MJD wrote a few years ago about a problem on Math.SE: "John has 77 boxes each having dimensions 3x3x1. Is it possible for John to build one big box with dimensions 7x9x11?"
He solves the problem by noticing that a single face can be filled only by rectangles of area 3 or 9, but one face has area 77–not a multiple of three.
This solution reminds me of the mutilated chessboard problem. Can a chessboard be tiled with dominoes, if two opposite corners are cut out?