, 2024

Solution:

For convenience, say the edge of length $20$ is vertical.

Consider some vertical line inside the rectangle. It must pass through rectangles of some heights adding to $20$. In particular, these heights correspond to ways to add up to $20$ with fours and fives, which is either $4+4+4+4+4$ or $5+5+5+5$. These options correspond to columns of rectangles with width $5$ or $4$, respectively. In particular, we need to span the width of the original $20 \times 24$ rectangle using these columns, meaning that we can just count the number of ways to add to $24$ with fours and fives.

There are two ways to do this: either $4+4+4+4+4+4$ or $4+5+5+5+5$. Considering the orders in which the second sum can be written, we get an answer of $1+5=6$.