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Consider a square grid of size n×n. Your task is to dissect it into four congruent polyominos, each of which has the same perimeter as the square itself. What’s the smallest value of n for which you can do this?

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  • $\begingroup$ Considering one polyomino, does it have to be connected? $\endgroup$ Commented May 10 at 3:04
  • $\begingroup$ @Will.Octagon.Gibson. I thought any polyomino is connected by definition. If not, then yes, it should be connected. $\endgroup$ Commented May 10 at 3:08
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    $\begingroup$ Polyominoes are defined by Golomb as a collection of squares joined together along their edges, so they are connected by definition. $\endgroup$ Commented May 10 at 6:26

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The minimum is 8:
polyomino dissection

For proof $n<8$ is not possible:
Each polyomino must have area $A=n^2/4$ (which must be an integer, so $n$ is even) and perimeter $P=4n$. But for polyominoes, $P\le2A+2$ so $2, 4, 6$ are ruled out.

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  • $\begingroup$ Correct! I came up with a different polyomino for n=8: i.sstatic.net/658OjnCB.png $\endgroup$ Commented May 10 at 9:39
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    $\begingroup$ @Pranay You're right, I forgot it has to be exact. $\endgroup$ Commented May 10 at 18:44

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