The Mutando of Insanity

Abstract: Puzzles based on coloured cubes and other coloured geometrical figures have a long history in the recreational mathematical literature. One of the most commercially famous of these puzzles is the Instant Insanity that consists of four cubes. Their faces are coloured with four different colours in such a way that each colour is present in each one of the four cubes. To solve the puzzle, one needs to stack the cubes in a tower in such a way that each one of the colours appears exactly once in the four long faces of the tower. The main purpose of this paper is to study the combinatorial richness of a mathematical model of this puzzle by analysing all possible ways of colouring the cubes to form a puzzle analogous to the Instant Insanity. We have done this analysis for $n$ cubes and $n$ colours for $n=4, 5, 6$. This combinatorial analysis allowed us to design the Mutando of Insanity, a puzzle that we presented at Gathering for Gardner 12 (G4G12).