Crazy-Cuts: From Theory to App

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg, 380, Stanford, CA 94305-2125, USA e-mail: [email protected] C razy-Cut puzzles are fascinating and often quite challenging: Given a planar shape (as for example depicted inFigure 1a), finda cutting curve thatdivides the shape into two parts, identical up to Euclidean transformations (rotations and translations). The solution, as seen in Figure 1b, is not trivial to find. Several further Crazy-Cut dissectionpuzzles are shown inFigures 2and12.Apparently, suchpuzzles were invented byHenry Dudeney (see Eriksson [1]) about one hundred years ago, but it was Martin Gardner whopopularized them in the 1970s in his Scientific American column [2] and in his books [3]. Gardner also posed the challenge of finding a formal algorithm for solving Crazy-Cut puzzles. In this article, we describe some recent work on algorithms for solving Crazy-Cut challenges that led to the designofapuzzlegameforsmart-phonesandpadcomputers. Eriksson was the first to propose an algorithm for solving polygonal Crazy-Cut challenges [1]. In Eriksson’s algorithm, two points on the perimeter of the shape are selected, and, starting from these points, two congruent paths are constructed, one of them (the ‘‘master’’) leads by following the boundaryof the shape,whereas the secondpath (the ‘‘slave’’) is led by the congruence constraint and is allowed to cross the shape. If the two starting points happen to be in ‘‘correspondence’’ when there is a solution to the challenge, the ‘‘slave’’ path will split the shape into two identical pieces; see an example in Figure 3. Eriksson proved that by checking all possible pairs of points, one will find the correct cut if such a cut exists or will determine that the shape cannot be cut into two identical parts. Following Eriksson’s work, Rote, with co-workers, improved and further elaborated upon Eriksson’s algorithm [4, 5]. El-Khechen et al. then proposed an algorithm for a variant of theproblem inwhich the twopieces are required to be mirror congruent, that is, identical up to reflection, rotation, and translation [6]. A previous work by the second and third authors of this article reconsidered Crazy-Cut puzzles from a different point of view. Quoting [7]: ‘‘We first analyze the inverse problem of assembling a planar shape from two identical shapes that have partially matching boundaries. This problem may be regarded as solving a simple jigsaw puzzle of two pieces (with no drawings on them).’’ Then the self-docking analysis readily provides a Crazy-Cut algorithm, and more importantly for us here, the insights also provide the mathematical basis required to design Crazy-Cut riddles systematically. In this work, we improve on our previous analysis, thereby enabling a simpler Crazy-Cut algorithm. Furthermore, based on the improved analysis, a formal method to design Crazy-Cut riddles is proposed.