How many ways are there to cover an 8 by 8 square with 32 identical 2 by 1 rectangles? This is different from this question because the layout can be recursive on two dimensions. I would start from a 2 by 2 square, then 3 by 3. However, from there I got lost how to write the recursion equation. Also, what if the area to cover is not a square, but a rectangle say 12 by 8?
2026-03-27 00:04:51.1774569891
How many ways are there to cover an 8 by 8 square with 32 identical 2 by 1 rectangles?
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For $m\times n$ with $m,n$ odd, you cannot cover the rectangle with dominoes because there are an odd number of little squares and each domino covers two squares. For general $m\times n$ the sequence is given here, and for the special case of $2n\times 2n$ here. Unfortunately, there is no "nice" formula even in the $2n\times 2n$ case. The answer for the $8\times 8$ case that you asked is $12988816$.