Distinct Diagonals #1 (anti-Queen)

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Distinct Diagonals #1 (anti-Queen)

Postby Mathimagics » Tue Jun 08, 2021 10:11 am

Code: Select all
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   |   | 2 |   |   |   |   |   |   | 3 |   | C |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   |   | 5 |   | 7 |   |   |   |   |   |   | A |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   |   | C |   | 9 |   |   |   |   |   |   |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   | 4 |   | 1 |   |   | 6 |   |   |   | 2 | 8 |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   | 3 | B |   | 6 |   |   | 4 |   |   | 5 |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   |   |   | A |   | B |   |   |   |   |   | 6 |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   |   |   | 9 |   |   | 4 |   |   |   | C |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   | A |   |   | 4 | 2 |   | 3 |   | 6 |   |   |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   | 7 |   |   | 3 |   |   | C | 9 |   | 6 |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   |   | 9 |   | 6 |   |   |   |   |   |   |   |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   | 2 | B |   |   |   |   |   |   |   |   |   | 9 |
   +---+---+---+---+---+---+---+---+---+---+---+---+
   | 1 |   | 6 | B |   |   |   |   |   |   |   |   |
   +---+---+---+---+---+---+---+---+---+---+---+---+

Code: Select all
..2......3.C..5.7......A..C.9........4.1..6...28.3B.6..4..5....A.B.....6...9..4...C.A..42.3.6....7..3..C9.6..9.6........2B.........91.6B........


DD squares are like Pandiagonal squares, but much simpler ...

  • fill the grid so that each row and column has the digits 1 to 12, shown here using the {1-9ABC} system
  • no value can appear twice in any row, col or diagonal
  • diagonals are like those on a chessboard, they do NOT wrap around the grid

So, we have a slightly smaller grid (12 x 12), and since the diagonals don't wrap around, there should be no problems for P&P solvers. It might be helpful to use a grid that has checkerboard shading ...

DD squares might simply be called "anti-Queen", which implies both the Latin square and the DD properties.

This puzzle, as far as I can tell, is very close to singles-only, but not quite.

I have chose N=12 as the grid size because the only solution grids for N < 12 are all pandiagonal grids (N=5,7,11), and as we have seen these are all cyclic, and scarce.

N=12 seems to be in the "Goldilocks" zone. It 454 distinct solution grids, none of which can be cyclic, since there are no pandiagonal squares for N=12.

N=13 might be interesting to explore ... with 12,386 pandiagonal grids, I would expect millions of DD grids.
Last edited by Mathimagics on Wed Jun 09, 2021 2:34 pm, edited 1 time in total.
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Re: Distinct Diagonals #1

Postby Mathimagics » Tue Jun 08, 2021 10:33 am

N=13 might be interesting to explore ... with 12,386 pandiagonal grids, I would expect millions of DD grids.

Hmmm, perhaps not. The set of templates is the same for both PD and DD squares, since any template is a solution to the N-queens problem.

It follows then, that all DD squares are pandiagonal, when N is not a multiple of 2 or 3.

For other values of N, none of the DD squares are pandiagonal.
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Re: Distinct Diagonals #1

Postby creint » Wed Jun 09, 2021 8:02 am

Mathimagics wrote:This puzzle, as far as I can tell, is very close to singles-only, but not quite.


Can be solved with singles only. Larger bottleneck so could be easier than normal sudoku. Most of the diagonals don't have to be used.
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Re: Distinct Diagonals #1 (anti-Queen)

Postby Mathimagics » Thu Jun 10, 2021 1:58 pm

creint, can you give me the first (presumably a hidden?) single you find?
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Re: Distinct Diagonals #1 (anti-Queen)

Postby Mathimagics » Fri Jun 11, 2021 2:36 pm

(bump)
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Re: Distinct Diagonals #1 (anti-Queen)

Postby creint » Fri Jun 11, 2021 6:47 pm

First pass 9 singles:
Code: Select all
. . 2 . . . . . . 3 . C
. . 5 3 7 . . . . . 9 A
. . C . 9 . . . . . . 4
. 4 9 1 . . 6 . . . 2 8
. 3 B 7 6 . . 4 . . 5 .
. . 4 A . B . . . . 3 6
. . . 9 . . 4 . . . C .
A . . 4 2 . 3 . 6 . . .
. 7 . . 3 . . C 9 . 6 .
. 9 3 6 . . . . . . . .
2 B . C . . . . . . . 9
1 . 6 B . . . . . . . .


First one: row 3 digit 4 in last column
First 8 using normal row/column constraints
Last one 9r1c12 using the largest diagonal
Pasting it in my solver and adding diagonal and anti-diagonal constraint will solve it, did not try that?

Second pass 13 singles
Code: Select all
. . 2 . . . . 9 . 3 . C
. . 5 3 7 . . . . . 9 A
. . C 2 9 . . . . . . 4
. 4 9 1 C . 6 3 . . 2 8
9 3 B 7 6 . . 4 . . 5 .
. . 4 A . B . . . 9 3 6
3 . . 9 . . 4 . . . C .
A . . 4 2 9 3 . 6 . . .
. 7 A 5 3 . . C 9 . 6 .
. 9 3 6 . . . . . . . .
2 B . C . . . . 3 . . 9
1 . 6 B . . 9 . . . . 3
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Re: Distinct Diagonals #1 (anti-Queen)

Postby Mathimagics » Sat Jun 12, 2021 3:07 am

Thanks, creint! Found and fixed the problem ...
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Re: Distinct Diagonals #1 (anti-Queen)

Postby 1to9only » Sat Jun 12, 2021 6:57 am

Code: Select all
..2......3.C..5.7......A..C.9........4.1..6...28.3B.6..4..5....A.B.....6...9..4...C.A..42.3.6....7..3..C9.6..9.6........2B.........91.6B........ ED=1.5/1.5/1.5
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Re: Distinct Diagonals #1 (anti-Queen)

Postby Mathimagics » Sat Jun 12, 2021 8:41 am

Great!

I'll prepare some reductions, and we'll see if we can find some puzzles of interest ...
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Re: Distinct Diagonals #1 (anti-Queen)

Postby Mathimagics » Sat Jun 12, 2021 3:20 pm

Do these show a decent range of difficulty?

I have tagged with #of givens + #of singles:
Code: Select all
..2......3.C..5.7......A..C.9........4.1..6...2..3B.6..4..5......B.....6...9..4...C.A...2...6....7..3...9....9.6........2B.........91.6B........ # 36 + 7
..2......3....5.7......A..C.9........4....6....8..B.6..4..5......B.....6...9..4...C.A..42.3.6....7.....C..6....6........2B.........9..6B........ # 33 + 2
..2......3....5.7......A..C.9........4....6....8..B.6..4..5......B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 30 + 3
..2......3....5.7.........C.9........4.........8..B.6..4.........B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 27 + 2
.........3....5.7.........C.9........4.........8..B.6..4.........B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 26 + 1
..............5.7.........C.9........4.........8..B.6..4.........B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 25 + 1
..............5.7...........9........4.........8..B.6..4.........B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 24 + 1
..............5.7....................4.........8..B.6..4.........B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 23 + 0
..............5.7..............................8..B.6..4.........B.....6...9..4...C.A..42.3.6..................6........2B.........9..6B........ # 22 + 0
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