Cube and Dice — Fundamental Concepts

Fundamental Concepts

Begin with the geometry. Three faces around one corner are pairwise adjacent. Opposite faces cannot share an edge or a vertex. If a label has four confirmed distinct neighbours, the remaining label is its opposite. Fewer neighbours may leave more than one possibility, so distinguish partial information from a complete arrangement.

For a net, hold one square fixed and track each fold through a right angle. In the original example A–B–C–D, with E above B and F below B, holding B as front gives A left, C right, D back, E top and F bottom. The opposite pairs are A–C, B–D and E–F. Distances on the flat drawing are not a substitute for this mapping.

For a fully painted cube divided into n parts along each edge, count by position. With n at least two, corners contribute eight three-painted cubes; non-corner edges contribute 12(n − 2) two-painted cubes; face interiors contribute 6(n − 2)² one-painted cubes; and the internal block contributes (n − 2)³ unpainted cubes.

At n = 4 the counts are 8, 24, 24 and 8, summing to 64. Changed painting conditions require a fresh classification. If a small cube is selected at random, use these counts as probability numerators only after confirming uniform selection.