SudokuSolver Forum

Hi all,

I am glad to post my first puzzles in this forum , so be nice and understanding ! I spent exciting hours to solve the Assassins sent in this Forum (and Ruud's website), and I hope all that will continue for a long time.

I wish you will have fun with these puzzles and may be the further ones
(some puzzles are yet ready for future Assassins).
I have personnally tried to solve them on paper and check they can be solved without
(at least too much) brute force. I am eager to know your ideas for solving them and your walkthroughs.
I wait for your opinion and every remark will be welcome

Many thanks to Ronnie, Andrew an Afmob for the assistance and the kindness they provided as soon as I have joined this forum.


So, let's begin.

This V1 is rated by SS V 3.3.0 at 1.19, and I would say it's a hard 1.

ASSASSIN 140 V.1


PS code


Solution :
134692857
759138462
268754193
923876541
871945236
546213789
395427618
612589374
487361925


The following V 1.5 is rated at 1.51. It might not to be much more difficult than V1 (close version) : my solving path was similar.

ASSASSIN 140 V1.5


EDIT : R5C4678 is a split 19(4) cage

PS code


(Same solution)

The last one might keep you busy for hours Only one minor change in the cages at N7, but it
makes really a difference !
The solving path you have used for the previous versions might not work for this one. SSolver rates it at 2.4 .......
It seems to be a very long series of little steps...I did not find any quick way to crack it

ASSASSIN 140 V.2


EDIT : R5C4678 is a split 19(4) cage

PS code

(same solution)


Have a good week !

Thanks for a fun puzzle! Excellent for your first one.

Edit. Afmob and Manu have both pointed out that my original step 8a was flawed; it should have been 8 locked, not 7 locked. That makes a big difference.

As I commented to both of them "Interesting that I got to the correct solution using a flawed step. It would have been better if I'd reached an impossible position and had to re-check my walkthrough to find where I'd gone wrong."

Manu and Afmob have both given A140 the correct rating of Hard 1.0. However in my rework after the flawed step I found it harder. I'll rate my walkthrough at 1.25 because of the clashes in step 18a and step 22.

It was interesting to see that Afmob and I used very different steps to reach the first placements. Our first placements were in the same two cells but in the opposite order.

Here is my walkthrough. I've added step 7e, corrected step 8a and step 9 and the rest of it is complete rework.

Prelims

a) R23C6 = {39/48/57}, no 1,2,6
b) R67C8 = {18/27/36/45}, no 9
c) R78C3 = {16/25/34}, no 7,8,9
d) R9C23 = {69/78}
e) R1C789 = {389/479/569/578}, no 1,2
f) 19(3) cage in N3 = {289/379/469/478/568}, no 1
g) 10(3) cage at R3C7 = {127/136/145/235}, no 8,9
h) 11(3) cage in N7 = {128/137/146/236/245}, no 9
i) 19(3) cage in N9 = {289/379/469/478/568}, no 1
j) 26(4) at R3C5 = {2789/3689/4589/4679/5678}, no 1
k) 32(5) cage at R4C1 = {26789/35789/45689}, no 1
l) 15(5) cage at R6C5 = {12345}
m) 23(6) cage in N1 = {123458/123467}, no 9, 1,2,3,4 locked for N1

1. 45 rule on N3 3 innies R2C9 + R3C79 = 6 = {123}, locked for N3

2. 45 rule on N8 2 outies R6C56 = 4 = {13}, locked for R6, N5 and 15(5) cage at R6C5, clean-up: no 6,8 in R7C8
2a. Naked triple {245} in R7C45 + R8C4, locked for N8

3. 45 rule on N7 2 innies R7C12 = 12 = {39/48/57}, no 1,2,6

4. 45 rule on N9 2 outies R6C78 = 15 = [78/87/96], clean-up: no 4,5,7 in R7C8

5. 45 rule on N47 2 outies R3C3 + R6C4 = 10 = [55/64/82]
5a. 9 in N1 locked in R2C23, locked for R2 and 22(4) cage at R2C2, no 9 in R3C4, clean-up: no 3 in R3C6
5b. Naked triple {245} in R678C4, locked for C4

6. 45 rule on C789 2 innies R5C78 = 5 = {14/23}

7. 45 rule on N1 2 outies R23C4 = 1 innie R3C3
7a. R23C4 cannot total 5,6 -> R3C3 = 8, R6C4 = 2 (step 4), R23C4 = 8 = {17}, locked for C4, N2 and 22(4) cage at R2C2, clean-up: no 4,5 in R2C6, no 5 in R3C6, no 7 in R9C2
7b. R2C23 = 14 = {59}, locked for R2 and N1
7c. R7C5 = 2 (hidden single in N8), R1C6 = 2 (hidden single in C6), clean-up: no 7 in R6C8, no 8 in R6C7 (step 4), no 5 in R8C3
7d. R3C3 = 8 -> R4C23 = 5 = {14/23}
7e. 1 in R1 locked in R1C123, locked for N1

8. 45 rule on N2 1 remaining innie R3C5 = 5
8a. R4C456 = 21 = {489/678}, 8 locked for R4 and N5

9. R5C6 = 5 (hidden single in C6), R5C45 = 13 = [67/94]

10. 19(3) cage in N3 = {469/478}, 4 locked for N3
10a. 9 of {469} must be in R3C8 -> no 6 in R3C8

11. 45 rule on R9 4 innies R9C1456 = 14 = {1238/1346} (cannot be {1247/1256/2345} because 2,4,5 only in R9C1), no 5,7,9
11a. 2,4 only in R9C1 -> R9C1 = {24}
11b. R9C456 = {136/138}, 1,3 locked for R9 and N8
11c. Killer pair 6,8 in R9C23 and R9C456, locked for R9
11d. R9C789 = {259/457}, 5 locked for N9

12. 45 rule on N9 3 innies R7C78 + R8C7 = 10 = {136} (cannot be {127} because R678C7 cannot be [772]), locked for N9, 6 locked in R78C7, locked for C7
12a. 10(3) cage at R3C7 = {127/145/235} (cannot be {136} because R34C7 = {13} clashes with R78C7), no 6

13. 45 rule on C9 2 innies R19C9 = 1 outie R8C8 + 5, IOU no 5 in R1C9
13a. Min R19C9 = 8 -> no 2 in R8C8
13b. Max R8C8 = 9 -> R9C19 = 14 -> no 9 in R9C9

14. 19(3) cage in N9 = {289/478}
14a. 2 of {289} must be in R8C9 -> no 9 in R8C9

15. 11(3) cage in N7 = {128/146/245} (cannot be {137} because R9C1 only contains 2,4, cannot be {236} which clashes with R78C3), no 3,7
15a. 4 of {245} must be in R9C1 (R8C12 cannot be {45} which clashes with R8C4), no 4 in R8C12

16. 45 rule on R1 3 innies R1C123 = 1 outie R2C5 + 5
16a. R1C123, which must contain 1, cannot total 9,13 -> no 4,8 in R2C5

17. 45 rule on R4 4 remaining innies R4C1789 = 19 = {1459/1567/2359} (cannot be {1279/1369/2467/3457} which clash with R4C23)
17a. 4,5 of {1459} must be in R4C78 (because max R3C7 = 3) -> no 4 in R4C19
17b. 1,7 of {1567} must be in R4C78 (because max R3C7 = 3) -> no 7 in R4C19

18. Hidden killer pair 8,9 in R456C9 and R6C78 for N6 -> R456C9 must contain one of 8,9
18a. 1,3 in C9 locked in R23456C9 = {12369/13458} (cannot be {12378} which clashes with 19(3) cage in N9, cannot be {13467} which doesn’t contain 8 or 9), no 7
18b. R1789C9 (from R23456C9) = {2679/4578}
18c. 6 of {2679} must be in R1C9 -> no 9 in R1C9
18d. 2,9 of {2679} must be in R78C9 (because of 19(3) cage combinations) -> no 2 in R9C9
18e. 5 of {4578} must be in R9C9 -> no 4 in R9C9
18f. 4 of {4578} must be in R78C9 -> no 4 in R8C8 (because of 19(3) cage combinations)

19. R1C789 = {569/578}
19a. 6 of {569} must be in R1C9 -> no 6 in R1C8

20. 20(4) cage in N2 = {2369/2468}
20a. 4 of {2468} must be in R1C5 -> no 8 in R1C5

21. Hidden killer pair 7,9 in R7C12 and R9C23 for N7 -> R7C12 must contain one of 7,9 = {39/57} (step 3), no 4,8

22. 22(5) cage at R5C3 = {12469/12478/23458/23467} (cannot be {12379} which clashes with R6C7, cannot be {12568} which clashes with R6C8), 4 locked in R5C3 + R6C23, locked for N4, clean-up: no 1 in R4C23 (step 7d)
22a. Naked pair {23} in R4C23, locked for R4 and N4

23. R5C3 = 1 (hidden single in N4), clean-up: no 4 in R5C78 (step 6), no 6 in R78C3
23a. Naked pair {23} in R5C78, locked for R5
23b. 3 in C9 locked in R23C9, locked for N3
23c. 4 in N4 locked in R6C23, locked for R6

24. 22(5) cage at R5C3 (step 22) = {12469/12478}, no 3,5, clean-up: no 7,9 in R7C1 (step 3)
24a. R7C2 = {79} -> no 7,9 in R6C23
24b. Killer pair 6,8 in R6C23 and R6C8, locked for R6

25. 32(5) cage at R4C1 = {35789} (only remaining combination), no 6 -> R7C1 = 3, R7C2 = 9 (step 3), R7C78 = [61], R8C7 = 3, R6C8 = 8, R6C7 = 7 (step 4), R5C78 = [23], R3C7 = 1, R2C23 = [59], R23C4 = [17], clean-up: no 4 in R78C3, no 6 in R9C23
25a. R78C3 = [52], R78C4 = [45], R9C123 = [487], R4C23 = [23], R9C789 = [925], R8C8 = 7, R78C9 = [84], R7C6 = 7, R456C9 = [169], R1C9 = 7, R5C12 = [87], R46C1 = [95], R5C45 = [94]

26. R1C9 = 7 -> R1C78 = 13 = [85]

and the rest is naked singles

I look forward to trying V1.5 to see what effect the slightly different cage pattern makes.

Thanks Manu for delivering this week's Assassin! I really liked all those potential Innies and Outies you could use. Maybe they become more useful in the harder versions?

I'll agree with manu's rating since I had to apply a Hidden Killer pair to crack it.

A140 Walkthrough:

1. R789 !
a) Outies N8 = 4(2) = {13} locked for R6+N5+15(5)
b) 15(5) = {12345} -> 2,4,5 locked for N8
c) ! Innies+Outies R9: 20 = R7C6+R8C56 - R9C1 -> R9C1 = (1234); R7C6+R8C56 <> 1,3
d) 34(6) = {136789} -> 1,3 locked for R9
e) 5 locked in 16(3) @ R9 for N9; 16(3) = 5{29/47}
f) Innies N9 = 10(3) <> 8,9
g) 8 locked in 19(3) @ N9 = 8{29/47}
h) 6 locked in Innies N9 = 10(3) = {136}
i) 16(3) = 6{19/37} because R78C7 = (136) -> R6C7 = (79); 6 locked for C7+N9

2. C1234
a) Innies N1 = 22(3) = 9{58/67}
b) Outies N47 = 10(1+1) = [55/64/82]
c) Naked triple (245) locked in R678C4 for C4
d) 9 locked in 22(4) @ N1 for R2+22(4) -> 22(4) = {1579}
e) Naked pair (17) locked in R23C4 for C4+N2+22(4)
f) Outies C123 = 10(3) = {127} -> R6C4 = 2
g) Outie N47 = R3C3 = 8

3. R123
a) Innie N12 = R3C5 = 5
b) Hidden Single: R7C5 = 2 @ N8, R1C6 = 2 @ N2, R5C6 = 5 @ N5
c) Innies N3 = 6(3) = {123} locked for N3

4. N47 !
a) 11(3) <> 7 because R9C1 = (24)
b) ! Hidden Killer pair (79) in Innies N7 because 15(2) can only have one of (79)
-> Innies N7 = 12(2) = {39/57}
c) 13(3) = 8{14/23}
d) 32(5) = 89{267/357/456} -> 8 locked for N4
e) 22(5) = 2{1379/1469/3467} <> 5
f) 5 locked in 32(5) @ N4 for C1 -> 32(5) = 589{37/46}
g) Innies N7 = 12(2) = {39} locked for R7+N7
h) 15(2) = {78} -> R9C3 = 7, R9C2 = 8
i) 2 locked in 13(3) @ N4 = {238} -> 2,3 locked for R4+N4

5. C789
a) Hidden Single: R8C7 = 3 @ R8
b) R7C8 = 1 -> R6C8 = 8, R7C7 = 6 -> R6C7 = 7
c) 10(3) = {145} -> R3C7 = 1; {45} locked for R4+N6
d) 16(3) @ R9 = {259} locked for R9+N9
e) R9C1 = 4, R7C3 = 5 -> R8C3 = 2

6. N25
a) Hidden Single: R5C5 = 4 @ N5
b) 23(5) = {23549} -> R5C7 = 2, R5C8 = 3, R5C4 = 9
c) 20(4) = {2369} -> R1C5 = 9; 3 locked for N2

7. Rest is singles.

Rating: (Hard) 1.0. I used a Hidden Killer pair.

V1.5 was a bit more difficult than V1 since it needed more than my previous Hidden Killer pair to be cracked. Though at first I thought it was not as difficult as V1 because the first placements felt easier.

A140 V1.5 Walkthrough:

1. R789
a) Outies N8 = 4(2) = {13} locked for R6+N5+15(5)
b) 15(5) = {12345} -> 2,4,5 locked for N8
c) Innies+Outies R9: 20 = R7C6+R8C56 - R9C1 -> R9C1 = (1234); R7C6+R8C56 <> 1,3
d) 34(6) = {136789} -> 1,3 locked for R9
e) 5 locked in 16(3) @ R9 for N9; 16(3) = 5{29/47}
f) Innies N9 = 10(3) <> 8,9
g) 8 locked in 19(3) @ N9 = 8{29/47}
h) 6 locked in Innies N9 = 10(3) = {136}
i) 16(3) = 6{19/37} because R78C7 = (136) -> R6C7 = (79); 6 locked for C7+N9

2. C1234
a) Innies N1 = 22(3) = 9{58/67}
b) Outies N47 = 10(1+1) = [55/64/82]
c) Naked triple (245) locked in R678C4 for C4
d) 9 locked in 22(4) @ N1 for R2+22(4) -> 22(4) = {1579}
e) Naked pair (17) locked in R23C4 for C4+N2+22(4)
f) Outies C123 = 10(3) = {127} -> R6C4 = 2
g) Outie N47 = R3C3 = 8

3. R123
a) Innie N12 = R3C5 = 5
b) 30(5) = {45678} -> 4,6,7,8 locked for N5
c) R5C4 = 9, R5C6 = 5
d) Hidden Single: R7C5 = 2 @ N8, R1C6 = 2 @ N2
e) Innies N3 = 6(3) = {123} locked for N3

4. N479
a) 9(2) = [63/81]
b) 11(3) <> 7 because R9C1 = (24)
c) Hidden Killer pair (79) in Innies N7 because 15(2) can only have one of (79)
-> Innies N7 = 12(2) = {39/57}

5. R456 !
a) ! Innies+Outies R1234: -2 = R5C5 - R234C9 -> R4C4 <> 8,9 since R5C5 <= 8
b) 9 locked in 22(4) @ R4 for N4 -> 22(4) = 89{14/23}
c) 22(5) = 2{1379/1469/1478/1568/3458/3467}
d) 22(4) = {2389} because (14) is a Killer pair of 22(5) -> 2,3 locked for R4+N4
e) 1 locked in R4C789 @ R4 for N6
f) 19(4) = {2359} -> 2,3 locked for R5
g) ! 22(5) = 24{169/178/358/367} because (39) only possible @ R7C2 and {12568} blocked by R6C8 = (68)
-> 4 locked for N4
h) 23(4): R7C1 <> 5 because (39) only possible there
i) 22(5): R7C2 <> 5 because 3 only possible there

6. N7
a) Innies N7 = 12(2) = {39} locked for R7+N7
b) 15(2) = {78} -> R9C3 = 7, R9C2 = 8
c) R7C8 = 1 -> R6C8 = 8, R7C7 = 6, R8C7 = 3 -> R6C7 = 7
d) 16(3) @ R9 = {259} locked for R9+N9
e) R9C1 = 4, R7C3 = 5 -> R8C3 = 2
f) 22(5) = {12469} -> R5C3 = 1, R7C2 = 9; {46} locked for R6+N4

7. N256
a) 10(3) = {145} -> R3C7 = 1; {45} locked for R4+N6
b) Hidden Single: R5C5 = 4 @ N5
c) 20(4) = {2369} -> R1C5 = 9; 3 locked for N2

8. Rest is singles.

Rating: Hard 1.0 - Easy 1.25. I used Killer pairs and a cage blocker.

Wow, It's been a while since I tackled something as difficult as this! The moves I used took me quite some time to find though they weren't of 2.0 rating.

A140 V2 Walkthrough:

1. R789
a) Outies N8 = 4(2) = {13} locked for R6+N5+15(5)
b) 15(5) = {12345} -> 2,4,5 locked for N8
c) Innies N7 = 12(2) <> 1,2,6
d) Outies N9 = 15(2) = [78/87/96]
e) Innies N9 = 10(3) <> 8,9
f) 9(2) = [63/72/81]

2. C1234
a) Innies N1 = 22(3) = 9{58/67}
b) Outies N47 = 10(1+1) = [55/64/82]
c) Naked triple (245) locked in R678C4 for C4
d) 9 locked in 22(4) @ N1 for R2+22(4) -> 22(4) = {1579}; 5 locked for R2+N1
e) Naked pair (17) locked in R23C4 for C4+N2+22(4)
f) Outies C123 = 10(3) = {127} -> R6C4 = 2
g) Outie N47 = R3C3 = 8

3. R123
a) Innie N12 = R3C5 = 5
b) 30(5) = {45678} -> 4,6,7,8 locked for N5
c) R5C4 = 9, R5C6 = 5
d) Hidden Single: R7C5 = 2 @ N8, R1C6 = 2 @ N2
e) Innies N3 = 6(3) = {123} locked for N3
f) 19(3) = 4{69/78} -> 4 locked for N3
g) 1 locked in R1C123 @ R1 for N1
h) Innies+Outies R1: -5 = R2C5 - R1C123: R2C5 <> 4,8 because R1C123 cannot be 9/13 using 1
i) 12(2) = [39/84]
j) 20(4) = 26{39/48} -> R1C5 = (49)

4. R456
a) 19(4) = 95{14/23}
b) 9(2) <> 7
c) Outies N9 = 15(2): R6C7 <> 8
d) 16(3): R78C7 <> 7 because R6C7 = (79)
e) Innies+Outies R1234: -2 = R5C5 - R234C9 -> R4C9 <> 8,9 since R5C5 <= 8
f) 9 locked in 22(4) @ R4 for N4 -> 22(4) = 89{14/23}
g) 8 locked in R4C456 @ R4 for N5
h) 5 locked in R6C123 @ N4 for R6

5. R789
a) Hidden Killer triple in Innies N7 + R7C69 for R7 because Innies N7 can only have one of (789)
-> R7C69 = (789)
b) 19(3) <> 5 because {568} blocked by Killer pair (56) of Innies N9

6. R123
a) 19(3): R3C8 <> 6 because 9 only possible there
b) 6 locked in R3C12 @ R3 for N1
c) Innies+Outies R1: -5 = R2C5 - R1C123; R2C5 = (36) -> R1C123 = 8/11(3) = 13{4/7} -> 3 locked for R1+N1
d) Hidden Single: R9C4 = 3 @ C4

7. C789
a) Innies+Outies C9: -5 = R8C8 - R19C9 -> R1C9 <> 5 (IOU @ N9); R9C9 <> 9 because R1C9 >= 6
b) Hidden Killer pair (89) in 21(5) for N6 since Outies N9 can only have one of (89)
c) 21(5) = 1{2369/2378/2459/3458} because {12468} blocked by R6C8 = (68) -> 1 locked for C9
d) 21(5): R6C9 <> 6 because 9 only possible there

8. C123
a) 5 locked in 23(4) @ C1 = 5{189/279/369/378/468}
b) 22(5): R7C2 <> 5 because {23458} blocked by Killer pair (34) of 22(4) and {12568} blocked by R6C8 = (68)
c) 7(2): R8C3 <> 5
d) 10(2): R8C1 <> 7

9. R789 !
a) ! Innies N9 = 10(3) <> {235} because R7C478 = {345} blocked by Killer triple (345) of Innies N7
-> Innies N9 = 1{36/45} -> 1 locked for N9
b) 19(3) <> 6 because {469} blocked by Killer pair (46) of 16(3) @ R6C7

10. R456 !
a) Hidden Killer pairs (12,13) in R5C123 for N5 since 22(4) can not have both of them
b) Killer pairs (12,13) locked in R5C123 + 19(4) for R5
c) ! Outies R1234 = 19(2+1) = 4+[69/78/87] / 6+[49] / 7+{48} because 6+[67] not placeable in 21(5)
-> CPE: R5C78 <> 4
d) 19(4) = {2359} -> 2,3 locked for R5+N6
e) 2 locked in 22(4) @ R4 -> 22(4) = {2389}

11. R456
a) Hidden Killer pair (48) in R5C12 for 23(4) since it cannot have more than one of (167)
b) Killer pair (48) locked in Outies R1234 + R5C12 for R5 (step 10c)
c) 22(5) = 2{1469/1478/1568/3467} because R5C3 = (167) and (39) only possible @ R7C2
d) 23(4) = 58{19/37/46} because (39) only possible @ R7C1
e) ! Hidden Killer triple (157) in 23(4) for N4 since 22(5) cannot have all three of (157)
f) 23(4): R7C1 <> 5 because R5C12+R6C1 cannot be {468}
g) Hidden Single: R6C1 = 5 @ C1

12. N7
a) Innies N7 = 12(2) = {39/48}
b) 7(2) = {16/25} since (34) is a Killer pair of Innies N7
c) 16(3) @ N7 <> {268} because (26) is a Killer pair of 7(2)

13. R789 !
a) ! Consider placement of 7 in N7 -> 16(3) @ R9C7 <> 7:
- i) 7 in R9C123 locked for R9 -> R9C789 <> 7
- ii) 7 in R8C2 a) 16(3) @ R8C2 = 7{18} locked for N9 -> 7(2) = [52] -> 19(3) = 7{39/48} -> 7 locked for N9
- ii) 7 in R8C2 b) 16(3) @ R8C2 = 7{45}; 4,5 locked for R9 -> 16(3) @ R9C7 <> {457}
b) 16(3) = 2{59/68} -> 2 locked for R9+N9
c) 4 locked in R9C123 @ R9 for N7
d) Innies N7 = 12(2) = {39} locked for R7+N7
e) R7C8 = 1 -> R6C8 = 8
f) Outies N9 = R6C7 = 7
g) 10(2) = [28/64]
h) 16(3) @ N7 = 7{18/45}

14. C789
a) Hidden Single: R6C9 = 9 @ N6
b) 19(3) @ N9: R8C8 <> 3 because 9 only possible there
c) Hidden Single: R5C8 = 3 @ C8, R9C8 = 2 @ C8
d) Innies+Outies C9: -5 = R8C8 - R19C9 -> R8C8 <> 4 because R19C9 >= 11
e) 20(3): R1C8 <> 6 because R1C9 <> 5,9
f) 19(3) = 7{39/48} -> R8C9 = (34)

15. R789 !
a) 16(3) @ R6C7 = 7{36/45}: R8C7 <> 6
b) Naked triple (345) locked in R8C479 for R8
c) ! R9C23 can only have one of (458) because two of (458) already in R9C1 and 16(3) @ R9C7
-> 16(3) @ R8C2 = {178} locked for N7; 8 locked for C2
d) R9C1 = 4 -> R8C1 = 6

16. Rest is singles.

Rating: 1.75. I used some Killer subsets and a 3-way-forcing chain.

By the way, I'll be quite busy with university so expect less participation (I'll probably make no new Assassins) from me over the next few weeks.

Hi,

Thanks for such activity on these puzzles : v1 and v1.5 doesn not seem to have resist a long time to the experts !

I did not find an easier walkthrough than Afmobs to solve V2. I have however written a WT for this puzzle since my solving way was slightly different from Afmob's. I have to use a "chain" , in fact a short contradiction move, (I am not very OK with solving techniques terminology) in step 22, I would be very interested if someone could do this move more directly (I solve puzzles on paper, and I sometime feel more at my ease with a short contradition move than long combinations checking).
Tell me if step 10a. is really correct I have used a unicity argument and I don't feel comfortable with it in a Killer Sudoku puzzle.
I hope my notations and terminology are correct : some step might be shortly done sometimes. Tell me ...

Walktrough A 140 v2

Preliminaries
Cage 7(2) n7 - cells do not use 789
Cage 12(2) n2 - cells do not use 126
Cage 9(2) n69 - cells do not use 9
Cage 10(2) n7 - cells do not use 5
Cage 10(3) n36 - cells do not use 89
Cage 20(3) n3 - cells do not use 12
Cage 19(3) n9 - cells do not use 1
Cage 19(3) n3 - cells do not use 1
Cage 15(5) n58 - cells ={12345}
Cage 23(6) n1 - cells do not use 9



1. 45 Rule on n8: r6c56 = {1 3} / r7c45, r8c4 = {2 4 5}


2. 45 Rule on n369 split cage 19(4) n56 : r5c7 + r5c8 = 5 :
r5c78 = {1 2 3 4}


3 45 Rule on n7 - innies r7c12 total 12 : h cage 12(2) (hcage = hidden cage)

4. 45 Rule on n36 - innies r6c78 total 15
r6c7 = {7 8 9}
r6c8 = { 678}

5. r2c23 + r3c3 = 22 implies r3c3 = {5 6 7 8 9}

6. 45 Rule on n47 : r3c3 + r6c4 = 10 and r6c56 = {1 3} implies r6c4 = { 245} r3c3 ={8 6 5 }
6a. Naked triple {245} found at r678c4

7. innies and outies on n1 : r2c4 +r3c4 total 5, 6 or 8 : totals 6 and 5 are impossible because of 6a. So r3c3 = 8 r23c4 = {1 7} and r6c4 =2: r7c5=2 r1c6=2. r2c23 = {5 9}
r6c7 = {7 9} r6c8 ={6 8}

8. 45Rule on n2 : r3c5 = 5 r5c6 = 5 (single)
r5c46 = 15 so r5c4=9

9. 12-cage at n2 : r2c6 ={3 8} r3c6 = {4 9}

10. Cage 16(3) at n69 : only combinations ( look at r7c8 too) 9{16}, 9{52}, 7{63} and 7{54}.
10a. ! Unique rectangle at r78c4 and r78c7 : combination 745 impossible
10b. 952 impossible since it implies r7c478 = [4 5 3] that blocks h cage 12(2) at n7 (step 3)
Cage 16(3) at n69 : only combinations : 9{16} and 7{63}.
10c. hcage 10(3) in n9 = {1 3 6}.


11. hiden triple 789 in r7c12 (hcage 12(2)) r7c6 r7c9.

12. hcage 19(3) at r5c5 r56c9 and r5c5 <=8 implies r56c9 >= 11 so
r234c9 <= 10 (
No 8 9 at r4c9.
No 8 9 at r4c89 (because of cage 10(3).)

13. 8 locked in r4c345 for r4 and n5 : no 8 at r5c5. 9 locked
in r4c123 for r4 and n4. Only combination for r4c123 : {9 2 3}
or {9 1 4 }

14. combination for r4c123 {9 1 4 } impossible since
it implies r5c5 =4 and r6c9 = 4, which makes impossible r5c5 + r56c9 =19.
r4c123 = {9 2 3}

15. Candidate 1 locked in r1 for n1 nowhere else in n1

16. 2 and 3 locked in r5c789 for r5, so r5c78 = {23} (step 2).
16a. candidate 5 locked in r6c123 for n4, so is nowhere else in r6 r6c9<>5.
16b. Candidate 5 in r1 must be in cage 20(3) n3 : only combinations :
{5 6 9 } {5 8 7}.

17. 3-Innies at n3 : r3c79, r2c9 = {1 2 3}

18. Limited placement of candidates in cage 19(3) n3 : r3c8 <>6, r3c8 = { 4 7 9}
Candidate 6 locked in r3 for n1 nowhere else in n1.

19. . Limited placement of candidates in cage 20(4) n2
Removed 3 from r1c5

20. r7c2 cannot be 5 since r7c1 would be 7 (step 3) which would force cage 22(5) to contain
all of {2 5 7}. (r4c123 does not contain 5 nor 7)
r7c2 cannot be 7 since r7c1 would be 5 (step 3) which would force cage 22(5) to contain
all of {2 5 7}.
r7c12 = {3 4 8 9}

21. r1c4 cannot be 3, since r1c123 would be {1 4 7} , cage 20(3) in n3
would be { 5 6 9} (step 16b.) so r1c5 =8 and {3 8} blocks all combinations of cage 12(2)
in n2. Remove 3 from r1c4.
21a. Hidden single in c4 : r9c4 = 3.
21b. Candidate 3 locked in n2 for r2 nowhere else in r2

!! 22. If r7c12 = {8 4} : then r78c4 =[5 4] so cage 16(3) in n9 must contain 4,
so 16(3) = {4 5 7 } which would force r8c2 to be 7 (hidden single) and would block every combinations of cage 16(3) in n7. So r7c12 = {3 9}.

23 Naked single r7c8 = 1, r6c8 =8 r 6c7 = 7 (step 4) r7c7 = 6 and r8c7 = 3.
23a. r5c7 = 2 r5c8 = 3 r3c7 = 1 r23c9 = [2 3] (naked singles)
23b hidden single : r5c5 = 4 r3c6 = 4 (hidden single) r2c6 = 8 r3c4 =7 r2c4 =1 r3c8 = 9
23c. Cage 19(3) in n3 : r2c7 =4 r2 c6 = 6 r4c7 = 5 r4c8 =4.

24 Hidden singles : r6c1 = 5 .....

Rest is singles (or very basic elimination)

Cheers,

manu

I agree with that. However my V1 walkthrough used a couple of steps that were harder than necessary and I've avoided the harder of those two steps for V1.5.

I'll rate A140 v1.5 at Easy 1.25 because I didn't use step 18a of my A140 walkthrough. I appear to have used the same breakthrough moves (steps 16, 19 and 21 in my walkthrough) that Afmob did.

The quoted SS score of 1.51 looks much too high; A140 v1.5 is only slightly harder than A140.

Here is my walkthrough.

Prelims

a) R23C6 = {39/48/57}, no 1,2,6
b) R67C8 = {18/27/36/45}, no 9
c) R78C3 = {16/25/34}, no 7,8,9
d) R9C23 = {69/78}
e) R1C789 = {389/479/569/578}, no 1,2
f) 19(3) cage in N3 = {289/379/469/478/568}, no 1
g) 10(3) cage at R3C7 = {127/136/145/235}, no 8,9
h) 11(3) cage in N7 = {128/137/146/236/245}, no 9
i) 19(3) cage in N9 = {289/379/469/478/568}, no 1
j) 15(5) cage at R6C5 = {12345}
k) 23(6) cage in N1 = {123458/123467}, no 9, 1,2,3,4 locked for N1

1. 45 rule on N3 3 innies R2C9 + R3C79 = 6 = {123}, locked for N3

2. 45 rule on N8 2 outies R6C56 = 4 = {13}, locked for R6, N5 and 15(5) cage at R6C5, clean-up: no 6,8 in R7C8
2a. Naked triple {245} in R7C45 + R8C4, locked for N8

3. 45 rule on N7 2 innies R7C12 = 12 = {39/48/57}, no 1,2,6

4. 45 rule on N9 2 outies R6C78 = 15 = [78/87/96], clean-up: no 4,5,7 in R7C8

5. 45 rule on N47 2 outies R3C3 + R6C4 = 10 = [55/64/82]
5a. 9 in N1 locked in R2C23, locked for R2 and 22(4) cage at R2C2, no 9 in R3C4, clean-up: no 3 in R3C6
5b. Naked triple {245} in R678C4, locked for C4

6. 45 rule on C789 2 innies R5C78 = 5 = {14/23}, R5C46 = 14 = [68/86/95]

7. 45 rule on N1 2 outies R23C4 = 1 innie R3C3
7a. R23C4 cannot total 5,6 -> R3C3 = 8, R6C4 = 2 (step 4), R23C4 = 8 = {17}, locked for C4, N2 and 22(4) cage at R2C2, clean-up: no 4,5 in R2C6, no 5 in R3C6, no 7 in R9C2
7b. R2C23 = 14 = {59}, locked for R2 and N1
[Alternatively, as in Afmob’s V1 walkthrough, 9 locked in 22(4) cage at R2C2 = {1579} (only remaining combination) -> R2C23 = {59}, R23C4 = {17} …]
7c. R7C5 = 2 (hidden single in N8), R1C6 = 2 (hidden single in C6), clean-up: no 7 in R6C8, no 8 in R6C7 (step 4), no 5 in R8C3
7d. 1 in R1 locked in R1C123, locked for N1

8. 45 rule on N2 1 remaining innie R3C5 = 5
8a. R5C6 = 5 (hidden single in C6), R5C4 = 9 (step 6)

9. 19(3) cage in N3 = {469/478}, 4 locked for N3
9a. 9 of {469} must be in R3C8 -> no 6 in R3C8

10. 45 rule on R9 4 innies R9C1456 = 14 = {1238/1346} (cannot be {1247/1256/2345} because 2,4,5 only in R9C1), no 5,7,9
10a. 2,4 only in R9C1 -> R9C1 = {24}
10b. R9C456 = {136/138}, 1,3 locked for R9 and N8
10c. Killer pair 6,8 in R9C23 and R9C456, locked for R9
10d. R9C789 = {259/457}, 5 locked for N9

11. 45 rule on N9 3 innies R7C78 + R8C7 = 10 = {136} (cannot be {127} because R678C7 cannot be [772]), locked for N9, 6 locked in R78C7, locked for C7
[Alternatively R7C78 + R8C7 = 10 = {127/136}, no 8,9; 8 locked in 19(3) cage in N9 = {289/478}, no 3, R7C789 + R8C7 = {136} (hidden triple for N9 …]
11a. 10(3) cage at R3C7 = {127/145/235} (cannot be {136} because R34C7 = {13} clashes with R78C7), no 6

12. 11(3) cage in N7 = {128/146/245} (cannot be {137} because R9C1 only contains 2,4, cannot be {236} which clashes with R78C3), no 3,7
12a. 4 of {245} must be in R9C1 (R8C12 cannot be {45} which clashes with R8C4), no 4 in R8C12

13. 45 rule on C9 2 innies R19C9 = 1 outie R8C8 + 5, IOU no 5 in R1C9
13a. Min R19C9 = 8 -> no 2 in R8C8
13b. Max R8C8 = 9 -> R9C19 = 14 -> no 9 in R9C9

14. 19(3) cage in N9 = {289/478}
14a. 2 of {289} must be in R8C9 -> no 9 in R8C9

15. R1C789 = {569/578}
15a. Hidden killer pair 8,9 in R1C45 and R1C789 -> R1C45 must contain one of 8,9
15b. 20(4) cage in N3 = {2369/2468}
15c. One of 8,9 must be in R1C45 -> no 8 in R2C5
15d. 4,9 must be in R1C5 (R1C45 cannot be {68} which clashes with R1C789) -> R1C5 = {49}, no 4 in R2C5

16. Hidden killer pair 7,9 in R7C12 and R9C23 for N7 -> R7C12 must contain one of 7,9 = {39/57} (step 3), no 4,8

17. Hidden killer pair 8,9 in R456C9 and R6C78 for N6 -> R456C9 must contain one of 8,9
17a. 45 rule on C9 4 innies R1789C9 = 24 = {2679/4569/4578} (cannot be {2589} which clashes with R456C9
17b. 6 of {2679/4569} must be in R1C9 -> no 9 in R1C9
17c. R1C789 (step 15) = {569/578}
17d. 6 in {569} must be in R1C9 -> no 6 in R1C8

18. 45 rule on R1234 3(2+1) outies R5C59 + R6C9 = 19
18a. Max R5C5 + R6C9 = 17 -> no 1 in R5C9

19. 45 rule on R1234 3 innies R234C9 = 1 outie R5C5 + 2
19a. Max R5C5 = 8 -> max R234C9 = 10, no 8,9 in R4C9

20. 9 in R4 locked in R4C123, locked for N4
20a. 8,9 locked in 22(4) cage at R3C3 = {1489/2389}, no 5,6,7

21. 22(5) cage at R5C3 = {12469/12478/23458/23467} (cannot be {12379} because 1,3,9 only in R5C3 + R7C2, cannot be {12568} which clashes with R6C8), 4 locked in R5C3 + R6C23, locked for N4, clean-up: no 1 in R4C123 (step 20a)

22. Naked triple {239} in R4C123, locked for R4 and N4
22a. 1 in R4 locked in R4C789, locked for N6, clean-up: no 4 in R5C78 (step 6)
22b. Naked pair {23} in R5C78, locked for R5

23. 23(4) cage at R5C1 = {1589/1679/3578} (cannot be {3569} because 3,9 only in R7C1)
23a. 3,9 only in R7C1 -> R7C1 = {39}, clean-up: no 5,7 in R7C2 (step 3)
23b. Naked pair {39} in R7C12, locked for R7 and N7 -> R7C8 = 1, R6C8 = 8, R6C7 = 7 (step 4), R78C7 = [63], R5C78 = [23], R3C7 = 1, R23C4 = [17], clean-up: no 4 in R7C3, no 1,4,6 in R8C3, no 6 in R9C23

24. R78C3 = [52], R78C4 = [45], R9C123 = [487], R2C23 = [59], R4C3 = 3

25. Naked pair {46} in R6C23, locked for R6 and N4 -> R5C3 = 1, R5C12 = [87], R6C1 = 5, R6C9 = 9, R7C2 = 9 (step 21), R7C1 = 3, R4C12 = [92]

26. Naked pair {45} in R4C78, locked for R4 and N6 -> R45C9 = [16], R5C5 = 4, R1C5 = 9, R3C6 = 4, R2C6 = 8

and the rest is naked singles

And now another V2 from my Unfinished folder. They seem to be getting harder! Congratulations manu for producing a really challenging variant of your first Assassin!

That change made a very big difference. It took me hours, spread over several days.

I agree with Afmob's assessment of the difficulty of this variant.

I liked the killer triple in Afmob's step 9a, manu had a similar step. I wish I'd spotted that step which would have made my solving path a bit shorter.

My solving path is somewhat different than those of Afmob and manu but all three depended on interactions between N7 and N9.


Here is my walkthrough for A140 V2.

I found that I hadn’t tried the V2 when this puzzle first appeared. I’ve now started from the beginning, without looking at how I solved the V1 and V1.5, so hope I haven’t missed any easier steps which I saw then.

Prelims

a) R23C6 = {39/48/57}, no 1,2,6
b) R67C8 = {18/27/36/45}, no 9
c) R78C3 = {16/25/34}, no 7,8,9
d) R89C1 = {19/28/37/46}, no 5
e) 20(3) cage in N3 = {389/479/569/578}, no 1,2
f) 19(3) cage in N3 = {289/379/469/478/568}, no 1
g) 10(3) cage at R3C7 = {127/136/145/235}, no 8,9
h) 19(3) cage in N9 = {289/379/469/478/568}, no 1
i) 15(5) cage at R6C5 = {12345}
j) 23(6) cage in N1 = {123458/123467}, no 9, 1,2,3,4 locked for N1

1. 45 rule on N3 3 innies R2C9 + R3C79 = 6 = {123}, locked for N3

2. 45 rule on C123 3 outies R236C4 = 10 = {127/136/145/235}, no 8,9

3. 45 rule on N12 2 innies R3C35 = 13 = {58/67}/[94], no 1,2,3,9 in R3C5

4. 45 rule on N7 2 innies R7C12 = 12 = {39/48/57}, no 1,2,6

5. 45 rule on N8 2 outies R6C56 = 4 = {13}, locked for R6, N5 and 15(5) cage at R6C5, clean-up: no 6,8 in R7C8
5a. Naked triple {245} in R7C45 + R8C4, locked for N8

6. 45 rule on N9 2 outies R6C78 = 15 = [78/87/96], clean-up: R7C8 = {123}

7. 45 rule on C789 2 innies R5C78 = 5 = {14/23}
7a. 45 rule on C789 2 outies R5C46 = 14 = {59/68}

8. 45 rule on N47 2 outies R3C3 + R6C4 = 10 = [55/64/82], clean-up: no 4,6 in R3C5 (step 3)
8a. Naked triple {245} in R678C4, locked for C4, clean-up: no 9 in R5C6 (step 7a)

9. 45 rule on N1 2 outies R23C4 = 1 innie R3C3 -> R3C3 = 8 (because R23C4 cannot total 5 or 6), R3C5 = 5 (step 3), R6C4 = 2 (step 8), clean-up: no 4,7 in R2C6, no 7 in R3C6
9a. R3C3 = 8 -> R23C4 = {17}, locked for C4, N2 and 22(4) cage at R2C2, no 7 in R2C23
9b. 1 in R1 only in R1C123, locked for N1

10. R7C5 = 2 (hidden single in N8), clean-up: no 7 in R6C8, no 8 in R6C7 (step 6), no 5 in R8C3

11. R5C6 = 5 (hidden single in N5), R5C4 = 9 (step 7a)

12. 45 rule on N1 2 remaining innies R2C23 = 14 = {59}, locked for R2 and N1, clean-up: no 3 in R3C6

13. 19(3) cage in N3 = {469/478}, 4 locked for N3
13a. 9 of {469} must be in R3C8 -> no 6 in R3C8
13b. 6 in R3 only in R3C12, locked for N1

14. R1C6 = 2 (hidden single in C6)
14a. 20(4) cage in R1C4 = {2369/2468}
14b. 9 of {2369} must be in R1C5 -> no 3 in R1C5
14c. 20(3) cage in N3 = {569/578}
14d. Hidden killer pair 8,9 in R1C45 and 20(3) cage at R1C7 for R1, 20(3) cage contains one of 8,9 -> R1C45 must contain one of 8,9
14e. 8,9 of 20(4) cage in R1C4 must be in R1C45 -> no 8 in R2C5

15. 45 rule on N9 3 innies R7C78 + R8C7 = 10 = {127/136/145/235}, no 8,9

16. 16(3) cage at R6C7 cannot contain both of 7,9, R6C7 = {79} -> no 7 in R78C7

17. 9 in R3 only in R3C68
17a. R3C6 = 9 => R2C6 = 3
or R3C8 = 9 => R2C78 = {46} (step 13) => R2C5 = 3
-> 3 must be in R2C56, locked for R2 and N2
17b. 3 in N3 only in R3C79, locked for R3

18. Killer pair 6,8 in R1C4 and 20(3) cage at R1C7, locked for R1
18a. Naked pair {49} in R1C5 + R2C6, locked for N2

19. R9C4 = 3 (hidden single in C4), clean-up: no 7 in R8C1

20. Hidden killer triple 7,8,9 in R7C12, R7C6 and R7C9 for R7, R7C12 contains one of 7,8,9 -> R7C6 = {789}, R7C9 = {789}

21. 45 rule on C6 2 remaining outies R89C5 = 2 innies R46C6 + 5
21a. 1 in N5 only in R6C56, 1 in N8 only in R89C56
21b. R46C6 = [41] = 5 => R89C5 = 10 = {19}, R46C6 cannot be [61/71/81] because R89C5 containing 1 cannot total more than 10
R46C6 cannot be [43/83] which clash with R23C6
-> R46C6 = [41/63/73], no 8 in R4C6

22. R6C4 = 2 -> R5C3 + R6C23 + R7C2 = 20 = {1469/1478/1568/3458/3467} (cannot be {1379} which clashes with R6C7) contains 1 in R5C3 or 3 in R5C3 + R7C2
22a. R5C3 = 1 => R5C78 = {23} (step 7)
22b. -> 3 in R5C3 + R7C2 or R5C78, CPE no 3 in R5C2

23. Max R3C7 = 3 -> min R4C78 = 7
23a. R3C3 = 8 -> R4C123 = 14
23b. 45 rule on R4 3 remaining innies R4C789 = 1 outie R5C5 + 6
23c. Max R5C5 = 8 -> max R4C789 = 14 -> max R4C9 = 7
23d. 8 in R4 only in R4C45, locked for N5

24. 9 in R4 only in R4C123, locked for N4
24a. R4C123 = 14 (step 23a) = {149/239}, no 5,6,7

25. 5 in R4 only in R4C789, locked for N6
25a. R4C789 = R5C5 + 6 (step 23b)
25b. R5C5 = {467} -> R4C789 = 10,12,13 = {145/235/156/157} (cannot be {345} which clashes with R4C123, cannot be {256} because 7 must be in R4C789 = 13 when R5C5 = 7)
25c. 1,2 of {145/235} must be in R4C9 (because min R4C78 = 7, step 23) -> no 3,4 in R4C9
25d. 5 of {156/157} must be in R4C9 (because min R4C78 = 7, max R4C78 = 9) -> no 6,7 in R4C9

26. Hidden killer pair 8,9 in R56C9 and R6C78 for N6, R6C78 contains one of 8,9 -> R56C9 must contain one of 8,9
26a. 21(5) cage at R2C9 = {12369/12378/12459/13458} (cannot be {12468} which clashes with R6C8, cannot be {12567/13467/23457} which don’t contain 8 or 9), 1 locked for C9
26b. R234C9 contain three of 1,2,3,5 -> no 1,2,3 in R5C9
26c. 9 of {12369} must be in R6C9 -> no 6 in R6C9

27. 5 in R6 only in R6C123
27a. R5C3 + R6C23 + R7C2 (step 22) = {1469/1478/1568/3458/3467}
27b. 7 of {1478/3467} cannot be in R7C2 (because R7C12 = [57] (step 4) clashes with R6C1 = 5, hidden single in R6) -> no 7 in R7C2, clean-up: no 5 in R7C1 (step 4)

28. R6C1 = 5 (hidden single in C1)
28a. R5C3 + R6C23 + R7C2 (step 22) = {1469/1478/3458/3467} (cannot be {1568} which clashes with R6C8), CPE no 4 in R5C2
28b. 1 of {1469/1478} must be in R5C3, 3 of {3458} must be in R5C3, 6 or 7 of {3467} must be in R5C3 (R6C23 cannot be {67} which clashes with R6C78) -> no 4 in R5C3

29. 23(4) cage at R5C1 contains 5 = {1589/2579/3569/3578/4568}
29a. 6 of {3569} must be in R5C2, {4568} = [4658/8654] (cannot be [6854] because R5C2 + R7C1 = 12 clashes with R7C12 = 12 (step 4), CCC) -> no 6 in R5C1

30. 23(4) cage at R5C1 cannot be {2579}, here’s how
{2579} => R7C1 = 9 => R7C2 = 3 (step 4) => {2579} clashes with R5C3 + R6C23 + R7C2 (step 22) = {3467} (only combination with 3 in R7C2)
30a. 23(4) cage at R5C1 (step 29) = {1589/3569/3578/4568}, no 2

31. 2 in R5 only in R5C78 = {23} (step 7), locked for R5 and N6
31a. R4C123 = {239} (hidden triple in N4)

32. 23(4) cage at R5C1 (step 30a) = {1589/3578/4568} (cannot be {3569} because 3,9 only in R7C1)
32a. 3 of {3578} must be in R7C1 -> no 7 in R7C1, clean-up: no 5 in R7C2 (step 4)

33. R7C12 (step 4) = 12 = {39/48}
33a. R78C3 = [16/52/61] (cannot be {34} which clashes with R7C12), no 3,4

34. 16(3) cage at R9C7 = {169/178/259/268/457}
34a. 16(3) cage in N7 = {169/178/259/367/457} (cannot be {268} which clashes with R78C3, cannot be {349/358} which clash with R7C12)
34b. 16(3) cage in N7 cannot be {367}, here’s how
{367} = 3{67} => 16(3) cage at R9C7 = {259} => R9C1 = 4 (hidden single in R9) => R8C1 = 6 clashes with {367}
34c. -> 16(3) cage in N7 = {169/178/259/457}, no 3
34d. {169/259} => no 9 in R89C1
{178/457} => R7C12 (step 4) = {39} => no 9 in R89C1
34e. -> no 9 in R89C1, clean-up no 1 in R89C1
[Steps 34d and 34e can be expressed as a combined cage R7C12 + 16(3) cage, if preferred. Alternatively the same result can be obtained from combining the 7(2) and 16(3) cages; in that case the combined cage must contain 1.]

35. 19(3) cage in N9 = {289/379/469/478} (cannot be {568} which clashes with R7C78 + R8C7), no 5
35a. 2 of {289} must be in R8C9 (R78C9 cannot be {89} which clashes with 21(5) cage at R2C9), no 2 in R8C8

[It looks like R7C78 + R8C7 (step 15) cannot be {145} because of UR for R78C47, although I know one has to be careful about URs for killers. However I don’t use URs because they rely on there being a unique solution and seem to bypass part of the solving path. Then I found a contradiction move.]

36. 16(3) cage at R9C7 cannot be {268} here’s how
{268} => 5 in R9 must be in R9C23 => 16(3) cage in N7 (step 34a) = {259} (cannot be {457} because no remaining candidates in R9C1) = 2{59} => R78C3 = {16} => cannot place 4 in R9
36a. -> 16(3) cage at R9C7 (step 34) = {169/178/259/457}

37. 19(3) cage in N9 = {289/469/478} (cannot be {379} which clashes with 16(3) cage), no 3
37a. 2 of {289} must be in R8C9, 9 of {469} must be in R8C9 -> no 9 in R8C9

38. 3 in N9 must be in R7C78 + R8C7 -> R7C78 + R8C7 (step 15) = {136/235}, no 4

39. 16(3) cage at R6C7 = {169/259/367}
39a. 2 of {259} must be in R8C7 -> no 5 in R8C7

40. R7C12 (step 33) = {39/48}
40a. R7C12 = {39} => R7C8 = 1 => R7C78 + R8C7 (step 38) = {136}
R7C12 = {48} => R7C4 = 5
-> no 5 in R7C7

41. R7C78 + R8C7 (step 38) = {136} (only remaining combination), locked for N9, 6 also locked for C7
41a. Killer triple 1,2,3 in R3C7, R5C7 and R78C7, locked for C7

42. 5 in N9 only in 16(3) cage at R9C7, locked for R9
42a. 16(3) cage at R9C7 (step 36a) = {259/457}, no 8
42b. 8 in C7 only in R12C7, locked for N3

43. R3C9 = 3 (hidden single in C9)
43a. 10(3) cage at R3C7 = {127/145}, no 6
43b. 6 in R4 only in R4C456, locked for N5

44. 19(3) cage in N9 (step 37) = {289/478}
44a. 21(5) cage at R2C9 (step 26a) = {12369/13458} (cannot be {12378} which clashes with 19(3) cage, ALS block for one permutation), no 7

45. 19(3) cage in N9 (step 44) = {289/478}
45a. 9 of {289} must be in R7C9 (R78C9 cannot be [82] which clashes with 21(5) cage at R2C9) -> no 9 in R8C8
45b. Killer pair 2,9 in 21(5) cage at R2C9 and 19(3) cage in N9 (2,9 must be in one or other cage, not the usual killer pair; maybe this is really a combined cage but I saw it as a killer pair), locked for C9

46. 10(3) cage at R3C7 (step 43a) = {127/145}
46a. 7 of {127} must be in R4C7 -> no 7 in R4C8
46b. 7 in N6 only in R46C7, locked for C7

47. 16(3) cage at R9C7 (step 42a) = {259/457}
47a. 2 of {259} must be in R9C8 -> no 9 in R9C8
47b. 9 in C8 only in R13C8, locked for N3

48. 20(3) cage in N3 (step 14c) = {569/578}
48a. 9 of {569} must be in R1C8 -> no 6 in R1C8

49. 16(3) cage in N7 (step 34c) = {169/178/259/457}
49a. Cannot be {259}, here’s how
{259} = 5{29} => 16(3) cage at R9C7 = {457}, R78C3 = {16} => R9C1 = 8, R8C1 = 2 clashes with 16(3) cage
49b. -> 16(3) cage in N7 = {169/178/457}, no 2
49c. 5 of {457} must be in R8C2 -> no 4 in R8C2

50. 21(5) cage at R2C9 (step 44a) = {12369/13458}
50a. 19(3) cage in N9 (step 37) = {289/478}
50b. 19(3) cage = {289} => 21(5) cage = {13458}, 4 locked for C9
or 19(3) cage = {478}, 4 locked for N9
-> no 4 in R9C9
50c. Hidden killer pair 2,4 in 21(5) cage at R2C9 and R8C9 for C9, 21(5) cage contains one of 2,4 -> R8C9 = {24}
50d. 19(3) cage in N9 = {289/478}
50e. R8C9 = {24} -> no 4 in R8C8

51. Hidden killer pair 8,9 in 21(5) cage at R2C9 and R7C9 for C9, 21(5) cage contains one of 8,9 -> R7C9 = {89}

52. R7C6 = 7 (hidden single in R7)

53. R7C12 (step 33) = {39/48}
R7C12 = {39} => R7C78 = [61] => R7C3 = 5 => 16(3) cage in N7 (step 49b) = {178}
or R7C12 = {48}
-> 8 must be in R7C12 or 16(3) cage, locked for N7, clean-up: no 2 in R89C1

54. R8C3 = 2 (hidden single in N7), R7C3 = 5, R78C4 = [45], R8C9 = 4, clean-up: no 8 in R7C12 (step 33), other clean-ups omitted

55. Naked pair {39} in R7C12, locked for R7 and N7 -> R7C78 = [61], R8C7 = 3, R7C9 = 8, R8C8 = 7, R56C9 = [69], R6C78 = [78], R8C1 = 6, R9C1 = 4, R9C789 = [925], R4C9 = 1, R4C78 = {45}, locked for R4 -> R4C6 = 6

56. R2C23 = [59], R2C9 = 2, R23C1 = [72], R4C3 = 3, R4C1 = 9, R7C1 = 3, R1C1 = 1, R5C1 = 8, R5C2 = 7 (cage sum from step 30a)

and the rest is naked singles.