"But how about the famous expression 1^2 + 2^2 + 3^2 + ... + L^2 = N^2 for L = 24 and N = 70, the unique nontrivial such equation. QUESTION: --------- Can a 70x70 square torus be tiled with one KxK square tile for each K in the range 1 ? K ? 24 ??? Possibly in a non-parallel fashion? ?Dan " Tiling or dissecting a given shape, with successive single squares of sides 1,2,3... n seems difficult - there are no known examples, however if we consider a 'patch' of tiles, a collection of squares, adjacent to each other, except at the boundary of the patch then it is possible. Adam Ponting has a method for tiling 1 to n^2 squares where n is an odd number. The key is to create a chessboard like nxn matrix with entries 1 to n^2 and then translate adjacent matrix entries into adjacent squares. see https://demonstrations.wolfram.com/PontingSquarePacking/ http://www.adamponting.com/squaring-the-plane/ http://www.adamponting.com/square-packing/ http://www.adamponting.com/update-ponting-packing/ http://www.adamponting.com/rectiling-i/ Stuart