Candy bar

Susie has a chocolate candy bar consisting of 3 rows with 6 squares in each row. She notices that the area of the candy bar is 18 squares of chocolate and the perimeter of the bar is also 18 units. Are there other rectangular chocolate bars that also have the same number of square units for the area as they have units for the perimeter? (Andrea Peter-Koop)

Approaches

If the rectangle has integer dimensions a and b, we have ab =2a +2b, so a(b – 2) = 2b, making a = 2b/(b – 2) = 2 + 4/(b – 2). For a to be an integer, b – 2 must be a divisor of 4. So the pair (b, a) must be (3, 6), (4, 4), or (6, 3).

The edges of squares on the perimeter must match the squares themselves. Each square around the outside contributes one edge to the perimeter, except for the four corners, which contribute an extra edge each. Therefore there can be only four squares in the middle, contributing no edges to the perimeter. These four squares can be arranged in a row, as in the 3 x 6 rectangle, or in a 2 x 2 square making up the center of a 4 x 4 candy bar.