Pandiagonal Latin Squares

Programs which generate, solve, and analyze Sudoku puzzles

Re: Pandiagonal Latin Squares

Postby 1to9only » Wed Jun 09, 2021 11:30 pm

Mathimagics wrote:I have updated the list above for N=7.

Only 1 of them is rated hard:
Code: Select all
.............4........5....17.2.....6............ ED=7.7/7.7/2.9

The others (most) are rated ED=2.9, two are ED=3.0.

Besides those I posted a few days ago, I also generated a few more with other (medium!) ratings:
Code: Select all
........6..1......7.............5...3...........4 ED=4.0/4.0/2.9 <- Hidden Triplet
...4................2...5.3...........6......1... ED=4.2/4.2/2.9 <- XY-Wing
.1..........5.............6......4.3....7........ ED=4.4/2.0/2.0 <- XYZ-Wing
....5.2.............1.4........6.............3... ED=4.4/4.4/2.9 <- XYZ-Wing
.......7......3............1..2...........4.5.... ED=6.2/2.3/2.3 <- APE

Here are some ED=7.x from another batch run:
PanDiagonal LS, N=7: Show
Code: Select all
4753..............................1........2..... ED=7.7/7.7/2.9
1...........................5......23...6.4...... ED=7.7/7.7/2.9
......5........1..4...............3.6...7........ ED=7.7/7.7/2.9
......64...7.2......5......1..................... ED=7.7/7.7/2.9
...........4.......1.....5..........23......7.... ED=7.7/7.7/2.9
....2...........4.....6.......5...............37. ED=7.7/7.7/2.9
....2.....7.......4.............1.5...........3.. ED=7.7/7.7/2.9
............4....6.5.............2..........71... ED=7.5/7.5/2.9
....7.3..............1......6.............2...4.. ED=7.5/7.5/2.9
..............1....4..........52....63........... ED=7.5/7.5/2.9
.........5.2....71.6............3................ ED=7.5/7.5/2.9
..1.2................................5.....6.34.. ED=7.5/7.5/2.9
.........3......4...1.......5...2.6.............. ED=7.5/7.5/2.9
.........2...........1...4.7.6...3............... ED=7.5/7.5/2.9
...............5..6.........4.....3.1...2........ ED=7.5/7.5/2.9
.........3......75.......................14....2. ED=7.5/7.5/2.9
............7................14..25..6........... ED=7.1/7.1/2.9
3.............6....4........5..1.....2........... ED=7.1/7.1/2.9
...........................4....751.........6...2 ED=7.1/7.1/2.9
...7..1.........52......6.............4.......... ED=7.1/7.1/2.9
....4......71.....3................56............ ED=7.1/7.1/2.9
.........3.....65..2................7...4........ ED=7.1/7.1/2.9
..3..7........1.2.............6......4........... ED=7.1/7.1/2.9
.....4........5....3.......6.....2.............1. ED=7.1/7.1/2.9
.........3..7...2....1........5......4........... ED=7.1/7.1/2.9
.........2.............4.3.5.........1.7......... ED=7.1/7.1/2.9
...6.1.............2.......................7..43. ED=7.1/7.1/2.9
2.......6....41.3...........7.................... ED=7.1/7.1/2.9
.2.......4.................5.....16......7....... ED=7.1/7.1/2.9

I've not generated any higher than ED=7.7.
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Thu Jun 10, 2021 4:32 am

Mathimagics wrote:I have updated the list above for N=7.
N=11 will take longer, I have to use DLX as I only built the matching-pattern solver for N=13. Finding minimal puzzles hammers the solver ... and N=11 is where it starts to hurt ...
Ok, they are in place now ...

Hi Mathimagics,
Thanks

The hardest I found in the 7x7 list is in Z3:
Code: Select all
.............4........5....17.2.....6............

Code: Select all
Resolution state after Singles:
12356   12347   34567   2345    12346   1367    357     
12356   237     13567   1236    235     1357    4       
36      12347   1346    1234567 13457   235     23567   
2346    5       3467    347     2367    23467   1       
7       134     2       1356    13456   3456    356     
12345   6       1357    137     23457   123457  23     
1345    1237    345     234567  1367    12346   23567   
181 candidates, 1563 csp-links and 1563 links. Density = 9.59%

To be noticed: the high density (much higher than in Sudoku), implying a high branching factor.

The first steps are interesting as they show several kinds of whips[1] (but no dn):
Code: Select all
whip[1]: a4n4{r7c3 .} ==> r7c4 ≠ 4, r1c3 ≠ 4, r1c2 ≠ 4
whip[1]: r1n4{c5 .} ==> r4c1 ≠ 4
whip[1]: c1n4{r7 .} ==> r3c4 ≠ 4, r3c5 ≠ 4
whip[1]: a3n4{r6c1 .} ==> r4c3 ≠ 4, r6c6 ≠ 4
whip[1]: r4n4{c6 .} ==> r5c5 ≠ 4
whip[1]: c5n4{r6 .} ==> r7c6 ≠ 4
whip[1]: a6n7{r6c4 .} ==> r6c6 ≠ 7
whip[1]: a1n7{r7c7 .} ==> r7c4 ≠ 7
whip[1]: a6n1{r6c4 .} ==> r6c6 ≠ 1
whip[1]: c6n1{r7 .} ==> r1c5 ≠ 1
whip[1]: c7n6{r5 .} ==> r5c5 ≠ 6
whip[1]: c5n6{r7 .} ==> r4c1 ≠ 6
whip[1]: c7n2{r3 .} ==> r3c4 ≠ 2
whip[1]: c4n2{r7 .} ==> r1c5 ≠ 2

Resolution state after Singles and whips[1]:
12356 1237  3567  2345  346   1367  357   
12356 237   13567 1236  235   1357  4     
36    12347 1346  13567 1357  235   23567
23    5     367   347   2367  23467 1     
7     134   2     1356  135   3456  356   
12345 6     1357  137   23457 235   23   
1345  1237  345   2356  1367  1236  23567
163 candidates.

The rest is standard bivalue-chains, z-chains, t-whips and whips: Show
z-chain[2]: r1n1{c1 c6} - a5n1{r2c6 .} ==> r7c2 ≠ 1
whip[1]: c2n1{r5 .} ==> r3c4 ≠ 1
whip[1]: c4n1{r6 .} ==> r2c1 ≠ 1
z-chain[2]: r1n2{c2 c4} - a5n2{r7c4 .} ==> r2c1 ≠ 2
whip[1]: c1n2{r6 .} ==> r6c6 ≠ 2
whip[1]: c6n2{r7 .} ==> r7c2 ≠ 2
z-chain[2]: d1n7{r4c5 r6c3} - c6n7{r2 .} ==> r1c2 ≠ 7
whip[1]: r1n7{c7 .} ==> r4c3 ≠ 7
whip[1]: r4n7{c6 .} ==> r3c5 ≠ 7
z-chain[2]: r4n6{c6 c5} - a6n6{r7c5 .} ==> r1c3 ≠ 6
whip[1]: c3n6{r4 .} ==> r3c4 ≠ 6
whip[1]: c4n6{r7 .} ==> r7c6 ≠ 6
z-chain[2]: a6n3{r3c1 r7c5} - a4n3{r7c3 .} ==> r3c4 ≠ 3
t-whip[2]: a6n3{r7c5 r1c6} - a4n3{r1c4 .} ==> r7c4 ≠ 3
z-chain[3]: r4c3{n3 n6} - c6n6{r1 r5} - c6n4{r5 .} ==> r4c6 ≠ 3
whip[2]: r4n3{c5 c3} - a4n3{r7c3 .} ==> r1c1 ≠ 3
z-chain[3]: r4c3{n3 n6} - d2n6{r4c6 r3c7} - r3c1{n6 .} ==> r2c1 ≠ 3
z-chain[3]: r4c3{n3 n6} - c6n6{r4 r5} - r3c1{n6 .} ==> r1c6 ≠ 3
whip[1]: a6n3{r7c5 .} ==> r6c5 ≠ 3
z-chain[2]: d3n3{r5c6 r3c1} - a4n3{r3c6 .} ==> r2c3 ≠ 3
t-whip[2]: a6n3{r6c4 r7c5} - a4n3{r7c3 .} ==> r3c7 ≠ 3
t-whip[2]: d2n3{r7c3 r1c2} - d3n3{r5c6 .} ==> r7c5 ≠ 3
whip[1]: a6n3{r6c4 .} ==> r6c1 ≠ 3
z-chain[2]: a6n3{r6c4 r3c1} - r4n3{c1 .} ==> r5c4 ≠ 3
z-chain[2]: d1n3{r7c2 r6c3} - c1n3{r4 .} ==> r3c5 ≠ 3
z-chain[2]: d7n3{r7c1 r2c6} - r5n3{c6 .} ==> r7c7 ≠ 3
z-chain[2]: c7n3{r5 r6} - c1n3{r7 .} ==> r3c2 ≠ 3
z-chain[2]: a7n3{r4c3 r7c6} - r3n3{c6 .} ==> r1c3 ≠ 3
biv-chain[3]: r4c1{n2 n3} - a6n3{r3c1 r6c4} - r6c7{n3 n2} ==> r3c7 ≠ 2, r4c5 ≠ 2, r6c1 ≠ 2
whip[1]: r6n2{c7 .} ==> r7c6 ≠ 2
whip[1]: r7n2{c7 .} ==> r2c2 ≠ 2
whip[1]: r2n2{c5 .} ==> r1c4 ≠ 2
biv-chain[2]: a7n2{r6c5 r3c2} - a7n4{r3c2 r6c5} ==> r6c5 ≠ 5, r6c5 ≠ 7
whip[1]: c5n7{r7 .} ==> r2c3 ≠ 7, r7c2 ≠ 7
stte


As for the 11x11, they are not even in T&E(1) and therefore unsolvable by a manual player. It seems to be hard to find interesting puzzles.
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Thu Jun 10, 2021 5:27 am

1to9only wrote:Besides those I posted a few days ago, I also generated a few more with other (medium!) ratings:
[...]
Here are some ED=7.x from another batch run

All solved in W2 (except 1 in W3). Interesting Naked or Hidden Pairs for some, in diagonals or anti-diagonals.
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Re: Pandiagonal Latin Squares

Postby Mathimagics » Thu Jun 10, 2021 11:18 am

For Serg,

In case you missed my post on the previous page, the number of ED 13x13 pandiagonal latin squares is only 10.

The automorphism group size table is:

Code: Select all
    |A|     EDG
   ------------ 
      2       1
      6       1
     26       1
     52       1
     78       2
    156       2
   2028       1
   8112       1
  -------------     
             10

  NDG   = 12386
        = Sum ( EDG * |A| / 16224)


NDG is the number of different grids, |A| the number of automorphisms, and 16224 is the number of VPT's.
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Re: Pandiagonal Latin Squares

Postby coloin » Thu Jun 10, 2021 11:31 am

I wonder if there is a 12 in every ED grid ...
and what is the ED / whip count of the hardest !! ?
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Re: Pandiagonal Latin Squares

Postby 1to9only » Thu Jun 10, 2021 12:31 pm

I've modified the SE generator to randomly pick one of the 8 (N=11) or 10 (N=13) ED grids, and create a PanDiagonal puzzle thereof.
For 11x11 Pandiagonal, I can temporarily offer this (it has a single solution, but it is not minimal):
Code: Select all
.8.3......2.............9.........1......B..........3.............4..........................A.7......1.5...........2.... ED=7.9/2.9/2.9

SE (thus the generator) written in Java is painfully slow validating these puzzles.
I'll post the minimal puzzle later - currently an ED=8.1/2.9/2.9, SE is still churning away...
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Thu Jun 10, 2021 1:05 pm

1to9only wrote:I've modified the SE generator to randomly pick one of the 8 (N=11) or 10 (N=13) ED grids, and create a PanDiagonal puzzle thereof.
For 11x11 Pandiagonal, I can temporarily offer this (it has a single solution, but it is not minimal):
Code: Select all
.8.3......2.............9.........1......B..........3.............4..........................A.7......1.5...........2.... ED=7.9/2.9/2.9

This is in W2.
Total solving time by CSP-Rules: 6.55s. Unfortunately, for larger and harder puzzles, due to the high branching factor of Pandiags, it can take much longer (4 mins for the 13x13 puzzle in W4 I mentioned before.)


1to9only wrote:SE (thus the generator) written in Java is painfully slow validating these puzzles.
I'll post the minimal puzzle later - currently an ED=8.1/2.9/2.9, SE is still churning away...

Can you keep the intermediate puzzles? I'm interested in a 8.1 or higher. For a solver, minimality is not really relevant.
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Thu Jun 10, 2021 1:07 pm

coloin wrote:I wonder if there is a 12 in every ED grid ...
and what is the ED / whip count of the hardest !! ?

I can compute the W rating. But the hardest in which list?
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Re: Pandiagonal Latin Squares

Postby Mathimagics » Thu Jun 10, 2021 1:09 pm

coloin wrote:I wonder if there is a 12 in every ED grid ...
and what is the ED / whip count of the hardest !! ?

I have found a 12-clue puzzle for every ED grid:

12C: Show
Code: Select all
.......................................................C..................7.......4........9...........A.1......6.....8...............B.3.......5........2...............
....................................1..A8........5.........D..6........3......9..................4.....................................................2........7.......B
.........A........9B..........8.........................7C2..................4.................................5....3....................1..............................6
...........C......................D...........1..4...................2......................3..............7.............A......................B.................5..89..
.......................D.....8.....1...........2..........................................C67...........4....................A.......B....3.5............................
........................1......................5...........2.....D............4...8...................................3..6............AB.......................C.......7.
...........C...............................................56...A..............................3.......B.........4........9....1................8......2..............7..
...................9A.....8........4....................B....................1......3................2..D...........C......................5.7...........................
1.....78.............................................9..............D...........................5............A.......C............6.........3..B.....4...................
1.....7.................4................5.........2........................................B............3....8...C.....A................6..9............................


Not sure how to answer the second part of the question ... Denis indicated that even for N=11, the 10-clue puzzles I gave above couldn't be rated, and these N=13 puzzles are even harder
Last edited by Mathimagics on Thu Jun 10, 2021 1:22 pm, edited 1 time in total.
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Thu Jun 10, 2021 1:18 pm

Mathimagics wrote:
coloin wrote:I wonder if there is a 12 in every ED grid ...
and what is the ED / whip count of the hardest !! ?

I have found a 12-clue puzzle for every ED grid:
[...][/hidden]
Not sure how to answer the second part of the question ... Denis indicated that even for N=11, the 10-clue puzzles I gave above couldn't be rated, and these N=13 puzzles are even harder


Oh, that list! As they are not in T&E(1), the W rating is easy to compute: infinite.
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Re: Pandiagonal Latin Squares

Postby Mathimagics » Thu Jun 10, 2021 1:24 pm

denis_berthier wrote:As they are not in T&E(1), the W rating is easy to compute: infinite.

Ok, I'll try and find some harder ones, then :lol:
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Re: Pandiagonal Latin Squares

Postby denis_berthier » Thu Jun 10, 2021 2:57 pm

Mathimagics wrote:
denis_berthier wrote:As they are not in T&E(1), the W rating is easy to compute: infinite.

Ok, I'll try and find some harder ones, then :lol:

Good idea.

As you can somehow compute the SER, I think the right values for something not too easy but still solvable by a human would be between 8 and 9.
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Re: Pandiagonal Latin Squares

Postby Serg » Thu Jun 10, 2021 4:01 pm

Hi, Mathimagics!
Mathimagics wrote:... the number of ED 13x13 pandiagonal latin squares is only 10.

The automorphism group size table is:

Code: Select all
    |A|     EDG
   ------------ 
      2       1
      6       1
     26       1
     52       1
     78       2
    156       2
   2028       1
   8112       1
  -------------     
             10

  NDG   = 12386
        = Sum ( EDG * |A| / 16224)


NDG is the number of different grids, |A| the number of automorphisms, and 16224 is the number of VPT's.

I saw your post, thanks! I need some time to crosscheck your results.

Serg
Last edited by Serg on Fri Jun 11, 2021 2:53 pm, edited 1 time in total.
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Re: Pandiagonal Latin Squares

Postby 1to9only » Thu Jun 10, 2021 9:36 pm

1to9only wrote:For 11x11 Pandiagonal, I can temporarily offer this (it has a single solution, but it is not minimal):
Code: Select all
.8.3......2.............9.........1......B..........3.............4..........................A.7......1.5...........2.... ED=7.9/2.9/2.9


PanDiagonal Latin Squares, N=11:
Code: Select all
.8.3......2.............9.........1......B.........53..5...A......4.............7............A.7..1...1.5...........2....  18 ED=2.9/1.5/1.5
.8.3......2.............9.........1......B..........3..5...A......4.............7............A.7..1...1.5...........2....  17 ED=2.9/1.5/1.5
.8.3......2.............9.........1......B..........3......A......4.............7............A.7..1...1.5...........2....  16 ED=2.9/2.9/2.9
.8.3......2.............9.........1......B..........3......A......4.............7............A.7......1.5...........2....  15 ED=2.9/2.9/2.9
.8.3......2.............9.........1......B..........3......A......4..........................A.7......1.5...........2....  14 ED=7.7/2.9/2.9
.8.3......2.............9.........1......B..........3.............4..........................A.7......1.5...........2....  13 ED=7.9/2.9/2.9 <- posted

.8.3......2.............9.........1......B..........3.............4..........................A.7........5...........2....  12 ED=8.1/2.9/2.9


I've stopped the program after ED=8.1.

PanDiagonal Latin Squares, N=13:
Code: Select all
..3.......B..6....................67....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3........4....  29 ED=2.9/1.5/1.5
..3.......B..6....................67....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  28 ED=7.1/1.5/1.5
..3.......B..6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  27 ED=7.7/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  26 ED=7.8/1.5/1.5


The 25 clues grids are harder:
Code: Select all
.............6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.7/1.5/1.5
..3................................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.4/1.5/1.5
..3..........6..........................4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.7/1.5/1.5
..3..........6.....................7...................B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.0/1.5/1.5
..3..........6.....................7....4...........................3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.7/1.5/1.5
..3..........6.....................7....4..............B...................A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=8.1/1.5/1.5
..3..........6.....................7....4..............B............3..........6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.3/1.5/1.5
..3..........6.....................7....4..............B............3......A.......A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.7/1.5/1.5
..3..........6.....................7....4..............B............3......A...6..........4.....2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=8.4/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A............2.........4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=8.3/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4...............4.6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.6/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2...........6........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.1/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4..........7...BCD.2..5..D12....7........8..B....3.............  25 ED=9.4/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6............BCD.2..5..D12....7........8..B....3.............  25 ED=8.3/8.0/2.9
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7....CD.2..5..D12....7........8..B....3.............  25 ED=9.9/7.6/2.9
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...B.D.2..5..D12....7........8..B....3.............  25 unrated
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BC..2..5..D12....7........8..B....3.............  25 ED=9.1/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD....5..D12....7........8..B....3.............  25 ED=9.3/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2.....D12....7........8..B....3.............  25 unrated
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5...12....7........8..B....3.............  25 ED=9.0/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D.2....7........8..B....3.............  25 unrated
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D1.....7........8..B....3.............  25 ED=9.0/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12.............8..B....3.............  25 ED=9.1/1.5/1.5
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7...........B....3.............  25 unrated
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8.......3.............  25 unrated
..3..........6.....................7....4..............B............3......A...6...A......4.....2.........4.6........7...BCD.2..5..D12....7........8..B..................  25 ED=9.2/1.5/1.5


The first unrated grid crashed the program with an 'out of memory' error! The other unrated grids were stopped after about 5-10 minutes of processing!
No processing of 24 clues done.
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An Elegant 12C

Postby Mathimagics » Fri Jun 11, 2021 1:31 pm

.
I found this (PD13) 12-clue puzzle, which has 7 empty rows, 7 empty columns, 7 empty right-diagonals and 7 empty left-digonals. :shock:

So only 24 of the 52 houses are hit - it could well be even harder (computationally) than the others. Sadly nobody is in a position to make that assessment!

Code: Select all
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 . . . . . . D 6 . . . . .
 . . . . . . . . 1 . . . .
 . . . . . . . . . . . 2 .
 . . . 5 4 . . 9 8 . . . .
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 . . . . 3 . C . . . . . .
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 . . . . . . . . B . . A .
Last edited by Mathimagics on Fri Jun 11, 2021 6:45 pm, edited 1 time in total.
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Mathimagics
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