8 different rectangles with area in the teens forming an 11x11 square: AABBCCCCCCC AABBCCCCCCC AABBDDDDEEE AABBDDDDEEE AABBDDDDEEE AABBDDDDEEE AAFFFFFFGGG AAFFFFFFGGG AAFFFFFFGGG HHHHHHHHGGG HHHHHHHHGGG 11112222333 11112222333 11112222333 11112222333 44556677333 44556677888 44556677888 44556677888 44556677888 44556677888 44556677888 11111111333 11111111333 22222222333 22222222333 44556677333 44556677888 44556677888 44556677888 44556677888 44556677888 44556677888 You have to use exactly eight rectangles, although I don't know for sure how many different shapes can be used yet. This assumes you have integer sides. Some nice Mathematica code to simplify the search: (*teen areas that will fit inside an 11x11 rectangle*) teens = {14, 15, 16, 18}; (*The maximum number of any rectangle that will fit inside an 11x11 rectangle max = 121/teens//Floor; (*Brute force find all the combinations that add upto 121*) (*This is fast: there are only 2688 possibilities*) tab = Table[{i,j,k,l}.teens, Evaluate[Sequence@@Transpose[{i,j,k,l}, max}]]]; Position[tab, 121] yields {{3, 3, 1, 1}, {4, 1, 2, 1}} as the only combinations of areas that work. The two given above come from the second solution. Brett Champion