# Write digits 1 to 9 in the table, each pointing to next 3x3 grid

Write the digits from 1 to 9 in an empty Sudoku table with these rules (similar to 9-Queen problem):

• There are 9 boxes indexed from 1 to 9 in the image below. Each box has a 3x3 grid child.
• Every two digits should NOT be in the same row, column or diagonal.
• Pointing rule: You have to place the next number in an specific box. (e.g. if you start to place 1 in top-left box and middle-center child grid, you have to continue with 2 in middle-center box)
• You are free to begin from any box.
• Close the loop: The last number (9) should point to the box that you have placed the first number (1) in it.
• At the end there will be a Sudoku table with only 9 digits in it each in a separate 3x3 box. (other 72 cells will remain empty)

Write down the 9 digits with the rules, or if there is no such a solution, prove it!

See the following example that have some mistakes:

1. 8 and 9 are in the same row.
2. 8 and 5 are in a diagonal.
3. 5 and 2 are in the same row.
• Can you define box? Is the puzzle made up of 81 boxes or 9 boxes? Commented Sep 19, 2014 at 13:48
• there are 9 Boxes.
– Rafe
Commented Sep 19, 2014 at 14:03
• You say no duplicates in a row, column, or diagonal; do you mean "box" instead of "diagonal"? If not, does that mean the box rule doesn't necessarily apply? It's best to edit the question if you think something was unclear, so that people don't have to read the comments. Commented Sep 19, 2014 at 14:51
• @EnvisionAndDevelop Don't worry buddy! I will edit the question each time I see a misunderstanding. Edit added: * At the end there will be a Sudoku table with only 9 digits in it (other 72 cells will remain empty) each in a separate 3x3 box
– Rafe
Commented Sep 19, 2014 at 15:18
• Ah, I see, so it's almost like a "9 queens on a special chessboard" problem. I had assumed the goal was to create a valid sudoku board by following certain rules about how to fill in numbers. Commented Sep 19, 2014 at 16:01

It is not possible.

Any solution to this puzzle must also be a solution to the 9x9 Queens puzzle. Luckily, that is a well-known puzzle. It has 352 solutions, but due to symmetry, those 352 solutions can be reduced to 46 solutions. After that, it is just a matter of checking against those 46 solutions.

I found a page showing the solutions at: http://stamm-wilbrandt.de/en/xsl-list/n-queens/n-queens.xsl.xml

I then looked at the first answer in each of the 46 columns.

Items that quickly ruled a solution out:

1. Queen in a corner. Any queen in a corner would mean that the box referred to itself.
2. Queen in the center of a side. Again, that box referred to itself.
3. Queen in center box. Same reason.
4. 2 Queens in same 3x3 box / empty 3x3 box.

This left me with very few actual solutions to check. From that page, #'s 19, 20, 31, 39, 42.

To check, I simply started in box 1 and followed the pattern until I returned to a box that I'd already seen. None of those took all 9 boxes.

• Useful link. the closest answers were #20 and #39. where each queen is in separate box and is not pointing to itself.
– Rafe
Commented Sep 20, 2014 at 5:35