Vertex labeling of a half-cube to induce face labels in arithmetic progression

Abstract

A plane, passing through the center and orthogonal to a diagonal, slices a cube into two identical halves each having three triangles, three pentagons and a hexagon. If you label the ten vertices of a half-cube with numbers, then each face is said to receive an induced face label given by the sum of all vertices around it. Label the ten vertices of a half-cube with digits 0 through 9 so that the induced labels of the three triangles and the three pentagons constitute six consecutive multiples of a fixed number, or more generally six numbers in arithmetic progression.