I'm currently working on fitting together the nets of a cube into squares as efficiently as I can. If I have a $N$ by $N$ square grid, how many cube nets can I fit into it, assuming each square of the nets is the same as a square on the grid? Is there a specific equation, or is it trial and error and case by case? Are there any patterns that arise across different values of $N$? Is there a way to classify different sizes of squares, for example maybe every even value of $N$ results in an even value of nets, or maybe every prime value of $N$ results in an odd value of nets. This question is far beyond my comprehension and any help at all is appreciated.
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1Here is a related question. There won't be a single pattern that works for all N, but there may well be several patterns that together give solutions for all N. – Jaap Scherphuis Feb 22 '23 at 17:06
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1I had my computer solve the smaller cases N=4,5,...,12, which gave solutions with 2,3,4,7,9,12,15,19,22 nets. I don't know if the last one is optimal, but all the others are. – Jaap Scherphuis Feb 22 '23 at 17:50