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h3lix
 Post subject: Anti-Bishop Killer
PostPosted: Sat May 22, 2010 4:06 pm 
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Anti-Bishop is a series of several diagonals on a grid that I named to be consistent with my book, the diagonals are lightly drawn on all versions of the puzzle.

This one comes in 3 versions, but given the standards of this site, I'll only post the harder two.

MEDIUM version:
Image
3x3::k:3328:3328:3328:3843:2564:4101:3078:3078:3078:1033:2826:2826:3843:2564:4101:2575:2575:3857:1033:68:69:3843:72:4101:73:75:3857:76:77:79:80:81:82:83:84:85:86:87:88:89:90:91:92:93:94:95:96:97:98:99:100:101:102:103:3647:104:105:3897:106:4667:107:108:2119:3647:1089:1089:3897:1347:4667:2886:2886:2119:5194:5194:5194:3897:1347:4667:4174:4174:4174:
JSudoku file


This INSANE version is definitely the hardest of the Anti-Chess puzzles I've posted so far
INSANE version (cupid's arrow):
Image
3x3::k:3329:3329:3329:3842:0000:4100:3077:3077:3077:1030:2823:2823:3842:0000:4100:2568:2568:3849:1030:0000:0000:3842:0000:4100:0000:0000:3849:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:0000:3594:0000:0000:0000:0000:0000:0000:0000:2061:3594:1038:1038:0000:0000:0000:2827:2827:2061:5132:5132:5132:0000:0000:0000:4099:4099:4099:
JSudoku file


I have made another visual walkthrough for the INSANE version, which I will post on monday.

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Børge
PostPosted: Sun May 23, 2010 12:08 pm 
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Anti-Bishop MEDIUM, images with udosuk Style Killer Cages: 
Image     Image
Anti-Bishop INSANE, images with udosuk Style Killer Cages: 
Image     Image

Daniel, some small info so you do not get a wrong impression of my program.
I added the 8 extra diagonals in Paint. I first made the medium version and then in paint removed the 4 cages to get the insane version.

Depending on the cage structure, the two main diagonals may not be absolutely diagonal in some Excel cells, which in my opinion is an Excel bug.
Since you only see this when the image is blown up, I normally do not care, but sometimes I correct it in paint.
Below the unretouched screen shot of what Excel produces.
Hidden Text:
Image

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h3lix
 Post subject: Re: Anti-Bishop Killer
PostPosted: Sun May 23, 2010 1:15 pm 
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thank you for the images! I made my diagonals on a photoshop layer which I typically put below the cages and grid. I imagine that adapting this to these images and using the magic wand select to delete where they overlay cage information might be easier than using paint

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HATMAN
 Post subject: Re: Anti-Bishop Killer
PostPosted: Sun May 23, 2010 3:14 pm 
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You are getting the medium level ones just right for me to do on paper - another enjoyable one.
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h3lix
 Post subject: Re: Anti-Bishop Killer
PostPosted: Mon May 24, 2010 10:17 pm 
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As promised (though a little later than I expected since work ran long), here is my visual walkthrough for the INSANE version

Rather than try to convert it to conventional solving methods, I just showed my paper solving method which relies on "possibility chains"

Walkthrough for Cupid's Arrow:
Note: D/ and D\ refer to the main diagonals

Image
1.
4/2 cages = {13}
Innie in N1: R3C23 = 17 = {89}
=> Cage 13/3 in N1 = {247|256}, must contain 2
=> Cage 11/2 in N1 = {56|47}

Innie on R1: R1C456 = 20, cannot contain a 1
=> 1 in R1 locked in cage 12/3 = {138|147|156}

Innie in N3: R3C78 = 8 = {26|35}

Image
2.
Cage 12/3 in N3 must contain 1 and Cage 11/2 in R2 must contain either 4 or 6
=> Cage 10/2 in R2 = {28|37}, must contain either 7 or 8
=> Cage 15/2 in N3 = [96]
=> R3C78 = {35}
=> R23C1 = [31] & Cage 10/2 in N3 = {28}
=> Cage 12/3 in N3 = {147}
=> Cage 13/3 in R1 = {256}
=> R1C456 = {389} & Cage 11/2 in N1 = {47}

Image
3.
R2C456 = {156}, R3C456 = {247}
No combinations for 6/2 in R12C4
=> R1C4 != 9
No combinations for 7/2 in R12C6
=> R1C6 != 9
=> R1C5 = 9

Innie in N2: R23C5 = 5 = [14]
Odd combinations: R2C46 = [56]

Image
4.
Cage 14/2 and Cage 20/3 in N7 have sum of 34/5 = {46789}
=> Cage 14/2 = [86], Cage 20/3 = {479}, remaining cells in N7 = {25}

Innie in R9: R9C456 = 9 = {126|135}
=> Cage 16/3 in R9 = {268|358}
Cage 11/2 in N9 = {29|47}
Cage 8/3 in N9 = [17]|[35], must contain 3 or 7
Innie in N9: R7C78 = {19|46}

Linked Possibilities in N9: either
8/2 = [35], 16/3 = {268}, 11/2 = {47}, R7C78 = {19}
or
8/2 = [17], 11/2 = {29}, 16/3 = {358}, R7C78 = {46}

Image
5.
6 of D/ must be in either R5C5 or R6C4
6 of D\ must be in either R5C5 or R7C7
=> R9C5 != 6

In the red possibility chain, R5C5 must be 6 (naked single on D\). In the blue possibility chain, R9C4 = 6, so R5C5 must be 6.

In both cases R5C5 = 6.

(simpler version: R7C7 and R6C4 both being 6 would remove all possibilities from N8 => R5C5 = 6)

Image
6.
The blue possibility chain creates a sequence of naked singles for 1 on D\, R1, D/ and N6, removing all possibility in N4.

=> the blue possibility chain is invalid.

Image
7.
All singles from here.

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Andrew
 Post subject: Re: Anti-Bishop Killer
PostPosted: Tue May 25, 2010 8:23 pm 
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Interesting that an Anti-Bishop Killer only imposes restraints on diagonals between odd-numbered edge cells. That's like chess with only one Bishop. ;)

As with your other puzzles, I started with the easier one first because I hadn't done an Anti-Bishop Killer before.

On my first try I wasn't able to finish it; then I realised:
that I hadn't been looking for CPEs using the diagonals so I started again using those steps. My impression from my first try was that only one of those CPEs is actually necesary to reach a unique solution but applying them as they arise leads to a quicker solving path.

Rating Comment:
I originally rated this puzzle as HARD because the need for a CPE using a diagonal made it feel the same difficulty as your Anti-King Killer in the Killer forum. However having now tried some of the later puzzles I'll amend that to MEDIUM/HARD

Here is my walkthrough for Anti-Bishop Killer Medium:
No repeated candidates on the diagonals R1C1-R9C9, R1C3-R7C9, R1C5-R5C9, R3C1-R9C7, R5C1-R9C5, R1C5-R5C1, R1C7-R7C1, R1C9-R9C1, R3C9-R9C3 and R5C9-R9C5.

CPEs have been omitted when the eliminations are achieved within the same step.

Prelims (not including any cage interactions)

a) R12C5 = {19/28/37/46}, no 5
b) R23C1 = {13}
c) R2C23 = {29/38/47/56}, no 1
d) R2C78 = {19/28/37/46}, no 5
e) R23C9 = {69/78}
f) R78C1 = {59/68}
g) R78C9 = {17/26/35}, no 4,8,9
h) R8C23 = {13}
i) R89C5 = {14/23}
j) R8C78 = {29/38/47/56}, no 1
k) 20(3) cage in N7 = {389/479/569/578}, no 1,2

Steps resulting from Prelims
1a. Naked pair {13} in R23C1, locked for C1 and N1, clean-up: no 8 in R2C23
1b. Naked pair {13} in R8C23, locked for R8 and N7, clean-up: no 5,7 in R7C9, no 8 in R8C78, no 2,4 in R9C5

2. 45 rule on N2 1 innie R3C5 = 4, placed for R1C3-R7C9 and R1C7-R7C1 diagonals, clean-up: no 6 in R12C5

3. R8C5 = 2, R9C5 = 3, placed for R5C1-R9C5 and R5C9-R9C5 diagonals, clean-up: no 7,8 in R12C5, no 6 in R7C9, no 9 in R8C78

4. 45 rule on N8 1 innie R7C5 = 7, placed for R3C1-R9C7 and R3C9-R9C3 diagonals, clean-up: no 8 in R2C9

5. Naked pair {19} in R12C5, locked for C5 and N2
5a. Naked triple {568} in R456C5, locked for N5

6. 45 rule on N1 2 innies R3C23 = 17 = {89}, locked for R3 and N1 -> R3C9 = 6, placed for R3C9-R9C3 diagonal, R2C9 = 9, R2C5 = 1, R1C5 = 9, placed for R1C5-R5C1 and R1C5-R5C9 diagonals, R2C1 = 3, R3C1 = 1, placed for R3C1-R9C7 diagonal, R3C3 = 8, placed for R1C1-R9C9 and R1C5-R5C1 diagonals, R3C2 = 9, clean-up: no 2 in R2C23, no 4,7 in R2C78, no 2 in R7C9

7. Naked pair {28} in R2C78, locked for R2 and N3

8. 45 rule on N3 2 innies R3C78 = 8 = {35}, locked for R3 and N3, CPE no 3,5 in R4C8 using R1C5-R5C9 diagonal
8a. Naked pair {27} in R3C46, locked for N2
8b. Naked pair {56} in R2C46, locked for R2 and N2
8c. Naked pair {47} in R2C23, locked for N1

9. 45 rule on N7 2 innies R7C23 = 7 = {25}, locked for R7 and N7, CPE no 2,5 in R6C2 using R5C1-R9C5 diagonal, clean-up: no 9 in R78C1
9a. Naked pair {68} in R78C1, locked for C1 and N7
9b. Naked triple {479} in R9C123, locked for R9

10. 45 rule on R9 2 innies R9C46 = 6 = {15}, locked for R9 and N8 -> R9C9 = 2, placed for R1C1-R9C9 diagonal, R1C1 = 5, placed for R1C1-R9C9 diagonal, R5C5 = 6, placed for R1C1-R9C9 diagonal, clean-up: no 5,6 in R8C7
10a. Naked pair {47} in R8C78, locked for R8 and N9 -> R8C9 = 5, R7C9 = 3, placed for R1C3-R7C9 diagonal
10b. Naked pair {68} in R9C78, locked for N9
10c. Naked pair {19} in R7C78, locked for R7, CPE no 1,9 in R6C8 using R5C9-R9C5 diagonal

11. Naked pair {47} in R2C2 + R8C8, locked for R1C1-R9C9 diagonal
11a. 3 on R1C1-R9C9 diagonal only in R4C4 + R6C6, locked for N5
11b. Naked pair {47} in R29C2, locked for C2
11c. 2 in C1 only in R456C1, locked for N4

12. 15(3) cage at R7C4 = {168} (only remaining combination, cannot be {159} because 1,5 only in R9C4, cannot be {456} which clashes with R2C4) -> R9C4 = 1, R7C4 = 6, R8C4 = 8, placed for R3C9-R9C3 diagonal, R8C6 = 9, placed for R3C1-R9C7 and R5C9-R9C5 diagonals, R79C6 = [45]
12a. R7C7 = 1, placed for R1C1-R9C9 and R5C9-R9C5 diagonals, R7C8 = 9

13. R1C4 = 3, R2C4 = 5, placed for R1C3-R7C9 and R1C5-R5C1 diagonals, R3C4 = 7 (cage sum), R4C4 = 9, R6C6 = 3

14. R1C6 = 8, R2C6 = 6, placed for R1C7-R7C1 diagonal, R3C6 = 2

15. R1C7 = 7, R8C7 = 4, R8C8 = 7, placed for R1C1-R9C9 diagonal, R2C2 = 4, R2C3 = 7, R9C2 = 7

16. R2C8 = 8, R4C6 = 7, R9C1 = 9 (hidden singles on R1C9-R9C1 diagonal)

17. R7C1 = 8, placed for R1C7-R7C1 diagonal, R8C1 = 6

18. R6C2 = 1, R8C2 = 3, placed for R1C9-R9C1 diagonal, R3C7 = 5, placed for R1C9-R9C1 diagonal, R7C3 = 2, placed for R1C9-R9C1 and R5C1-R9C5 diagonals, R6C4 = 4, placed for R1C9-R9C1 diagonal

and the rest is naked singles without using diagonals.

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


Last edited by Andrew on Fri Jun 25, 2010 10:39 pm, edited 1 time in total.

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h3lix
 Post subject: Re: Anti-Bishop Killer
PostPosted: Wed May 26, 2010 1:19 am 
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Quote:
Interesting that an Anti-Bishop Killer only imposes restraints on diagonals between odd-numbered edge cells. That's like chess with only one Bishop. ;)

I tried doing every diagonal, but there is no solution, even if all nonets are removed! It would make more sense if I only used odd numbered diagonals and just made it one bishop. :)

Now that some time has passed, I might give the medium version another pass and see if my impression of the rating changes. Having done it right after the insane might have skewed my impression of it's difficulty.

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Andrew
 Post subject: Re: Anti-Bishop Killer
PostPosted: Sat Jun 26, 2010 4:44 am 
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Dan wrote:
This INSANE version is definitely the hardest of the Anti-Chess puzzles I've posted so far
I haven't tried the Vanilla Anti-Knight but it's definitely the hardest of your killers and I'm including Anti-Bishop Killer 2, which has now been posted.

I got as step 19, at which stage I realised that I would probably need chains to make further progress. I then took a break to try other puzzles and only came back to this puzzle after I had finished both versions of Anti-Bishop Killer 2.

Rating Comment:
Definitely INSANE! I used a lot of forcing chains plus a couple of short contradiction moves. It may not appear so from the walkthrough but I took a long time to find several of the forcing chains; that's one reason why I consider this puzzle to be harder than Anti-Bishop Killer 2 INSANE.

Here is my walkthrough for Anti-Bishop Killer INSANE:
No repeated candidates on the diagonals R1C1-R9C9, R1C3-R7C9, R1C5-R5C9, R3C1-R9C7, R5C1-R9C5, R1C5-R5C1, R1C7-R7C1, R1C9-R9C1, R3C9-R9C3 and R5C9-R9C5.

Prelims (not including any cage interactions)

a) R23C1 = {13}
b) R2C23 = {29/38/47/56}, no 1
c) R2C78 = {19/28/37/46}, no 5
d) R23C9 = {69/78}
e) R78C1 = {59/68}
f) R78C9 = {17/26/35}, no 4,8,9
g) R8C23 = {13}
h) R8C78 = {29/38/47/56}, no 1
i) 20(3) cage in N7 = {389/479/569/578}, no 1,2

Steps resulting from Prelims
1a. Naked pair {13} in R23C1, locked for C1 and N1, clean-up: no 8 in R2C23
1b. Naked pair {13} in R8C23, locked for R8 and N7, clean-up: no 5,7 in R7C9, no 8 in R8C78

2. 45 rule on N1 2 innies R3C23 = 17 = {89}, locked for R3 and N1, CPE no 8,9 in R4C2 using R1C5-R5C1 diagonal, clean-up: no 2 in R2C23, no 6,7 in R2C9
2a. R2C78 = {19/28/37} (cannot be {46} which clashes with R2C23), no 4,6
2b. 2 in N1 only in R1C123, locked for R1

3. 45 rule on N7 2 innies R7C23 = 7 = {25}, locked for R7 and N7, CPE no 2,5 in R6C2 using R5C1-R9C5 diagonal, clean-up: no 9 in R78C1, no 6 in R8C9

4. Naked pair {68} in R78C1, locked for C1 and N7
4a. Naked triple {479} in R9C123, locked for R9

5. 45 rule on R9 3 innies R9C456 = 9 = {126/135}, no 8, 1 locked for R9 and N8
5a. 8 in R9 only in R9C789, locked for N9

6. R1C123 = {247/256}
6a. 45 rule on R1 3 innies R1C456 = 20 = {389/479/569} (cannot be {578} which clashes with R1C123), no 1, 9 locked for R1 and N2
6b. 1 in R1 only in R1C789, locked for N3, clean-up: no 9 in R2C78

7. R2C9 = 9 (hidden single in R2), R3C9 = 6, placed for R3C9-R9C3 diagonal, clean-up: no 2 in R8C9

8. 45 rule on N3 2 innies R3C78 = 8 = {35}, locked for R3 and N3, R3C1 = 1, placed for R3C1-R9C7 diagonal, R2C1 = 3, clean-up: no 7 in R2C78
8a. Naked pair {35}, CPE no 3,5 in R4C8 using R1C5-R5C9 diagonal

9. Naked pair {28} in R2C78, locked for R2 and N3
9a. Naked triple {147} in R1C789, locked for R1
9b. Naked triple {256} in R1C123, locked for R1 and N1
9c. Naked pair {47} in R2C23, locked for R2

10. R123C4 = {258/357} (cannot be {159/168} because 1,5,6 only in R2C4, cannot be {249/267} because 2,4,7 only in R3C4, cannot be {348} because 3,8 only in R1C4, cannot be {456} because 5,6 only in R2C4) -> R2C4 = 5, placed for R1C3-R7C9 and R1C5-R5C1 diagonals, R1C4 = {38}, R3C4 = {27}

11. R123C6 = {178/268/367} (cannot be {169}because 1,6 only in R2C6, cannot be {349} because 3,9 only in R1C6), no 4,9

12. R1C5 = 9 (hidden single in R1), placed for R1C5-R5C1 and R1C5-R5C9 diagonals, R3C3 = 8, placed for R1C1-R9C9 diagonal, R3C2 = 9
12a. R3C5 = 4 (hidden single in R3), placed for R1C3-R7C9 and R1C7-R7C1 diagonals
12b. Naked pair {47} in R29C2, locked for C2

13. 45 rule on N2 1 remaining innie R2C5 = 1, R2C6 = 6, placed for R1C7-R7C1 diagonal, R7C1 = 8, placed for R1C7-R7C1 diagonal, R8C1 = 6, clean-up: no 5 in R8C78
13a. Naked pair {13} in R68C2, locked for C2, CPE no 3 in R6C4 using R1C9-R9C1 diagonal
13b. R5C2 = 8 (hidden single in C2)

14. R9C789 = {268/358}
14a. 2 of {268} must be in R9C9 -> no 2 in R9C78
14b. 8 in C9 only in R46C9, locked for N6

15. 45 rule on N9 2 innies R7C78 = 10 = {19/46} (cannot be {37} which clashes with R7C5), no 3,7
15a. 7 in R7 only in R7C456, locked for N8

16. 6 on R1C9-R9C1 diagonal only in R5C5 + R6C4, locked for N5
16a. 6 in C5 only in R59C5, CPE no 6 in R7C7 using R1C1-R9C9 and R5C9-R9C5 diagonals, clean-up: no 4 in R7C8 (step 15)

17. R5C5 = 6 (hidden single on R1C1-R9C9 diagonal)
17a. R9C456 (step 5) = {126/135}
17b. 2 of {126} must be in R9C5 -> no 2 in R9C46
17c. 2 in R9 only in R9C59, CPE no 2 in R5C9 using R5C9-R9C5 diagonal

18. Hidden killer pair 4,6 in R7C46 and R7C78 for R7, R7C78 must contain both or neither of 4,6 -> R7C46 must contain both or neither of 4,6
18a. 6 in R7C46 only in R7C4 -> no 4 in R7C4
18b. 4 in R7 only in R7C67, CPE no 4 in R6C6 using R1C1-R9C9 diagonal, no 4 in R8C6 using R5C9-R9C5 diagonal

19. 1 in R7 only in R7C789, CPE no 1 in R6C8 using R1C3-R7C9 and R5C9-R9C5 diagonals
[This has been there since step 5 but I’ve only just spotted it.]

20. Consider placements for R1C1
R1C1 = 2 => R1C3 = 6
R1C1 = 5 => R7C2 = 5 (hidden single in C2) => R7C3 = 2 => R1C3 = 6
-> R1C3 = 6, placed for R1C3-R7C9 diagonal

21. R4C2 = 6 (hidden single in C2), R6C7 = 6 (hidden single in R6)

22. R1C12 = {25}, R17C2 = {25}, R7C23 = {25} -> R1C1 + R7C3 = {25}, CPE no 2 in R5C1 using R5C1-R9C5 diagonal

23. Consider placements for R2C2
R2C2 = 4, placed for R1C1-R9C9 diagonal => R8C7 = 4 (hidden single in N9) => R8C8 = 7
or R2C2 = 7
-> R2C2 = 7 or R8C8 = 7, locked for R1C1-R9C9 diagonal

24. 2,4,8 in C9 only in R14569C9
24a. 45 rule on C9 5 innies R14569C9 = 22 = {12478/23458}
24b. Consider placements for R9C9
R9C9 = 2 => R8C78 = {47} => R78C9 = [35] => no 3,5 in R46C9
R9C9 = {35} => R14569C9 = {23458} => 2,8 only in R46C9 => no 3,5 in R46C9
-> no 3,5 in R46C9

25. Consider placements for R3C7
R3C7 = 3 => 3 in N6 only in R56C8 (no 3 in R5C9 using R1C5-R5C9 diagonal)
R3C7 = 5 => 5 in N6 only in R56C8 (no 5 in R5C9 using R1C5-R5C9 diagonal)
25a. Killer pair 3,5 in R3C8 and R56C8, locked for C8
[Alternatively R3C7 “sees” all of N6 except for R56C8 (R3C7 “sees” R5C9 using R1C5-R5C9 diagonal) -> R3C7 is “cloned” in R56C8, killer pair 3,5 in R3C8 and R56C8, locked for C8.]

26. 3,5 in C9 only in R59C9 or R78C9
R59C9 = {35} => R9C789 = {358} => R9C7 = {35} => naked pair {35} in R39C7, locked for C7 => no 3 in R5C7
R78C9 = [35], 3 placed for R1C3-R7C9 diagonal => no 3 in R5C7
-> no 3 in R5C7

27. 3,5 in C9 only in R59C9 or R78C9
27a. R9C9 cannot be 5, here’s how
R9C9 = 5 => R9C789 = [385] => R5C9 = 3, placed for R1C5-R5C9 diagonal => R3C7 = 5, R2C8 = 2, R2C8 + R3C7 = [25] clashes with R7C3 using R1C9-R9C1 diagonal
-> no 5 in R9C9, no 3 in R5C9
27b. 3 in C9 only in R79C9, locked for N9
27c. 3 in C7 only in R34C7, CPE no 3 in R4C6 using R1C9-R9C1 diagonal
27d. 5 in N9 only in R8C9 + R9C7, CPE no 5 in R8C6 using R3C1-R9C7 diagonal
27e. 5 on R1C1-R9C9 diagonal only in R1C1 + R6C6, CPE no 5 in R6C1

28. Consider placements for R9C9
R9C9 = 2 or
R9C9 = 3 => R9C789 = [583] => R2C8 = 2, placed for R1C9-R9C1 diagonal => R7C3 = 5 => R7C2 = 2 => R1C2 = 5 => R1C1 = 2
-> R1C1 = 2 or R9C9 = 2, locked for R1C1-R9C9 diagonal, clean-up: no 9 in R8C7

29. Consider placements for R9C9
R9C9 = 2, placed for R1C1-R9C9 diagonal => R1C1 = 5 => R1C2 = 2 => R7C2 = 5 => R7C3 = 2 => R3C7 = 5 (hidden single on R1C9-R9C1 diagonal) => no 5 in R4C7
R9C9 = 3 => R9C789 = [583] => no 5 in R4C7
-> no 5 in R4C7
29a. 5 in N6 only in R5C89, locked for R5

30. Consider placements for R9C9
R9C9 = 2 => 3 on R1C1-R9C9 diagonal only in R4C4 + R6C6, CPE no 3 in R6C2 using R1C7-R7C1 diagonal => R6C2 = 1 => R8C2 = 3, placed for R1C9-R9C1 diagonal => R3C7 = 5, placed for R1C9-R9C1 diagonal => R7C3 = 2
R9C9 = 3 => R9C789 = [583] => R2C8 = 2
-> R2C8 = 2 or R7C3 = 2, locked for R1C9-R9C1 diagonal

31. R1C7 cannot be 1, here’s how
R1C7 = 1, placed for R1C7-R7C1 diagonal => R6C2 = 3, placed for R1C7-R7C1 diagonal => R4C4 = 9 => R6C6 = 1 (hidden single on R1C1-R9C9 diagonal) => 2,3,5 on R1C1-R9C9 diagonal only in R1C1 + R9C9
-> no 1 in R1C7
31a. R1C7 = 7, placed for R1C7-R7C1 diagonal, clean-up: no 4 in R8C8
31b. 4 in N9 only in R78C7, locked for C7

32. Consider placements for R1C1
R1C1 = 2, placed for R1C1-R9C9 diagonal => R9C9 = 3 => R9C789 = [583] => no 5 in R9C5
R1C1 = 5, placed for R1C1-R9C9 diagonal => R9C6 = 5 (hidden single in C6) => no 5 in R9C5
-> no 5 in R9C5
32a. Naked pair {23} in R9C59, locked for R9

33. Consider placements for R9C9
R9C9 = 2, placed for R1C1-R9C9 diagonal => R1C1 = 5 => R1C2 = 2 => R7C2 = 5 = R7C3 = 2
R9C9 = 3 => R9C5 = 2
-> R7C3 = 2 or R9C5 = 2, locked for R5C1-R9C5 diagonal

34. Consider placements for R3C7
R3C7 = 3, placed for R1C9-R9C1 diagonal => R8C2 = 1 => R6C2 = 3
R3C7 = 5 => R9C7 = 8 => R9C89 = [62] => R9C5 = 3
-> R6C2 = 3 or R9C5 = 3, CPE no 3 in R6C5

35. Consider placements for R9C9
R9C9 = 2 => R9C789 = [862] => R2C7 = 2
R9C9 = 3 => R8C7 = 2 (hidden single in N9)
-> R2C7 = 2 or R8C7 = 2, locked for C7

36. Consider placements for R9C5
R9C5 = 2, placed for R5C1-R9C5 diagonal => R7C3 = 5, placed for R1C9-R9C1 diagonal => R3C7 = 3, placed for R1C9-R9C1 => R8C2 = 1 => R6C2 = 3
or R9C5 = 3
-> R6C2 = 3 or R9C5 = 3, CPE no 3 in R6C8 using R5C9-R9C5 diagonal
36a. R35C8 = {35} (hidden pair in C8)

37. Consider placements for R9C4
R9C4 = 1 => R9C56 = [35], 3 placed for R5C1-R9C5 diagonal => R6C2 = 1
R9C4 = 6 => R9C6 = 1 (hidden single in R9)
-> R6C2 = 1 or R9C6 = 1, CPE no 1 in R6C6

38. 1 on R1C1-R9C9 diagonal only in R4C4 + R7C7, CPE no 1 in R4C7
38a. 1 in C7 only in R57C7, CPE no 1 in R5C9 using R5C9-R9C5 diagonal

39. 9 in C7 only in R457C7, CPE no 9 in R6C8 using R5C9-R9C5 diagonal
[Just spotted this; it’s been possible since step 28.]
39a. 9 in N6 only in R45C7, locked for C7, clean-up: no 1 in R7C8
39b. R14C8 = {14} (hidden pair in C8)

40. 1 in R7 only in R7C79, CPE no 1 in R5C7 using R1C3-R7C9 diagonal
40a. R5C7 = 9, placed for R3C9-R9C3 diagonal, R4C7 = 3, R3C7 = 5, placed for R1C5-R5C9 and R1C9-R9C1 diagonals, R3C8 = 3, R9C7 = 8, placed for R3C1-R9C7 diagonal, R9C8 = 6, R7C8 = 9, R8C8 = 7, placed for R1C1-R9C9 diagonal, R2C7 = 2, R8C7 = 4, R7C7 = 1, placed for R1C1-R9C9 diagonal, R78C9 = [35], R9C9 = 2, placed for R1C1-R9C9 diagonal, R9C5 = 3, placed for R5C1-R9C5 diagonal

41. R8C4 = 8, R1C4 = 3, R3C4 = 7 (cage sum), R4C4 = 9, R6C4 = 4, placed for R1C9-R9C1 diagonal

and the rest is naked singles without using diagonals.

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