Solving Scramble Squares puzzles with repetitions

Abstract

A Scramble Squares puzzle consists of nine square pieces with half of an image on each side. To solve the puzzle the pieces are arranged in a $3\times 3$ grid so that sides of adjacent pieces form complete images. A repetition is a half-image that appears more than once on a piece. Previous research uses a graph-theoretical approach to establish necessary and sufficient conditions for solutions without repetitions to $2\times 2$ Scramble Squares puzzles. We use a similar approach to establish necessary and sufficient conditions for solutions with repetitions to $2\times 2$ Scramble Squares puzzles.