Tiling a rectangle with $n$ rectangles to maximize product of areas

Question

Consider a rectangle $R$ with integer width and integer height.

We want to tile $R$ using exactly $n$ rectangles with integer dimensions.

Now we carefully want to choose such rectangles so that $\max\ (w_1\times h_1)\times \ldots \times (w_n\times h_n)$ for every rectangle $(w_i, h_i)$.

What approaches are there to tackle this optimization problem! Thank you!

Answer 1

You have a set of $n$ numbers that add up to the area of $R$. To maximize their product, you would like them as equal as possible. If the area of $R$ is divisible by $n$, make all the rectangles the same size if possible. I don't know any other advice to give.