# No Numbers Sudoku: The Ball of Queens

Those are all of the numbers you need. No, I did not forget any.

• Normal sudoku rules apply.
• All numbers, except for 1's and 9's, are queens with respect to themselves.
• E.g. 2's have the queen's constraint against other 2's but not against 3's.
• All numbers must be rotationally symmetric with respect to their counter part on the other side of 5.
• Aka, 1 with 9, 2 with 8, 3 with 7, 4 with 6, and 5 with itself. See below image for an example.
• Not all 2's are next to non-queens in the top three sudoku squares. Aka, the white sudoku squares in the image below.
• All 2's must be next to a 4 in the three middle sudoku squares. Aka, the three blue sudoku squares in the image below.
• All 2's must be next to a 3 in the three bottom sudoku squares. Aka, the three red sudoku squares in the image below.
• If a 2 is surrounded by 3 non-queens, it will see two 9's.
• Top right corner is not a queen.
• The queens of 5 dance clockwise.

"Next to" includes above, below, to the side, or diagonally.

This is my first sudoku puzzle, so any feedback would be awesome. Good luck!

EDIT Apologies, I noticed another symmetry within the puzzle that needed defining to gift a unique solution. I've added 1 last rule at the bottom to account for this.

HINTS

r1c3, r2c9, r3c4, r4c8, r5c5

• Is "next to" also diagonally? – Jens Sep 20 '20 at 19:19
• Yes, I'll amend the description to be a bit clearer. Thanks. – user3303504 Sep 20 '20 at 19:22
• And "rotationally symmetric" means "at a multiple of 45 degrees within each 3x3 box", correct? – Jens Sep 20 '20 at 19:28
• No, it means the entire sudoku grid could be rotated and the numbers would fall onto their counterparts perfectly. According to Wikipedia its "Automorphic", but I wasn't sure about using jargon ( en.wikipedia.org/wiki/Mathematics_of_Sudoku#Automorphic_Sudokus ) – user3303504 Sep 20 '20 at 19:32
• So rotationally symmetric means a 2-fold symmetry (180 degrees), not a 4-fold (90 degrees) symmetry. What do you mean by "dance clockwise"? – Jaap Scherphuis Sep 21 '20 at 13:59

I wrote a program to solve this. I found 15 solutions. Here are two of them. As the solution should be unique, I tried to hunt for violation of the rules. But I could not found any.

  8  9  5 |  6  7  2 |  3  4  1
1  7  2 |  3  4  8 |  6  9  5
3  4  6 |  5  9  1 |  2  7  8
----------+----------+---------
7  2  3 |  1  8  6 |  9  5  4
4  8  9 |  7  5  3 |  1  2  6
6  5  1 |  4  2  9 |  7  8  3
----------+----------+---------
2  3  8 |  9  1  5 |  4  6  7
5  1  4 |  2  6  7 |  8  3  9
9  6  7 |  8  3  4 |  5  1  2

6  1  5 |  3  2  4 |  8  7  9
9  2  4 |  8  7  6 |  3  1  5
8  7  3 |  5  1  9 |  4  2  6
----------+----------+---------
2  4  8 |  9  6  3 |  1  5  7
7  6  1 |  2  5  8 |  9  4  3
3  5  9 |  7  4  1 |  2  6  8
----------+----------+---------
4  8  6 |  1  9  5 |  7  3  2
5  9  7 |  4  3  2 |  6  8  1
1  3  2 |  6  8  7 |  5  9  4


• "All numbers must be rotationally symmetric with respect to their counter part on the other side of 5." on the first line of your first solution you have "8 9 5 6 7" but 8 is not the counterpart of 7 and 9 is not the counterpart of 6. – Stef Sep 30 '20 at 11:06
• @Stef this is not how I read that condition. I though it meant that the top-left 8 has their counterpart in the bottom-right 2 etc – daw Sep 30 '20 at 11:15
• @daw okay, you put my programming skills to shame! I thought I had checked all possibilities as well but clearly not. FYI, the first sudoku you put up was the intended solution. – user3303504 Oct 2 '20 at 8:38