Cake Division

You have a rectangular cake of length $\text{length}$ and width $\text{width}$. You want to divide it into $\text{pieces}$ equal-area rectangular pieces. Each cut must be parallel to the sides of the cake, and must completely split one of the existing pieces into two. (To divide the cake into $n$ pieces, you need exactly n-1 cuts.) You prefer pieces that are as close to square as possible, i.e., with the smallest possible aspect ratio. The "aspect ratio" of a piece is defined as the ratio of its longer side to its shorter side. You need to cut the cake so as to minimize the maximum aspect ratio among all pieces.

For example, a 2 \times 3 cake can be divided into $6$ pieces of size 1 \times 1. Each has aspect ratio $1.0$, which is the minimum possible, so this is optimal. Another example: to cut a 5 \times 5 cake into $5$ pieces, one possible way is to first cut it into two pieces of sizes 2 \times 5 and 3 \times 5. Cut the smaller into two (2 \times 5/2 each), and the larger into three (3 \times 5/3 each). The largest aspect ratio is on the 3 \times 5/3 pieces: 3/(5/3) = 1.8. It is not possible to split a 5 \times 5 cake into $5$ equal-area pieces with a smaller maximum aspect ratio.