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I have created a mobile game called Bineroo where you need to fill in grids with black and white circles to succeed.

There are basically 3 rules to respect

  1. Equal number of whites and blacks on each column and row
  2. No more than two whites or blacks side-by-side vertically and horizontally
  3. Each column is unique. Each row is unique.

Here are examples of completed grids Grid example 1 Grid example 2

My question is the following one:

Depending on the grid size (which is necessarily an even number), what is the total number of possible grid combinations?

Thanks for your answers

DARGIL
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    I hope you don;t claim this puzzle idea as your own: https://www.binarypuzzle.com/ – Bram28 Nov 21 '22 at 16:53
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    I would do a computer brute force on $2\times 2$, $4\times 4$ and $6\times 6$ (maybe $8\times 8$ and $10\times 10$, depending on running time), then put the results into OEIS. – Arthur Nov 21 '22 at 17:18
  • Are your grids always square? Does (3) mean that no two columns can be identical, and no two rows can be identical, or does it mean that no row can be the tranpose of a column? – Paul Sinclair Nov 22 '22 at 17:55
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    @Bram28 No I absolutely don't claim that. If you look at the game description, it clearly states that this game is known under different names. – DARGIL Nov 23 '22 at 09:04
  • @PaulSinclair yes the grids are necessarily squares. Rule 3 means that we compare check equality between columns and rows separately. There is no equality check between columns and rows. I hope it is clearer :D – DARGIL Nov 23 '22 at 09:06
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    Thank you for the clarifications. I suggest you edit your post to make these points clear. Readers should not have to read through the comments to figure out the question is saying. – Paul Sinclair Nov 23 '22 at 12:20

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