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Ed
 Post subject: Assassin 431
PostPosted: Sat Apr 01, 2023 5:54 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
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1-9 cannot repeat on either diagonal

Assassin 431
Perfect hardness for me. Enjoyed it. Can crack it fairly quickly once you find the way. Jsudoku has a pretty hard time but SudokuSolver says 1.55.
triple click code:
3x3:d:k:1792:1793:1793:3842:7683:7683:7683:7683:3588:3333:1792:3842:3842:3842:7683:7683:3588:2566:3335:3333:7432:7432:8201:8201:8201:8201:2566:3335:7432:7432:1546:3595:8201:8201:1292:1292:3853:3853:7432:7432:1546:3595:7438:2831:2831:3853:7184:7184:7184:7438:7438:7438:7438:7438:4625:4625:3602:7184:3347:7188:5397:2326:2326:4625:3602:3351:7184:3347:7188:7188:5397:2326:3602:3351:3351:2840:2840:7188:7188:7188:5397:
solution: 
+-------+-------+-------+
| 3 1 6 | 7 2 9 | 5 4 8 |
| 8 4 2 | 1 5 3 | 7 6 9 |
| 7 5 9 | 4 6 8 | 3 2 1 |
+-------+-------+-------+
| 6 7 1 | 5 8 4 | 9 3 2 |
| 2 9 5 | 3 1 6 | 8 7 4 |
| 4 3 8 | 9 7 2 | 1 5 6 |
+-------+-------+-------+
| 9 8 7 | 2 4 5 | 6 1 3 |
| 1 2 3 | 6 9 7 | 4 8 5 |
| 5 6 4 | 8 3 1 | 2 9 7 |
+-------+-------+-------+
Cheers
Ed
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wellbeback
 Post subject: Re: Assassin 431
PostPosted: Fri Apr 07, 2023 8:04 pm 
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Posts: 282
Location: California, out of London
Thanks Ed! I liked the way I did it but it I'm not sure whether I found the quick way that you did. We'll see! :)
Assassin 431 WT:
1. Innies r12 = r2c19 = +17(2) = {89}
-> One of:
13(2)n1 = [85], 10(2)n3 = [91], 14(2)n3 = [86]
13(2)n1 = [94], 10(2)n3 = [82], 14(2)n3 = [95]

2. Innies D\ = [r3c3,r6c6] = +11(2)
One of:
6(2)D\ = {15}, 7(2)D\ = {34}, 21(3)D\ = {678}, H+11(2)D\ = {29}
6(2)D\ = {24}, 7(2)D\ = {16}, 21(3)D\ = {579}, H+11(2)D\ = {38}

3. IOD n6 -> r6c5 + r6c6 = r4c7
-> r4c7 > r6c6
-> H+11(2)D\ cannot be [29]

4. Outies r6789 = r5c127 = +19(3)
-> At least one of (89) in r5c127
Can H+11(2)D\ be [38]?
That would put 8 in r5c89, 9 in r4c7, and 9 in r5c12, which leaves no solution for 14(2)n5
-> H+11(2)D\ = [92] or [83]

5. Innies n1 = r2c3,r3c1,r3c3 = +18(3). It has a 7, and one of (89) in r3c3
-> Innies n1 = {279} or {378}
Since neither of (23) can go in r3c1 -> Innies n1 = [279] or [378]
-> 13(2)r3c1 = [76]

6. -> 7 in n3 in 30(6)n23
-> 7 in n2 in 15(4)n12
-> 15(4)n12 = {1347} or {1257} with one of (23) in r2c3.
-> (17) in n2 in 15(4)n12

7. Where are (89) in n5?
One of (89) in 14(2)n5
D\ in n5 = [{15}2] or [{24}3] (No 6789)
Outies r1234 = r5c3456 = +15(4)
Since r5c6 is Min 5 -> No (89) in r5c4
IOD n6 -> r6c56 is max +9(2)
Since r6c6 from (23) -> r6c5 is max 7
-> One of (89) in D/ in n5 in r4c6,r6c4

8. Where is 7 in n5?
Innies D/ = r3c7,r4c6,r5c5,r6c4 = +17(4)
Since one of (89) in r4c6,r6c4 -> 7 not in innies D/
-> 7 in n5 in r5c4 or r6c5

9! -> 7 in c6 only in r789c6
Innies n89 = r78c4 = +8(2) - Not {17} (Which could have been deduced from Step 6 Line 4)
-> 1 in n8 in r789c6
Outies n9 = r789c6 = +13(3) -> r789c6 = {157}
-> 14(2)n8 = {49}
-> 11(2)n8 = {38}
-> r78c4 = {26}

10! Also 1 in n9 in 9(3)n9
-> 4 in n9 in 28(6)n89
Innies n9 = r8c7,r9c78 = +13(3) cannot be {348} because of 11(2)n8 = {38}
-> r8c7,r9c78 = {249}
-> 9(3)n9 = {135}
-> 21(3)n9 = {678}
-> H+11(2)D\ = [92]
-> 6(2)D\ = {15}
-> 7(2)D\ = {34}

11. Finishing up
-> 13(2)n1 = [85]
-> 10(2)n3 = [91]
-> 14(2)n3 = [86]
Also r2c3 = 2 and r1c23 = {16}
Also 14(2)n5 = [86]
-> Remaining Outies r1234 = r5c345 = +9(3) (No 7)
-> r6c5 = 7
-> (IOD n6) r4c7 = 9
-> r6c4 = 9
Also (HS 7 in n6) 11(2)n6 = {47}
-> 5(2)n6 = {23}
-> r5c4 = 3
-> r4c6 = 4
-> (Innies D/) r5c5 = 1 and r3c7 = 3
etc.

After posting the above I realized it could be even quicker. After Step 6...
...much quicker finish:
7. Innies n89 = r78c4 = +8(2)
At least one of (17) in r12c4 -> r78c4 not {17}
-> 1 in n8 in r789c6
-> 1 in n9 in 9(3)n9 = {126} or {135}
-> 4 in n9 in that part of 28(6)n89 in n9 (Innies n9)
-> Innies n9 = +15(3) = {249} or {348}
-> Outies n9 = r789c6 = +13(3) = {157}
-> 13(2)n8 = {49}
-> 11(2)n8 = {38} and r78c4 = {26}
-> Innies n9 = {249}
-> 9(3)n9 = [135}
-> 21(3)n9 = {678}
etc.
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Andrew
 Post subject: Re: Assassin 431
PostPosted: Sun Apr 09, 2023 3:46 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Thanks Ed for your latest fairly challenging Assassin. It took me some time to crack it and get my first placements, then it crumbled. I'll be interested to see how you cracked it.

Here's how I solved Assassin 431:
Prelims

a) 7(2) cage at R1C1 = {16/25/34}, no 7,8,9
b) R1C23 = {16/25/34}, no 7,8,9
c) 14(2) cage at R1C9 = {59/68}
d) 13(2) cage at R2C1 = {49/58/67}, no 1,2,3
e) R23C9 = {19/28/37/46}, no 5
f) R34C1 = {49/58/67}, no 1,2,3
g) 6(2) cage at R4C4 = {15/24}
h) 14(2) cage at R4C5 = {59/68}
i) R4C89 = {14/23}
j) R5C89 = {29/38/47/56}, no 1
k) R78C5 = {49/58/67}, no 1,2,3
l) R9C45 = {29/38/47/56}, no 1
m) 21(3) cage at R7C7 = {489/579/678}, no 1,2,3
n) 9(3) cage at R7C8 = {126/135/234}, no 7,8,9

1a. Killer triple 4,5,6 in 7(2) cage at R1C1, R1C23 and 13(2) cage at R2C1, locked for N1, clean-up: no 7,8,9 in R4C1
1b. 7(2) cage at R1C1 = {16/34} (cannot be {25} which clashes with 6(2) cage at R4C4), no 2,5
1c. Killer pair 1,4 in 7(2) cage and 6(2) cage, locked for D\
1d. 45 rule on D\ 2 innies R3C3 + R6C6 = 11 = {29/38} (cannot be {56} which are only in R6C6)
1e. 21(3) cage at R7C7 = {579/678}, 7 locked for N9
1f. 45 rule on R12 2 outies R3C29 = 6 = [42/51] -> R2C19 = {89}, locked for R2, clean-up: no 5,6 in R1C9
1g. Naked pair {89} in R12C9, locked for C9 and N3, clean-up: no 2,3 in R5C8
1h. 6 in N1 only in 7(2) cage and R1C23 = {16}{25}/{16}{34}, 1 locked for N1
[Alternatively 45 rule on N1 3 innies R2C3 + R3C13 = 18 = {279/378} (cannot be {189} which clashes with R2C1), no 1]
1i. 45 rule on R789 2 innies R78C4 = 8 = {17/26/35}, no 4,8,9
1j. 45 rule on R789 3 outies R6C234 = 20 = {389/479/569/578}, no 1,2
1k. 45 rule on R6789 1 outie R5C7 = 1 innie R6C1 + 4, no 1,2,3,4 in R5C7, no 6,7,8,9 in R6C1
1l. 15(3) cage at R5C1 = {168/249/258/348/357} (cannot be {456} which clashes with R4C1, cannot be {159/267} which clash with R5C7 + R6C1)
1m. 1 of {168} must be in R6C1 -> no 1 in R5C12
1n. 45 rule on C1234 3 outies R259C5 = 9 = {126/135/234}, no 7,8,9, clean-up: no 2,3,4 in R9C4
1o. 45 rule on R1234 4 outies R5C3456 = 15 must contain 1 for R5 = {1239/1248/1257/1356} (cannot be {1347} because R5C6 only contains 5,6,8,9)
1p. 8,9 of {1239/1248} must be in R5C6 -> no 8,9 in R5C34
1q. 45 rule on N6 2 outies R6C56 = 1 innie R4C7
1r. Min R6C56 = 4 (cannot be [12] which clashes with 6(2) cage at R4C4) -> min R4C7 = 4
1s. Max R4C7 = 9 -> max R6C56 = 9, no 8,9 in R6C5, no 9 in R6C6, clean-up: no 2 in R3C3
1t. R2C3 + R3C13 = {279/378}
1u. 7 of {279} must be in R3C1 -> no 9 in R3C1, clean-up: no 4 in R4C1
1v. 2 on D\ only in R4C4 + R5C5 + R6C6, locked for N5

2a. 7 in R12 only in 15(4) cage at R1C4 and 30(6) cage at R1C5 -> each must contain 7
2b. 15(4) cage must contain 7 = {1257/1347}, no 6,8,9, 1 locked for N2
2c. 1 in R3 only in R3C789, locked for N3

3a. 45 rule on N9 3 innies R8C7 + R9C78 = 15 = {159/168/249/348} (cannot be {258/456} which clash with 9(3) cage at R7C8)
3b. 45 rule on N9 3 outies R789C6 = 13 = {157/247/256/346} (cannot be 139/148} which clash with R8C7 + R9C78, cannot be {238} which clashes with R6C6), no 8,9
3c. 1 in N8 only in R78C4 = {17} or in R789C6 -> R789C6 = {157/256/346} (cannot be {247} (locking out cages)
3d. R8C7 + R9C78 = {159/249/348} (cannot be {168} which clashes with R789C6), no 6
3e. R78C5 = {49/58} (cannot be {67} which clashes with R789C6), no 6,7
3f. Hidden killer pair 8,9 in R78C5 and R9C4 for N8, R78C5 contains one of 8,9 -> R9C4 = {89}, R9C5 = {23}
3g. Killer pair 4,5 in R78C5 and R789C6, locked for N8, clean-up: no 3 in R78C4
3h. R259C5 (step 1n) = {135/234}, 3 locked for C5
3i. Killer pair 4,5 in R259C5 and R78C5, locked for C5, clean-up: no 9 in R5C6
3j. Combined cage R78C5 + R9C45 = {49}[83]/{58}[92]
3k. R259C5 = {15}3/[342] (cannot be {24}3 which clashes with combined cage) -> R2C5 = {135}, R5C5 = {145}, clean-up: no 4 in R4C4

4a. R5C3456 (step 1o) = {1248/1356} (cannot be {1257} = [2715] which clashes with 6(2) cage at R4C4 = {15}, CCC), no 7
4b. 2 of {1248} must be in R5C3 -> no 4 in R5C3
4c. R5C89 = {47/56}/[92] (cannot be [83] which clashes with R5C3456), no 3,8
4d. 45 rule on R6789 3 outies R5C127 = 19 = {289/379/478} (cannot be {469/568} which clash with R5C3456), no 5,6, clean-up: no 1,2 in R6C1 (step 1k)
4e. Hidden killer pair 1,2 in R6C56 and R6C789 for R6, R6C45 cannot contain both of 1,2 which clashes with 6(2) cage at R4C4, CCC, R6C789 cannot contain both of 1,2 which clashes with R4C89 -> R6C56 and R6C789 each contain one of 1,2
4f. Killer pair 1,2 in 6(2) cage at R4C4 and R6C56, locked for N5
4g. Killer pair 1,2 in R4C89 and R6C789, locked for N6, clean-up: no 9 in R5C8
4h. 2 in R5 only in R5C123, locked for N4
4i. 3 in R5 only in R5C1234, CPE no 3 in R4C23
4j. 9 in R5 only in R5C127 = {289/279}, no 4

5a. Consider combinations for R5C3456 (step 4a) = {1248/1356}
R5C3456 = {1248} = [2418]
or R5C3456 = {1356}, 3 in R5C34 locked for 29(6) cage at R3C3, no 3 in R3C3 => no 8 in R6C6 (step 1d)
-> no 8 in R6C6, clean-up: no 3 in R3C3
[Cracked at last, first placement follows.]
5b. Naked pair {89} in R2C1 + R3C3, locked for N1 -> R3C1 = 7, R4C1 = 6, clean-up: no 1 in R2C2, no 8 in R5C6
5c. R5C3456 = {1356} (only remaining combination), locked for R5, 6 locked for N5, clean-up: no 2 in R4C4
5d. Naked pair {15} in 6(2) cage at R4C4, locked for N5 and D\
[Routine clean-ups omitted from here]
5e. R5C6 = 6 -> R4C5 = 8, R5C4 = 3, R6C5 = 7, R6C6 = 2 -> R3C3 = 9 (step 1d), placed for D\, R2C1 = 8 -> R3C2 = 5, R2C9 = 9 -> R3C9 = 1, R1C9 = 8 -> R2C8 = 6, both placed for D/
5f. Naked pair {34} in 7(2) cage at R1C1, locked for N1 -> R2C3 = 2
5g. Naked pair {16} in R1C23, locked for R1
5h. Naked pair {47} in R5C89, locked for N6, 7 locked for R5, clean-up: no 1 in R4C8
5i. R2C3 = 2 -> 15(4) cage at R1C4 = {1257} (only remaining combination), 5,7 locked for N2
5j. Naked triple {157} in R124C4, 1,7 locked for C4
5k. Naked pair {26} in R78C4, locked for C4, 2 locked for N8
5l. R9C5 = 3 -> R9C4 = 8, R3C4 = 4, R6C4 = 9, R4C6 = 4, both placed for D/
5m. R6C4 = 9, R78C4 = {26} -> R6C23 = 11 = {38}, locked for R6 and N4
5n. Naked pair {29} in R5C12, 9 locked for R5, R6C1 = 4 (cage sum), R1C1 = 3
5o. 14(3) cage at R7C3 = {257} (only remaining combination), locked for N7, 2,5 locked for D/
5p. Naked pair {19} in R78C1, locked for N7, 9 locked for C1, R7C2 = 8 (cage sum)

and the rest is naked singles, not using the diagonals.
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Ed
 Post subject: Re: Assassin 431
PostPosted: Fri Apr 14, 2023 10:15 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
I used a similiar contradiction move to wellbeback's step 4, then used his Much Quicker Finish

However, a really 'pure' way to use the feature behind our contradiction moves is after Andrew's step 4. Just takes a lot, lot longer to get there. Really glad you found that clean way through this puzzle Andrew! Will put it first.
alt Andrew's step 5: 
.-------------------------------.-------------------------------.-------------------------------.
| 1346      123456    123456    | 123457    26789     23456789  | 234567    234567    89        |
| 89        1346      237       | 123457    145       234567    | 234567    56        89        |
| 78        45        389       | 23456789  26789     23456789  | 1234567   1234567   12        |
:-------------------------------+-------------------------------+-------------------------------:
| 56        1456789   1456789   | 125       689       3456789   | 456789    1234      1234      |
| 23789     23789     12356     | 3456      145       568       | 789       4567      4567      |
| 345       3456789   3456789   | 3456789   167       238       | 123456789 123456789 1234567   |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 123456789 123456789 | 1267      4589      1234567   | 56789     123456    123456    |
| 123456789 123456789 123456789 | 1267      4589      1234567   | 1234589   56789     123456    |
| 123456789 123456789 123456789 | 89        23        1234567   | 1234589   1234589   567       |
'-------------------------------.-------------------------------.-------------------------------'
End Andrew's step 4 above, paste into A431 in SudokuSolver then

5. since r6c56 = r4c7 -> r6c56 must repeat in n6 in r45c89
5a. no 8 in r45c89 -> no 8 in r6c6 -> no 3 in r3c3
alt wellbeback step 4: paste candidates into A431 in SudokuSolver: 
.-------------------------------.-------------------------------.-------------------------------.
| 1346      123456    123456    | 123456789 123456789 123456789 | 123456789 123456789 89        |
| 89        1346      1234567   | 1234567   1234567   1234567   | 1234567   56        89        |
| 456789    45        389       | 123456789 123456789 123456789 | 123456789 123456789 12        |
:-------------------------------+-------------------------------+-------------------------------:
| 456789    123456789 123456789 | 1245      5689      123456789 | 123456789 1234      1234      |
| 123456789 123456789 123456789 | 123456789 1245      5689      | 123456789 23456789  234567    |
| 123456789 123456789 123456789 | 123456789 123456789 238       | 123456789 123456789 1234567   |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 123456789 123456789 | 123456789 456789    123456789 | 56789     123456    123456    |
| 123456789 123456789 123456789 | 123456789 456789    123456789 | 123456789 56789     123456    |
| 123456789 123456789 123456789 | 23456789  23456789  123456789 | 123456789 123456789 567       |
'-------------------------------.-------------------------------.-------------------------------'

End wellbeback's step 3 above, then

4. since r6c56 = r4c7 -> r6c56 must repeat in n6 in r45c89
4a.
i. if 1 in r6c5 -> 5(2)n6 = {14}: both locked for r4 and 6(2)n5 = [24]
ii. -> 4 outies r1234 at r5c3456 = 15 = [2346]: only permutation possible with 4 in r5c5
iii. -> 8 in r4c5

5. since r6c56 = r4c7 -> max. r6c56 = 9
5a. but can't be [18] (step 4)
5b. -> no 8 in r6c6
5c. -> no 3 in r3c3

on from there.
Another way of doing wellbeback's alt. step 7: 
.-------------------------------.-------------------------------.-------------------------------.
| 134       123456    123456    | 1457      2345689   2345689   | 123456789 123456789 89        |
| 89        1346      23        | 1457      1457      23456     | 1234567   56        89        |
| 7         45        89        | 2345689   2345689   2345689   | 12345689  12345689  12        |
:-------------------------------+-------------------------------+-------------------------------:
| 6         12345789  12345789  | 1245      589       12345789  | 12345789  1234      1234      |
| 234589    2345789   12345789  | 123456789 245       5689      | 123456789 23456789  234567    |
| 1234589   12345789  12345789  | 123456789 123456789 23        | 123456789 123456789 1234567   |
:-------------------------------+-------------------------------+-------------------------------:
| 1234589   123456789 123456789 | 123456789 456789    123456789 | 56789     123456    123456    |
| 1234589   123456789 123456789 | 123456789 456789    123456789 | 123456789 56789     123456    |
| 1234589   123456789 123456789 | 23456789  23456789  123456789 | 123456789 123456789 567       |
'-------------------------------.-------------------------------.-------------------------------'

End step 6 above, then

7. "45" on c1234: 3 outies r259c5 = 9 (no 7,8,9)

8. 7 in n2 only in c4: 7 locked for c4

9. "45" on n89: 2 innies r78c4 = 8 = {26/35}(no 1,4,7,8,9) (oops, thanks for the correction Andrew!)

Now back to wellbeback's step 7 line 3 with 1 in n8 only in r789c6
Cheers
Ed
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