| We consider the problem of tiling large rectangles using smaller rectangles with the prescribed dimensions 4 x 6 and 5 x 7. Problem B-3 on the 1991 William Lowell Putnam Examination asked "Does there exist a natural number L such that if m and n are integers greater than L, then an mxn rectangle may be expressed as a union of 4 x 6 and 5 x 7 rectangles, any two intersect at most along their boundaries?" Narayan and Schwenk showed in 2002 that all rectangles with length and width at least 34 can be partitioned into 4 x 6 and 5 x 7 rectangles. We investigate necessary and sufficient conditions for an m x n rectangle to be tiled with 4x6 and 5x7 rectangles. Ashley et al. answered this question for all but 37 cases. We use an integer linear programming approach to eliminate all but five of these cases. |
