 4 (step 16)
23a. R3C7 = 4 -> R1C3 = R2C6 = 4 (only common candidate in these cells), R3C3 = 3 (step 3), placed for NR1C1, R3C6 = 6 (cage sum)
23b. R1C3 = 4 -> R12C6 = 12 = {39/57}

24. 20(3) cage at R7C5 (step 13a) = {569/578}
24a. 6,7 only in R7C7 -> R7C7 = {67}
24b. Naked pair {67} in R47C7, locked for C7
24c. 15(4) cage at R1C7 (step 10) = {1248/1347/2346}
24d. 6,7 of {1347/2346} must be in R1C8 -> no 3 in R1C8

25. LoL for R789 three outies R6C234 must be exactly equal to three innies R7C456, no 6,7 in R7C456 -> no 6,7 in R6C234 -> R6C4 = 5, R7C3 = 8 (step 5), placed for NR6C2, R7C56 = {59}, locked for R7 and NR3C4, R7C2 = 7, placed for NR6C2, R7C7 = 6, placed for NR7C7, R9C7 = 9 (step 2), R4C78 = [76], clean-up: no 8 in R1C6 (step 8), no 7 in R12C4 (step 23b)
25a. Naked pair {39} in R12C4, locked for C4 and NR1C4 -> R1C6 = 5, R7C56 = [59]
[Routine clean-ups omitted from here]

26. 15(3) cage at R4C5 = {348} (only remaining combination), locked for C5 and NR3C4 -> R7C4 = 1, R8C4 = 8 (cage sum), placed for NR8C3, R4C4 = 4, R5C4 = 6 (step 11a), both placed for NR4C1
26a. Naked pair {37} in R89C6, locked for C6 and NR8C3 -> R9C4 = 2, placed for NR8C3, R89C5 = 10 = [91], placed for NR8C3, R3C45 = [72]
26b. Naked pair {67} in R12C5, locked for NR1C4
26c. Naked pair {56} in R89C3, locked for C3 and NR8C3 -> R9C2 = 4, R4C3 = 1, placed for NR4C1
26d. R4C34 + R5C4 = [146] = 11 -> R5C23 = 9 = [27], placed for NR4C1

27. R7C2 = 7 -> R6C23 = 10 = [19], placed NR6C2 -> R6C6 = 8, placed for NR4C9, R5C6 = 1, R6C1 = 3, R6C7 = 2
27a. R5C6 + R6C67 = [182] = 11 -> R5C78 = 8 = [35], placed for NR4C9 -> R4C9 = 9

28. Naked pair {23} in R7C89, locked for R7 and NR7C7 -> R7C1 = 4, R8C9 = 1
28a. Naked triple {358} in R123C9, locked for C9 and NR1C9 -> R2C8 = 1, R7C89 = [32], R9C9 = 7

29. 14(3) cage at R2C1 = {158} (only remaining combination) -> R3C1 = 1, R2C1 + R3C2 = {58}, locked for NR1C1

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