Logic Masters Deutschland e.V.

Queens on 7x7 Torus

(Eingestellt am 13. August 2020, 00:00 Uhr von SudokuExplorer)

I lately have had lots of fun solving sudokus on a torus as well as on klein bottles. This inspired me to create this sudoku on a torus. Do check out Rotstein's stunning toroidal chess sudoku series. I feel that some of them deserve more solves. If you want another anti-queen sudoku I recommend trying emmettcito's lovely imperfect anti-queen sudoku (which coincidentally also has 7 queens).

Anyone who finds this easy, can try the larger Queens on 11x11 Torus sudoku. If you want a similar puzzle but with a twist, do try my Gossiping Queens on 7x7 Torus puzzle. Below this sudoku, there is also a bonus puzzle to do with this construction. There is also a bonus bonus puzzle about existence. Do give them a try and tell me how you find them (in a hidden comment).

Normal sudoku rules apply (each row, column and region contains the numbers 1 to 7).

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 the "edges" of the sudoku. For example, in this sudoku, the given digits 5 and 6 are both a queen's move away from 2.

You can try this on Penpa-Edit. Have fun on this torus!
Bonus Puzzle:
Show that there are two anti-queen patterns on this 7x7 torus (after permutations of the digits 1 to 7).
The above sudoku gives one pattern, what is special about the other one?

Bonus Bonus Puzzle:
Show that if a n x n toroidal sudoku has the anti-queen constraint, then n is coprime to 6.
Hint: Express the anti-queen constraint in the language of permutations modulo n. You might first want to show that n must be odd.

Lösungscode: Enter the 5th and last rows (left to right).

Zuletzt geändert am 31. Juli 2021, 20:32 Uhr

Gelöst von Greg, NikolaZ, aneoid, panthchesh, glum_hippo, skywalker, zorant, adam001, MartinR, Narayana, Qodec, marcmees, cdwg2000, CHalb, Isa, kamea92, saskia-daniela, zzzzz, zhergan, mackerel, Zzzyxas, moss, ... Uhu, Joo M.Y, ManuH, Matt, rimodech, Thomster, Storm, Bobby, AsilG, Gotroch, Raistlen, ParaNox, flaemmchen, Vebby, derKrampus, zrbakhtiar, Jordan Timm, Gniffel, Awoobledoople, Crul, shadow-nexus
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Kommentare

am 31. Juli 2021, 20:32 Uhr von SudokuExplorer
Fixed labels/tags

am 21. September 2020, 19:16 Uhr von SudokuExplorer
Added toroidal tag (it seems to have magically disappeared)

am 11. September 2020, 19:49 Uhr von SudokuExplorer
Added link to recent Gossiping Queens on 7x7 Torus puzzle.

am 4. September 2020, 01:38 Uhr von SudokuExplorer
Added link to 11x11 Anti-Queen Toroidal Sudoku

am 16. August 2020, 16:20 Uhr von SudokuExplorer
Edited hint of second bonus puzzle

Zuletzt geändert am 14. August 2020, 20:11 Uhr

am 14. August 2020, 14:29 Uhr von SudokuExplorer
@glum_hippo I have been thinking a bit about an 11x11 version. So far I have two possible 11x11 toroidal anti-queen latin squares (after translation, rotation, reflections and digit permutations). I'll also think about a 13x13 version (to see if I can find more interesting patterns). Maybe I'll try a bit more number theory and see how far I can get, unless anyone else has different ideas to construct such patterns. Be prepared to scream in two weeks time! ;-)

am 14. August 2020, 14:04 Uhr von glum_hippo
It's a lovely idea. If I see an 11x11 all-anti-queen sudoku I will scream, however.

am 14. August 2020, 14:01 Uhr von SudokuExplorer
Added hint for bonus bonus puzzle.
Do think about the two bonus puzzles.

am 13. August 2020, 21:13 Uhr von SudokuExplorer
Added bonus bonus puzzle

Zuletzt geändert am 13. August 2020, 21:09 Uhr

am 13. August 2020, 21:08 Uhr von SudokuExplorer
@panthchesh Btw I just did a bit of number theory, and managed to show that if a n x n toroidal sudoku has n queens, then n has to be coprime to 6. I was really lucky to construct this 7x7 anti-queen toroidal sudoku :-D

Zuletzt geändert am 13. August 2020, 11:24 Uhr

am 13. August 2020, 09:18 Uhr von SudokuExplorer
@panthchesh and @aneoid I'm pleased you enjoyed it :-)
I'm trying to make a couple of more approachable puzzles, after my last few harder sudokus. You might see a bigger version in two weeks time (next week I will post my 9x9 killer pentuplet sudoku).

am 13. August 2020, 03:50 Uhr von panthchesh
Thanks that was fun! Next time, consider a larger version! :D

am 13. August 2020, 03:31 Uhr von aneoid
Thanks for another fun puzzle, SE.

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ID:00042B

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