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Toroidal Killer Sudoku

 
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Para
Yokozuna
Yokozuna


Joined: 08 Nov 2006
Posts: 384
Location: The Netherlands

PostPosted: Thu Nov 08, 2007 9:43 pm    Post subject: Toroidal Killer Sudoku Reply with quote

Hi all

I have been planning to make a toroidal killer sudoku for a while. Well here is my first attempt. So instead of 9 3X3 nonets, we have 9 jigsaw nonets that wrap around the grid. I have coloured the nonets to make it easier to recognize which cells belong together in a nonet. There is also a SumoCue-code available but this makes it a bit harder to spot the seperate nonets.
It was fun to solve. It's not the hardest puzzle, but the first one doesn't really have to be. The toroidal cages make the solving process a bit complex already.


Toroidal Killer Sudoku #1



SumoCueV1=19J0+0J0=15J0+2J1=9J1=14J2+5J3=12J4+7J4+0J0=11J1+10J1+2J1+4J2+5J2=19J5+15J4+7J0=18J1+10J1=20J2+20J2=24J2=20J5+23J5+15J6=8J0+18J1+18J2=17J2+20J5+22J5+23J5=14J6+26J6+26J7=7J8+36J2+29J5+29J5+22J6+33J6+33J6=16J7+43J7=10J8+45J3+29J5=16J6+22J6=17J7+33J7=17J7+52J8+45J3=22J3+48J4+48J6+22J7+50J7+50J8=18J8+52J8=14J3+55J4+55J4=11J0=12J7=8J8+61J8+61J3=17J3+63J4+63J4+66J0+66J0+67J1+68J8+68J3+71J3+71J4

Hope you enjoy it.

greetings

Para
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Caida
Hooked
Hooked


Joined: 03 Nov 2007
Posts: 38
Location: Canada

PostPosted: Fri Nov 09, 2007 6:05 pm    Post subject: Walkthrough for Toridal Killer Sudoku #1 Reply with quote

Hi,

I found this one really interesting. Below is my walkthrough.

I'm putting it in really small text in case anyone else is planning to do a walkthrough and doesn't want to look.

I would love to have any comments/suggestions/corrections on my walkthrough.


edited to add: I've fixed my walkthrough - my original step 5 was faulty logic and wishful thinking - turns out I didn't actually need it! (Thanks Para!)
As always, would appreciate any comments/suggestions/corrections Smile

edited to add: I've made some changes based on suggestions/corrections received (Thanks Mike!!)


Cheers,

Caida


Toroidal Killer Sudoku #1

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

Prelims
a. 19(3)r1c1 and r2c7 = {289/379/469/478/568} (no 1)
b. 9(2) r1c5 = {18/27/36/45} (no 9)
c. 11(3)r2c2 and r8c4 = {128/137/146/236/245} (no 9)
d. 20(3)r3c3 and r3c6 = {389/479/569/578} (no 1,2)
e. 8(3)r3c9 and r8c6 = {125/134} (no 6..9) no 1 in r8c7 (CPE)
f. 14(4)r4c7 = {1238/1247/1256/1346/2345} (no 9)
g. 7(2) r5c1 = {16/25/34} (no 7..9)
h. 16(2)r5c8 = {79} Note {79} now locked for n8 and r5
i. 10(3)r6c1 = {127/136/145/235} (no 8,9)
j. 22(3)r7c2 = {589/679} (no 1..4) no 9 in r7c3 r8c189 r9c2 (CPE)
k. 12(2) r8c5 = {39/48/57} (no 1,2,6)

cleanup
l. r9c5 no 3,5

Walkthrough
1. Outies c12: r28c3 = 11(2) = [29/38/47/56/65] (no 1)
1a. -> r2c3 no 7,8

2. Outies c89: r28c7 = 17(2) = {89} (locked for c7)
2a. -> 20(3)r3c6 no 3 as combo {389} blocked by r2c7
2b. 20(3)r3c6 requires either an 8 or a 9
2c. -> killer pair {89} locked for n6 in r2c7 and 20(3)r3c6 also no 8, 9 in r2c6 (CPE)

3. Outies r12: r3c28 = 9(2) = {18/27/36/45} (no 9)
3a. -> r3c2 no 8

4. Outies r89: r7c28 = 12(2) = {39/48/57} (no 1,2,6)
4a. ->r7c2 no 5 (combo 22(3) = [5]{89} blocked by r8c7)
4b. ->r7c2 no 7 (combo 22(3) = [7]{69} would require r7c8 to equal 5 (step 4) and r8c7 to equal 8, this would given an invalid combo in 18(3)r7c8 of [5]{85})
4c. ->r7c8 no 5,7,8,9
4d. 18(3)r7c8 = [396/486]
4e. -> can’t be [387] as this is blocked by 22(3) = [9]{58/67}
4f. -> can’t be [495] as this is blocked by 22(3) = [8]{59}
4g. ->r8c8 = 6
4h. -> 22(3)r7c2 = [9]{58}/[8]{59}; [9]{67} blocked because 6 is unavailable due to r8c8
4i. cleanup: r8c23 no 7; 22(3)r7c2 = {589}
4j. 5 locked for n5 in r8c23
4k. r8c1, r7c3 and r9c2 no 5,8,9 (can see all of 22(3))
4l. {589} locked for r8 in c237 (naked triple)
4m. cleanup: r9c5 no 4,7: (12(2) = [39/48]

5. 17(3)r8c9: r89c9 can’t be {79} (blocked by r5c9
5a. r9c8 no 1
5b. 17(3)r6c8: r67c9 can’t be {79}
5c. r6c8 no 1

6. 20(3)r3c3: r3c34 no 3 (needs both 8 and 9 and there is no 8/9 in r4c4 b/c step 2c)

7. 18(3)r3c1: r34c1 can’t be {89} (blocked by r9c5 – all in n2)
7a. r4c2 no 1

8. {89} is locked in n4 at r7c2 and r9c8 (no 2..7 in r9c8) (hidden pair)
8a. {89} locked in r9 at r9c58 (naked pair)

9. 19(3)r2c7 = [8]{29}/[9]{28}/[8]{47}
9a. -> [9]{37} blocked as this would lead to r8c8=8, r7c8=4, r7c2 = 8(step4), r9c8 = 9(step 8), r5c8 = 7
9b. -> [9]{46} and [8]{56} blocked by r8c8
9c. -> r23c8 no 3,5
9d. killer triple {789} in r2359c8 (19(3) requires either 7, 8, or 9 in r23) no 7, 8, 9 elsewhere in c8

10. 14(3) r8c1 = {167/347} (no 2)

11. 17(3)r6c8: max of r6c8 = 5
11a. -> min of r67c9 = 12(2) = {93/84}
11b. -> r67c9 no 1,2

12. 8 in r5 locked in r5c56 (I had originally written this as: r5c56 contains only 8s in r5)
12a. -> 8 locked in r5 for n7 (in r5c56)
12b. r2c8 no 2 (as requires either an 8 or 9 in r3c8)
12c. 19(3)r2c7 requires an 8 (step 9)
12d. -> 8 is either in r2c7 or r2c8 (locked for r2)
12e. cleanup: r1c5 no 1 (as no 8 in r2c5)


13. 8 in n1 locked in r1 -> not elsewhere in r1 (I had originally written this as 8 is locked in n1 r1c123 (nowhere else in n1 has option of an 8 so no other n can have an 8 in r1))
13a. cleanup: r2c5 no 1 (as no 8 in r1c5)


14. 8 in n7 is only in c56; 8 in n8 is only in c56
14a. LOL – no other n can have an 8 in c56
Mike suggests writing this as “grouped X-Wing "8 in n78 locked in c56 -> not elsewhere in c56” but I am not yet adept at seeing things as “X-Wings” and still need to see them as LOL
14b. r9c5 = 9, r8c5 = 3
14c. r9c8 = 8; r7c2 = 9
14d. r2c7 = 8; r8c7 = 9; r7c8 = 3

15. r3c4 = 8 as it is the only 8 in c4 hidden single, c4
15a. r4c1 = 8 as it is the only 8 in n2
15b. 18(3)r3c1 = {378/468} (no 1,2,5)
15c. 20(3)r3c3 = {389/578} (no 4,6)
15d. cleanup: r3c3 no 3 pointed out by Mike that the 3 is already gone – so this step is not needed

16a. r6c4 = 9 as it is the only 9 in c4 hidden single, c4
16b. 9 is in only r12 (no other r) for both n1 and n5, no 9 in other ns in r12
Note that this is another X-Wing: grouped X-Wing "9 in n15 locked in r12 -> not elsewhere in r12" and could be written as 9 in n6 locked in c6 -> not elsewhere in c6
16c. 9 in n3 locked for in c3, not elsewhere in c3
16d. n1 is the only n with a 9 available in c1, no other c in n1 can have a 9 Better written as: 9 in c1 locked in n1 -> not elsewhere in n1

17. 16(3)r6c4 = [9]{16/25} (no 4,7)
17a. cleanup: r7c4 no 2
17b. cleanup: r12c5 no 6 missed from step 14b
17c. r9c7 no 4 (would need a {13} in r89c6)

18. 17(3)r8c9 = [8]{27}
18a. -> {27} locked for c9
18b. -> r5c89 = [79]
18c. -> r23c8 = [92]

19. 17(3)r6c8 = {458} no 6
19a. -> 6 in c9 locked in 12(3)r1c8
19b. -> 12(3)r1c8 = [165]

20. 8(3)r3c9 = [341]
20a. cleanup: r7c3 = 2; r6c8 = 5; r9c9 = 7; r8c9 = 2

21. r9c1 = 3, hidden single
21a.
14(3)r8c1 = [734]

22. 8(3) r8c6 = [125]
22a. r8c4 = 4


23. {48} locked for n9 in c9
23a. -> r7c7 no 4; r56c1 no 4

everything from now on is all singles and cage sums


24 (all singles): r6c1 = 6; r123c1 = [924; r5c1 = 5; r7c1 = 1; r6c2 = 3; r1c7 = 4; r78c7 = [27]; r345c7 = [631]; r6c6 = 7
24a. 19(3)r1c1 = [982]
24b. r1c3 = 7; r8c23 = [58]

25. 9(2)r1c5 = [54]
25a. r34c5 = [12]; r1c6 = 3; r1c4 = 2; r3c2 = 7; r4c2 = 6; r2c2 = 1; r5c2 = 2; r5c4 = 3; r5c3 = 4; r6c3 = 1; r9c3 = 6; r9c4 = 1; r2c34 = [36]; r2c6 = 7; r7c4 = 5; r4c4 = 7;

26. 20(3)r3c3 = [587]
26a. 20(3)r3c6 = [965]
26b. 14(4)r4c7 = [3812]
26c. 24(5)r3c5 = [12678]
26d. 17(3)r6c6 = [467]
26e. 17(3)r6c8 = [584
26f. r4c3 = 9





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


Last edited by Caida on Tue Dec 11, 2007 3:46 am; edited 2 times in total
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mhparker
Grandmaster
Grandmaster


Joined: 20 Jan 2007
Posts: 345
Location: Germany

PostPosted: Mon Nov 19, 2007 10:50 am    Post subject: Toroidal Killer 1 Walkthrough Reply with quote

Hi folks,

Well, I finally did it! Very Happy Got stuck for some time in the middle, though, so hopefully the difficulty of subsequent toroidal killers can be kept at around this level. Wink

BTW, these toroidal killers are a great invention (thanks, Para!). Some of those CPE moves have to be seen to be believed!

This puzzle appears to have quite an elegant, definite, solving path ("text book" solution). Caida did a better job of finding it than I did. The text in the box below explains why.

Quote:
I got stuck immediately after step 14. I finally managed to find step 15 (and the follow-up steps) to break the deadlock. However, the "intended" way forward at this point was probably the killer triple on {789} in C8, which sets up a grouped X-Wing on 8 in N78 and C56, which in turn fixes 12(2) at R89C5. This corresponds to the route Caida took. I eventually saw both of these moves (steps 17b and 24), but too late. Nevertheless, I've decided to post my WT, in order to show an alternative path.

Other interesting features about this puzzle were:
  • Not many innie/outie ("45") moves available on the nonets, because they were very thin, with (consequently) lots of protruding cages.
  • No productive LoL moves, because the nonets were orientated in a diagonal direction.


Now to the WT:

Edited to include feedback from Para. Many thanks.

Toroidal Killer 1 Walkthrough

Nonet Layout:

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

Prelims:

a) 19(3) at R1C1 and R2C7 = {289/379/469/478/568} (no 1)
b) 9(2) at R1C5 = {18/27/36/45} (no 9)
c) 11(3) at R2C2 and R8C4 = {128/137/146/236/245} (no 9)
d) 20(3) at R3C3 and R3C6 = {389/479/569/578} (no 1,2)
e) 8(3) at R3C9 and R8C6 = {125/134} (no 6..9) -> no 1 in R78C7 (CPE)
f) 14(4) at R4C7 = {1238/1247/1256/1346/2345} (no 9)
g) 7(2) at R5C1 = {16/25/34} (no 7..9)
h) 16(2) at R5C8 = {79}, locked for R5 and N8
i) 10(3) at R6C1 = {127/136/145/235} (no 8,9)
j) 17(3) at R6C6 = {269/278/359/368/458/467} (no 1); no 2 in R7C7
(Note: {179} blocked, and no 2 in R7C7, because {79} only available in R7C7)
k) 17(3) at R6C8 = {179/269/278/359/368/458/467}; no 1 in R67C9
(Note: no 1 in R67C9 because {79} unavailable in R6C8)
l) 22(3) at R7C2 = {589/679} (no 1..4) -> no 9 in R7C3, R8C189, R9C2 (CPE!)
m) 12(2) at R8C5 = [39/48/57/84] (no 1,2,6; no 3,5 in R9C5)

1. Outies R89: R28C7 = 17(2) = {89}, locked for C7
1a. {89} in R2C7 blocks {389} combo for 20(3) at R3C6 (Prelim d)
1b. -> 20(3) at R3C6 = {479/569/578} (no 3) = {(8/9)..}
1c. R2C7 and 20(3) at R3C6 form killer pair on {89} within N6
1d. -> no 8,9 elsewhere in N6
1e. CPE(N6): no 8,9 in R2C6

2. {89} now unavailable in R4C4 -> {389} combo for 20(3) at R3C3 (prelim d) must have 3 in R4C4
2a. -> no 3 in R3C34

3. 8 in R5 locked in N7 -> not elsewhere in N7

4. Outies R12: R3C28 = 9(2) = {27/36/45} (no 1,8,9)
(Note: {18} blocked, because neither of these 2 digits available in R3C8)
4a. 19(3) at R2C7 = {289/379/469/478/568}
4b. {89} only available in R2C78
4c. -> no 2 in R2C8

5. Outies C12: R28C3 = 11(2) = [29/38/47/56/65] (no 1,7,8 in R2C3)
5a. 11(3) at R2C2 = {137/146/236/245} (no 8)
(Note: {128} blocked, because {18} only available in R2C2)
5b. 1 only available in R2C2
5c. -> no 7 in R2C2

6. Outies R89: R7C28 = 12(2) = [57/75/84/93] (no 1,2,6; no 8,9 in R7C8)

7. Innies R89: R8C2378 = 28(4) = {(47/56)89} (no 1,2,3)
7a. 8 locked for R8
7b. {12} now unavailable to 18(3) at R7C8 = {369/378/459/468}
(Note: {567} blocked, because 18(3) must have 1 of {89} due to R8C7)
7c. possible permutations are: [396/387/495/594/486]
7d. -> no 7 in R7C8, no 8 in R8C8
7e. cleanup: no 5 in R7C2 (step 6), no 4 in R9C5

8. Hidden pair (HP) in N4 at R7C2+R9C8 = {89} (no 1..7)
8a. -> no 8 in R9C2 (CPE)
8b. cleanup: no 5 in R7C8 (step 6)
8c. 18(3) at R7C8 (step 7c) = [396/387/495/486]
8d. -> no 4 in R8C8

9. 4 now unavailable to R89 innies (step 7)
9a. -> R8C2378 = {5689} (no 7)
9b. {56} locked for R8
9c. cleanup: no 7 in R9C5, no 4 in R2C3 (step 5)

10. 22(3) at R7C2 = {589} (no 6) (last combo)
10a. 5 locked in R8C23 for N5 and R8
10b. no 8 in R7C3 (CPE)
10c. cleanup: no 5 in R2C3 (step 5)

11. Naked single (NS) in R8 at R8C8 = 6

12. Naked pair (NP) on {89} at R9C58 -> no 8,9 elsewhere in R9

13. {589} unavailable to 14(3) at R8C1
13a. -> 14(3) at R8C1 = {(16/34)7} (no 2)

14. 17(3) at R8C9 = {179/269/278/368} (no 4)
(Note: {467} blocked because 17(3) requires 1 of {89} due to R9C8)
14a. 6 only available in R9C9
14b. -> no 3 in R9C9

15. 12(2) at R8C5 requires either 3 in R8 or 8 in R9
15a. -> [386] permutation blocked for 17(3) at R8C9
15b. -> no 3 in R8C9
15c. 17(3) at R8C9 = {179/269/278} = {(1/2)..}

At this point, Para commented (via PM):

"Glad you picked up on this step, as it was in my intended walk-through as well, except you missed the follow-through. The 17(3) now either needed a 9 in C8 or a 7 in C9, which blocked the [97] combination for 16(2) in R6C89. Found that combo of moves pretty cool."

True. Wish I'd seen that. Ah well, what might have been...


16. 8(3) at R3C9 cannot have both of {12} within C9 due to 17(3) at R8C9 (step 15c)
16a. -> no 5 in R4C8

17. 6 unavailable to 19(3) at R2C7 (step 4a) = {289/379/478} (no 5)
17a. -> 19(3) must have 2 of {789}, 1 of which must be within R23C8
17b. -> R23C8, R5C8 and R9C8 form killer triple on {789} within C8
17c. -> no 7,8,9 elsewhere in C8

18. HS in C8 at R6C8 = 5
18a. split 12(2) at R67C9 = {39/48} (no 2,6,7) = {(3/4)..}
18b. -> R67C9 and R7C8 form killer pair on {34} within N9
18c. -> no 3,4 elsewhere in N9
18d. cleanup: no 3,4 in R5C2

19. 5 in C9 locked in N1 -> not elsewhere in N1

20. 8(3) at R8C6 = {125} (no 3,4) (last combo)
20a. no 5 in R7C7 (CPE)

21. {34} in R9 locked in R9C1234
21a. R9C34 cannot contain both of {34} due to 11(3) cage sum
21b. -> R9C12 must contain at least 1 of {34}
21c. -> 14(3) at R8C1 (step 13a) = {347} (no 1,6)

22. CPE: R7C7 sees all 6's in N8
22a. -> no 6 in R7C7

23. Naked single (NS) at R7C7 = 7
23a. -> split 10(2) at R67C6 = {28/46} (no 3)

24. Grouped X-Wing on 8 as follows:
24a. 8 in N78 locked in C56
24b. -> no 8 elsewhere in C56
24c. cleanup: no 1 in R12C5

25. R89C5 = [39]
25a. cleanup: no 6 in R12C5

26. NS at R9C8 = 8
26a. -> split 9(2) at R89C9 = {27} (last combo), locked for C9
26b. no 2 in R9C7 (CPE)

27. R5C89 = [79]

28. NS at R7C2 = 9
28a. -> split 13(2) at R8C23 = {58}, 8 locked for R8 and N5

29. NS at R8C7 = 9
29a. -> R7C8 = 3 (cage sum)

30. NS at R2C7 = 8
30a. -> split 11(2) at R238 = [92] (last permutation)

31. 2 in 8(3) at R8C6 locked in R89C6 -> not elsewhere in C6 and N9
31a. -> R67C6 (step 23a) = {46} (no 8), locked for C6 and N8
31b. cleanup: no 5 in R5C2

32. R4C89 = [41]
32a. -> R3C9 = 3 (cage sum)

The rest is now naked singles and a few cage sums.
_________________
Cheers,
Mike
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