Christian wrote: << Frank Rubin asks this interesting problem: --------------------------------- "It is possible to write a square on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a square. For example, write 1 on two opposite faces, and 4 on the other 4 faces. The sum of the 3 faces surrounding each vertex is then 1+4+4=9, a square. Can this be done using the squares of 6 distinct whole numbers? If so, what set of 6 such squares has the smallest sum?" . . .
On the principle that it's always easier to ask more questions than to answer one, how about the other 4 regular polyhedra? How about the 13 Archimedean solids? How about the regular and Archimedean tilings of the plane? And last (and least), how about regular polygons? (I.e., for a given n >= 3, do there exist distinct nonnegative integers {X_k | 0 <= k < n} such that for each index k we have (X_k)^2 + (X_(k+1))^2 is a square number?) -------------------------------------------------------------------------------- And there's no reason not to extend these questions to other regular tilings of surfaces or even higher-dimensional things: E.g., given the tiling of a torus by a rosette of 7 hexagons, same question for the 14 vertices. Or given a 4D regular solid -- say the 24-cell (composed of 24 octahedra, 6 per vertex), can we assign a square number to each of the 24 octahedra such that the sum of the 6 squares in the octahedra containing any vertex v add up to a square number? (Or why stick with only square numbers; why not higher powers?) --Dan