There are three types of rectangular tiles : 3' × 3', 3' × 7' and 3' × 11'. An area of rectangular shape of dimensions 3' × 100' is to be covered using these tiles without breaking them. If x and y are the maximum and minimum numbers of tiles of various sizes, respectively, that can be used to cover the area exactly, then x - y is<br/>

Let the numbers of 3', 7', 11' tiles be a, b, c. Then 3a + 7b + 11c = 100. Total tiles n = a + b + c = (100 - 4b - 8c)/3. So n is maximized by minimizing 4b + 8c: take b = 1, c = 0, giving a = 31 and n = 32. For minimum n, use as many longer tiles as possible. 10 or 11 tiles are impossible, but 12 tiles are possible: c = 8, a = 4, b = 0 gives 88 + 12 = 100. Hence x = 32, y = 12, so x - y = 20.