Queens on 11x11 Torus
(Published on 1. September 2020, 11:11 by SudokuExplorer)
If you had a go at the bonus puzzles at the bottom of my 7x7 sudoku, you would have realised that no 8x8, 9x9 or 10x10 toroidal sudoku can have the anti-queen constraint. There is a purely logical path which may use some slightly unusual logic. (I did add a few extra givens to avoid any trial and error).
We also have the toroidal constraint.
On a torus, when you walk along a row you end up back to the start of that row. Whilst if you climb up a column, you end up back to the bottom of that column. To visualise the regions in this sudoku, you may have to walk/climb through the thick grey lines at the "edges" of the sudoku. The thick black lines are the edges of the regions.
Moreover, every digit has the anti-queen constraint. This constraint is toroidal, meaning that you can move diagonally across from one side to the other side of the sudoku. For example, the 2 in the 1st row is a queen's move away from the 9 in the 10th row.
You can try this on Penpa-Edit. Have fun on this larger torus!
With a checkerboard coloured grid, you can try here on Penpa-Edit.
(Note that, apart from the two main diagonals, the diagonal changes colour when you move diagonally across from one side to the other side of the sudoku).
Hint:
The toroidal diagonals contain each of the numbers 0 to 10, because of the toroidal anti-queen constraint. This provides extra regions.
Hint 2:
A standard trick is to find two regions that have digits A in exactly two rows/columns, then that would imply that there are no more As elsewhere on those rows/columns. A similar trick may be needed but for diagonals instead of rows/columns.
Solution code: Enter the first column (top to bottom) followed by the bottom row (left to right) (using the roman numeral X in place of the number 10, 22 characters in total).
Comments
on 31. July 2021, 20:33 by SudokuExplorer
Fixed labels/tags
on 12. December 2020, 17:27 by SudokuExplorer
@Dina + @Storm: Thanks for giving it a go and for your feedback. It seems like I overestimated its difficulty, but I'm glad you enjoyed it :-)
on 12. December 2020, 05:48 by Dina
The size seemed daunting at first, but it turned out to be quite easy and very fun!
on 12. December 2020, 01:05 by Storm
@SudokuExplorer: Fun puzzle!
@dm_litv: Thanks for the interesting link :-)
on 8. October 2020, 23:49 by SudokuExplorer
@dm_litv Wow! Thanks for the link. I just had a brief look at some of the results and conjectures, and they look very interesting. I'm not too surprised that a Hungarian mathematician noticed this property. Thanks again :-)
on 8. October 2020, 22:48 by dm_litv
This fact (N must be coprime with 6) was first noticed and proven by György Pólya in 1918. The proof using the sum of the values and the squares of the permutation values is the simplest and clearest.
A good overview of the current state of the art in placement of queens problem (not just on a toroidal board) can be found here
https://www.sciencedirect.com/science/article/pii/S0012365X07010394
on 21. September 2020, 19:18 by SudokuExplorer
Added toroidal tag (it seems to have magically disappeared)
on 18. September 2020, 01:56 by SudokuExplorer
Adjusted image of sudoku
on 6. September 2020, 19:53 by SudokuExplorer
Added two hints about anti-queen constraint
on 5. September 2020, 11:09 by SudokuExplorer
If anyone does try this puzzle, do tell me what you like or dislike about the solving experience. This will help greatly if I decide to set a 13x13 puzzle.
on 2. September 2020, 16:59 by SudokuExplorer
@panthchesh Your wish was my command! I'm glad you had fun :-)
on 2. September 2020, 03:51 by panthchesh
That was very interesting! Thanks for the puzzle! :D
on 1. September 2020, 18:13 by SudokuExplorer
Added a link with a checkerboard colouring to visually aid you during the solve.
