A rectangle is cut by one horizontal line and two vertical lines into six small rectangles, arranged in 2 rows and 3 columns. Counting rectangles of every size, how many rectangles does the figure contain?

A rectangle is fixed by choosing 2 of the 3 horizontal lines and 2 of the 4 vertical lines. There are 3 ways to pick the pair of horizontal lines and 6 ways to pick the pair of vertical lines, giving $3\times 6=18$ rectangles. Counting by size gives the same: six $1\times 1$, four $1\times 2$ wide, two $1\times 3$ wide, three $2\times 1$ tall, two $2\times 2$ and one $2\times 3$, that is 6 + 4 + 2 + 3 + 2 + 1 = 18. The trap 6 counts only the smallest cells; 12 misses the rectangles that are two rows tall.