# uniform rectangle packing

I'm wondering if there is a formula or algorithm how many uniform rectangles of size $$a\times b$$ I can put in a square of size $$c\times c$$.

Does anyone have a hint for literature?

• A necessary condition is that $N \times a \times b=c^2$ for a certain $N$, i.e., $c^2$ has to be divisible by $a \times b$. Is it a sufficient condition ? Probably, but I am not sure. Jun 26, 2020 at 21:21
• See slides 27 and following here (obtained by using query "generalized domino tilings") Jun 26, 2020 at 21:49
• Packing problems are hard. It doesn't do this one, concentrating on circles, but packomania.com will give you an idea that the configurations of interest are not always regular. This one has more regularity because of the rectangles. Jun 27, 2020 at 1:40
• @JeanMarie: I wish I had heard the talk. Thanks. I think OP doesn't mean to require that we cover the whole rectangle, just to find the maximum number of tiles that will fit. The talk falsifies your condition by stating that $10 \times 15$ cannot be tiled by $1 \times 6$ Jun 27, 2020 at 1:48
• See as well the reference given in the answer by Robert Israel here Jun 27, 2020 at 13:45