On a generalization of "Eight Blocks to Madness" puzzle

We consider a puzzle such that a set of colored cubes is given as an instance. Any cube is the unit length on each edge and its surface is colored so that what we call Surface Color Condition is satisfied. Given a palette of 6 colors, the condition requires that each face should be assigned exactly one color and all faces should be assigned 6 different colors from each other. The puzzle asks to compose a 2×2×2 cube that satisfies Surface Color Condition from 8 suitable cubes in the instance. Note that cubes and solutions have 30 varieties respectively. In this paper, we present a collection of our research results on this puzzle. First we present a novel necessary and sufficient condition that a solid of a given variety is composable. This condition immediately leads us to succinct procedures to identify the composability and to decide the assignment of cubes to the corners of the solid. Next we create two extreme instances: a universal instance (i.e., one having all 30 solutions) and an infeasible instance (i.e., one having no solution). We show the minimum size of the universal instance to be 12 and the maximum size of the infeasible instance to be 23. The latter result affirmatively solves one of the open problems that were raised in [E. Berkove et al., “An Analysis of the (Colored Cubes) Puzzle,” Discrete Mathematics, 308 (2008) pp. 1033–1045].