r/mathriddles • u/pichutarius • 2d ago

Medium torus

(this is a follow up question from this question inspired by u/blungbat)

the image shows 3 rectangular grids on torus, i.e. (Z/nZ)^2 .

each torus is partitioned into two regions. each region is connected and does not contain 2x2 subregion.

unfortunately the second and third grid is partially destroyed. their size is 7x7 and 8x8 in an unknown order.

how many blue tiles are there in each torus?

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u/LaplacesDemonLab 2d ago

https://reddit.com/link/pdrrrkx/video/qagcfhs0veth1/player

You don’t need to reconstruct the missing board to count its tiles. Here’s a short visual proof for the third grid.

Those two checkerboard corners force an equal split: 32 blue tiles.!

Solution spoilers in the video.

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u/pichutarius 1d ago

What about the other torus?

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u/LaplacesDemonLab 10h ago

If you mean the other two boards: same principle. I used the third one because it’s the cleanest example, but none of them actually need to be reconstructed.

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u/pichutarius 7h ago

true, although that is not what was asked, the task is to find how many blue tiles in each torus.

solution is expected to give two numbers, one for each torus.

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u/LaplacesDemonLab 6h ago

You're right — I went back and checked the other torus properly.

The answers are 24 and 32 blue tiles.

The right-hand fragment must be the 8×8 torus: the two visible checkerboard 2×2 corners force both color adjacency graphs to have no cycles, so the areas are equal: 32 + 32.

That leaves the thin fragment as the 7×7 torus. There, the visible yellow cells already form a closed cycle around the torus — down one side of the surviving strip and back up the other through the periodic boundary. So yellow has one cycle and blue has none. This gives an area difference of one tile. Since there are 49 tiles total, that's 25 yellow and 24 blue.

So:

7×7 → 24 blue

8×8 → 32 blue

Thanks for making me look at the second one more carefully!

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u/pichutarius 6h ago

well done