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A rep-tile is a polygon that can be dissected into smaller copies (of thesame size) of the original polygon. A polyomino is a polygon that is formed byjoining one or more unit squares edge to edge. These two notions were firstintroduced and investigated by Solomon W. Golomb in the 1950s and popularizedby Martin Gardner in the 1960s. Since then, dozens of studies have been made incommunities of recreational mathematics and puzzles. In this study, we firstfocus on the specific rep-tiles that have been investigated in thesecommunities. Since the notion of rep-tiles is so simple that can be formulatedmathematically in a natural way, we can apply a representative puzzle solver, aMIP solver, and SAT-based solvers for solving the rep-tile problem in common.In comparing their performance, we can conclude that the puzzle solver is theweakest while the SAT-based solvers are the strongest in the context of simplepuzzle solving. We then turn to analyses of the specific rep-tiles. Using someproperties of the rep-tile patterns found by a solver, we can complete analysesof specific rep-tiles up to certain sizes. That is, up to certain sizes, we candetermine the existence of solutions, clarify the number of the solutions, orwe can enumerate all the solutions for each size. In the last case, we find newseries of solutions for the rep-tiles which have never been found in thecommunities.

Solving Rep-tile by Computers: Performance of Solvers and Analyses of Solutions