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SumoCueV1=8J0=16J0+1J0+1J0=23J1+4J2=14J2=13J2+7J2+0J0=24J0=9J0=15J1+4J1+6J3+6J2+7J2=9J2+10J0+10J1+11J1+12J1+4J3=22J3+23J3+23J3+17J2=10J0=15J1+11J4+12J4+23J3+23J3=20J3+33J3+33J2+27J1+28J1+28J4+28J4=24J4=18J4+41J4+41J5=10J5=19J6+45J7+45J7+40J7+40J7=8J4=18J4+41J5+44J8+45J6=23J7+55J7+40J7=14J7+50J5+51J5=21J5+61J8+55J6+55J6=15J6+65J7+58J5+50J5+51J8+61J8=10J8=10J6+72J6+65J6+58J6+58J5=17J8+77J8+77J8+71J8

5 6 8 2 7 9 1 4 3
3 7 1 9 4 8 5 6 2
9 8 2 5 3 6 4 1 7

4 3 6 1 9 2 7 5 8
6 1 7 4 8 5 2 3 9
2 4 5 7 6 3 9 8 1

8 2 9 3 1 4 6 7 5
7 5 4 8 2 1 3 9 6
1 9 3 6 5 7 8 2 4

Jigsaw nonets :

n1 at r1c1
n2 at r1c5
n3 at r1c6
n4 at r2c6
n5 at r4c3
n6 at r5c8
n7 at r6c1
n8 at r6c2
n9 at r6c9

1) a) Innies-outies for n1 : « 45 rule » => r3c2 = r2c3 + r4c1 + 3
r3c2=(789) => r2c3+r4c1=4/5/6 : r2c3=(12345), r4c1=(1234)
b) Cage sum for 10(2) ; r5c1=(6789)

2)a) Innies-outies for c12345 : r1c6=r4c5
b) hcage h23(4) = r1234c5 since r1c6=r4c5.
c) Outies for n34 : r12c5=h11(2) => r34c5= h12(2).
d) Outies for n78 : r589c5=h15(3).
e) Innies for c5 : r67c5= h7(2).

3)a) LOL for r1234 (or for n1234) : {r5c12}={r4c34}
b) LOL for c1234 (or for n1278) : {r1267c5} = {r4c34+r5c34}

The following is the cracking move :

c){r4c34} ={r5c12} from step 3)a), so digits at r4c34 must be different of any digit of {r12c5} since they share n4 with {r5c12}
=> {r4c34} = {r67c5} (from step 3)b)) = h7(2) (from step 2)e))
=> {r4c34}={r5c12}=h7(2) : r5c1=6, r5c2=1
d) r4c1=4 (cage sum)
e) Innie for r1234 : r4c2=3
f) From step 3)c) : r4c34={16} locked for r4 and n5 and r67c5={16} locked for c5 and n8.
g) Cage 20(3) at r4 : {578} locked for r4.


4)a) No 2 at h12(2)c5 : naked single : r4c5=9=r1c6. Cage sum for h12(2) : r3c5=3.
b) r4c6=2
c) Last combination for cage 22(5) at r3c6 = {12469}, {164} locked for r3 and n4.
d) r12c5=h11(2)={47} locked for c5 and n2.
e) 7 locked for n1 at cage 24(3) : last combination : r12c1= {35} locked for n1 and
c1.
f) Last combination 16(3) at r1c2 : {268} locked for r1 and n1.
g) Last place for cage 24(3) : r3c2=8 : r2c3=1
h) r4c34=[61], r3c3=2, r23c4={59} locked for c4.

The rest is not difficult.

Prelims

a) R12C1 = {17/26/35}, no 4,8,9
b) R23C9 = {18/27/36/45}, no 9
c) R45C1 = {19/28/37/46}, no 5
d) R56C9 = {19/28/37/46}, no 5
e) R89C9 = {19/28/37/46}, no 5
f) R9C12 = {19/28/37/46}, no 5
g) 24(3) cage at R2C2 = {789}
h) 9(3) cage at R2C3 = {126/135/234}, no 7,8,9
i) 20(3) cage at R4C7 = {389/479/569/578}, no 1,2
j) 8(3) cage at R6C6 = {125/134}
k) 21(3) cage at R7C8 = {489/579/678}, no 1,2,3
l) 14(4) cage at R7C5 = {1238/1247/1256/1346/2345}, no 9

1. 8(3) cage at R6C6 = {125/134}, 1 locked for C6
1a. 8(3) cage at R6C6 = {125/134}, CPE no 1 in R6C8

2. Naked triple {789} in 24(3) cage at R2C2, CPE no 7,8,9 in R1C2

3. 45 rule on R1234 2 innies R4C12 = 7 = {16/34}/[25], no 7,8,9, no 2 in R4C2, clean-up: no 1,2,3 in R5C1

4. 16(3) cage at R1C2 must contain one of 7,8,9
4a. Killer triple 7,8,9 in 16(3) cage and 24(3) cage, locked for NR1C1, clean-up: no 1 in R12C1
4b. 16(3) cage = {169/268/349/358/457} (cannot be {178} which clashes with 24(3) cage at R2C2, cannot be {259/367} which clash with R12C1)

5. 45 rule on NR1C6 + NR2C6 2 innies R1C6 + R3C5 = 12 = {39/48/57}, no 1,2,6
5a. 45 rule on NR1C6 + NR2C6 2 outies R12C5 = 11 = {29/38/47/56}, no 1

6. 45 rule on C6789 1 outie R4C5 = 1 innie R1C6 -> R4C5 = {345789}
6a. 1 in NR2C6 only in R3C78, locked for R3, clean-up: no 8 in R2C9
[Note. R1C6 + R3C5 = 12 (step 5), R4C5 = R1C6 (step 6) -> R34C5 = 12 may come in useful later.]

7. 45 rule on NR6C1 + NR6C2 1 outie R5C5 = 2 innies R7C5 + R9C4 + 1
7a. Min R7C5 + R9C4 = 3 -> min R5C5 = 4
7b. Max R7C5 + R9C4 = 8, no 8 in R7C5 + R9C4

8. 45 rule on C6789 4 outies R1234C5 = 23
[Alternatively, 23(4) cage at R1C5 and R1C6 = R4C5, step 6]
8a. 45 rule on NR6C1 + NR6C2 3 outies R589C5 = 15
8b. 45 rule on C5 2 remaining innies R67C5 = 7 = {16/25/34}, no 7,8,9

9. 45 rule on NR1C6 2 innies R1C6 + R4C9 = 1 outie R2C6 + 9, IOU no 9 in R4C9
9a. Max R1C6 + R4C9 = 17 -> max R2C6 = 8

10. 45 rule on NR1C1 1 outie R3C2 = 2 innies R2C3 + R4C1 + 3
10a. Max R2C3 + R4C1 = 6, no 6 in R2C3 + R4C1, clean-up: no 1 in R4C2 (step 2), no 4 in R5C1
10b. 1 in NR1C5 only in R2C4 + R5C2, CPE no 1 in R5C4

11. 45 rule on NR6C9 1 outie R7C8 = 2 innies R6C9 + R8C7 + 3
11a. Min R6C9 + R8C7 = 3 -> min R7C8 = 6
11b. Max R6C9 + R8C7 = 6, no 6,7,8,9 in R6C9 + R8C7, clean-up: no 1,2,3,4 in R5C9

12. 45 rule on NR6C9 2 outies R5C9 + R7C8 = 1 innie R8C7 + 13
12a. Max R5C9 + R7C8 = 17 -> max R8C7 = 4
12b. Max R8C7 = 4 -> min R67C7 = 14, no 1,2,3,4 in R67C7

13. 17(3) cage at R9C6 must contain one of 1,2,3,4
13a. Killer quad 1,2,3,4 in R6C9, R8C7, R89C9 and 17(3) cage, locked for NR6C9
13b. 21(3) cage at R7C8 = {579/678}, CPE no 7 in R9C8

14. Law of Leftovers (LoL) for R6789 two outies R5C89 must exactly equal two innies R6C67, R5C9 = R6C7 = {6789} (because no 6,7,8,9 in R6C6), R6C6 = R5C8 = {12345}

15. 45 rule on R6789 2 outies R5C59 = 1 innie R6C8 + 9, IOU no 9 in R5C5
15a. Max R5C59 = 17 -> max R6C8 = 8
15b. 9 in NR5C8 only in R5C9 + R7C78, CPE no 9 in R7C9

16. 9 in C5 only in R1234C5 (step 8) = 23 = {2489/2579/3479} (cannot be {3569} which clashes with R67C5)
16a. R12C5 = 11 (step 5a) -> R12C5 = {29/47}, no 3,5,6,8

17. LoL for C1234 four outies R1267C5 must exactly equal four innies R45C34, no 8 in R1267C5 -> no 8 in R45C34

18. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34, no 8 in R4C34 -> no 8 in R5C12, clean-up: no 2 in R4C1, no 5 in R4C2 (step 3)

19. 45 rule on C1234 2 outies R56C5 = 1 innie R9C4 + 8
19a. Max R56C5 = 14 -> max R9C4 = 6

20. 45 rule on C9 3 innies R147C9 = 16 = {268/358/367/457} (cannot be {169/259} because 1,2,9 only in R1C9, cannot be {178/349} which clash with the pair of 10(2) cages in C9), no 1,9
20a. 9 in C9 only in R56C9 = [91] or R89C9 = {19}, 1 locked for C9 (locking cages), clean-up: no 8 in R3C9
20b. 1 in C9 only in R689C9, locked for NR6C9

21. 45 rule on NR1C1 4 innies R2C23 + R34C1 = 21 = {1389/1479/1578/2478} (cannot be {2379} which clashes with R12C1, cannot be {3459} because R2C2 + R3C1 only contain 7,8,9)
21a. R2C23 + R34C1 = {1389/1479/2478} (cannot be {1578} because 24(3) cage at R2C2 clashes with R45C1 = [19], killer combo clash in NR1C5), no 5
21b. 3 of {1389} must be in R2C3 (cannot be 1{89}3 because 24(3) cage at R2C2 clashes with R45C1 = [37], killer combo clash in NR1C5), no 3 in R4C1, clean-up: no 4 in R4C2 (step 3), no 7 in R5C1

22. Hidden killer pair 5,6 in R12C1 and 16(3) cage at R1C2 for NR1C1, R12C1 contains one of 5,6 -> 16(3) cage must contain one of 5,6, 16(3) cage (step 4b) = {169/268/358/457} (cannot be {349} which doesn’t contain 5 or 6)

23. R4C1 = {14}, R4C12 = 7 (step 3) -> R4C12 + R5C1 = [169/436], 6 locked for NR1C5
23a. R4C12 + R5C1 = [169/436], CPE no 6 in R5C34
23b. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34, when 6 is in R5C1 there must be 6 in one of R4C34 -> 6 in R4C2 or R4C34, locked for R4

[Taking step 21a a bit further]
24. R2C23 + R34C1 (step 21a) = {1479/2478} (cannot be {1389} because 7 from 24(3) cage at R2C2 and 9 from R45C1 = [19] clash with R12C5 using NR1C5, killer combo clash), no 3, 4,7 locked for NR1C1 and 24(3) cage at R2C2, no 7 in R3C3
24a. 4 in NR1C1 only in R2C3 + R4C1, CPE no 4 in R4C3

[And further still]
25. R2C23 + R34C1 (step 24) = {1479} (cannot be {2478} because R4C12 (step 3) = [43] blocks the only combination for 9(3) cage at R2C3 containing 2 in R2C3) -> R2C3 + R4C1 = {14}, locked for NR1C1, R2C2 + R3C1 = {79}, locked for NR1C1 and 24(3) cage at R2C2 -> R3C3 = 8, placed for NR1C5, clean-up: no 2 in R9C1
25a. Naked pair {14} in R2C3 + R4C1, CPE no 1 in R4C3
25b. 16(3) cage at R1C2 (step 22) = {268/358}, 8 locked for R1
25c. 1 in R1 only in R1C78, locked for NR1C6
25d. 8 in C1 only in R6789C1, locked for NR6C1
25e. Clean-up: no 4 in R1C6 + R3C5 (step 5), no 4,8 in R4C5 (step 6)

26. R12C5 (step 16a) = {29/47}, R1C6 + R3C5 (step 5) = {39/57}, R1C6 = R4C5 (step 5) -> R34C5 = {39/57}
26a. Killer pair 7,9 in R12C5 and R34C5, locked for C5

27. 9(3) cage at R2C3 = {126/135/234}
27a. R2C3 = {14} -> no 4 in R3C3
27b. 9(3) cage = {126/234} (cannot be {135} which clashes with R4C12 (step 3) = [43], killer combo clash), no 5, 2 locked for C3
27c. 1 in NR1C1 only in R2C3 -> 9(3) cage = {126} = [126] or in R4C1 -> R4C12 (step 3) = [16] (locking cages) -> 6 must be in R4C23, locked for R4

28. 15(3) cage at R2C4 = {159/249} (cannot be {357} because R4C2 = 6, R5C1 = 9 => R4C4 = 9 (LoL, step 18) blocks {357}), no 3,7, 9 locked for C4
28a. 3 in NR1C5 only in R3C3 + R45C2, CPE no 3 in R5C3

29. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34
29a. No 7 in R4C34 -> no 7 in R5C2
29b. R5C12 and R4C34 cannot be [69] because R23C3 = [12] clashes with R4C2 + R5C34 = [312] -> no 9 in R5C2

30. Variable hidden killer pair 1,5 in 15(3) cage at R2C4 and R5C2 for NR1C5, R5C2 cannot contain both of 1,5 -> 15(3) cage must contain at least one of 1,5 -> 15(3) cage (step 28) = {159} (only remaining combination, cannot be {249} which doesn’t contain 1 or 5), locked for C4

31. LoL for R1234 two outies R5C12 must exactly equal two innies R4C34
31a. No 4 in R4C34 -> no 4 in R5C2

32. R12C5 = {47} (hidden pair in NR1C5), locked for C5 and 23(4) cage at R1C5, clean-up: no 5 in R1C6 + R3C5 (step 5), no 5 in R4C5 (step 6)
32a. Naked pair {39} in R34C5, locked for C5 and NR2C6
32b. 20(3) cage at R4C7 = {578} (only remaining combination), locked for R4

33. Killer pair 2,4 in R4C6 and 8(3) cage at R6C6, locked for C6

34. 1,2,4 in NR2C6 only in 22(5) cage at R3C6 = {12469} (only remaining combination), no 3,5,7 -> R3C6 = 6, R34C5 = [39], R3C3 = 2, placed for NR1C5, R1C6 = 9, placed for NR1C6, clean-up: no 3,6,7 in R2C9

35. Naked triple {159} in 15(3) cage at R2C4, 9 locked for NR1C5 -> R5C1 = 6, placed for NR1C5, R4C1 = 4, placed for NR1C1, R4C2 = 3, R24C3 = [16], R4C6 = 2, clean-up: no 2 in R12C1, no 5 in 8(3) cage at R6C6, no 4 in R6C9, no 7 in R9C1, no 4,6 in R9C2
35a. Naked pair {59} in R23C4, locked for NR1C5 -> R4C4 = 1, R5C2 = 1, clean-up: no 9 in R9C1
35b. LoL for R6789 two outies R5C89 must exactly equal two innies R6C67, no 2,5 in R5C8, no 1 in R6C6, no 6 in R6C7

36. Naked pair {35} in R12C1, locked for C1 and NR1C1, clean-up: no 7 in R9C2
36a. Naked triple {268} in 16(3) cage at R1C2, locked for R1

37. Naked triple {134} in 8(3) cage at R6C6, 1 also locked for NR5C8
37a. Naked triple {134} in 8(3) cage at R6C6, CPE no 3,4 in R6C8

38. 1 in C5 only in R67C5 (step 8b) = {16}, locked for C5 and NR6C2
38a. 2 in C5 only in R89C5, locked for NR5C8 and 14(4) cage at R7C5, no 2 in R9C4

39. 2 in C5 only in 14(4) cage at R7C5 = {1238/1256} (cannot be {2345} because R7C5 only contains 1,6) -> R7C5 = 1, R6C5 = 6, R9C4 = {36}
39a. R8C6 = 1 (hidden single in C6), clean-up: no 9 in R9C9
39b. Deleted, I did it earlier
39c. 6 in C9 only in R789C9, locked for NR6C9

40. R6C5 = 6 -> 24(4) cage at R5C5 = {3678} (only remaining combination) -> R5C5 = 8, placed for NR4C3, R67C4 = {37}, locked for C4 and NR6C2 -> R9C4 = 6, placed for NR6C1, clean-up: no 2 in R6C9, no 4 in R8C9

41. R45C2 = [31] = 4 -> R5C34 = 11 = [74/92]
41a. Naked pair {79} in R5C39, locked for R5 -> R5C6 = 5

42. R5C6 = 5, 3 in R5 only in R5C78 -> 18(4) cage at R5C5 = {2358} (only remaining combination) -> R5C7 = 2, R56C8 = [38], both placed for NR5C8, R67C6 = [34], R6C9 = 1, R5C9 = 9, placed for NR5C8, R5C34 = [74], 7 placed for NR4C3
42a. Naked pair {25} in R89C5, locked for NR5C8
42b. R6C7 = 9 -> R78C7 = 9 = [63], 3 placed for NR6C9, clean-up: no 7 in R89C9

43. 16(3) cage at R1C2 = [682], R8C4 = 8, clean-up: no 2 in R9C9

44. R67C4 = [73], R6C1 = 2, placed for NR6C1, R9C2 = 9, placed for NR6C1, R9C1 = 1

45. Naked pair {45} in R6C23, locked for NR6C2 -> R7C23 = [29]

46. R8C9 = 6 (hidden single in R8), R9C9 = 4, clean-up: no 5 in R23C9
46a. R23C9 = [27], 7 placed for NR1C6, R7C9 = 5 (hidden single in R7), placed for NR6C9, R1C9 = 3, R12C1 = [53]
46b. R1C9 = 3 -> R12C8 = 10 = [46]

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