B3. Some cells of a sheet of graph paper together form a rectangle. Of these cells, there are as many that do lie on the edge of the rectangle as do not.

How many cells does the rectangle contain in total? Give all possibilities.
B3. 48 and 60 The number of squares in the length of the rectangle we call $a$ and the number in the width we call $b$. We may assume that $a \geqslant b$. The total number of squares in the rectangle is $a b$ and the number of squares on the edge is equal to $2 a+2 b-4$. Given that half of the squares are on the edge, so $a b=2(2 a+2 b-4)$. Rewriting gives: $a b-4 a-4 b+16=8$. The left side can be factored, so we find that $(a-4)(b-4)=8$.

Since $a$ and $b$ are positive integers and $a \geqslant b$, the only possibilities are: $a-4=$ $8, b-4=1$ and $a-4=4, b-4=2$. The possibilities where $a-4$ and $b-4$ are negative are excluded, because then $b-4$ is at most -4 and $b$ is not positive. In other words: $a=12, b=5$ and $a=8, b=6$. For the rectangle, this gives $12 \times 5=60$ or $8 \times 6=48$ squares in total.