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On a Complex KenKen Problem David Nacin David Nacin ( is a professor of mathematics at William Paterson University of New Jersey in Wayne. He enjoys designing and studying puzzles that take into account the structure of groups and Lie algebras, various partition identities, the motion of chess pieces, and sometimes other puzzles themselves. The goal of KenKen, like Sudoku, is filling an n × n grid, usually from the set {1, 2, . . . , n}, so that each element appears exactly once in each row and in each column, i.e., a Latin square. The constraints in KenKen come from heavily outlined regions of cells called cages. Clues are given inside these cages, each consisting of an operation and a target value. When the operation is applied to all of the entries in the cage, the output must equal this target. Brian Hayes is an author and well-known popularizer of mathematics and computer science. He maintains the blog bit-player where he discusses accessible topics in these areas. One post was devoted to what he called KenKen-friendly sets: collections of numbers where, for some list of operations, the target always determines the opera- tion [1]. These sets are not the focus of this paper. Instead, we focus on a tangential comment in that blog post. In an aside in this entry, Hayes suggested: ". . . venturing a little farther afield, we might try a 4 by 4 puzzle with the candidates {1, −1, i, −i}." This was not directly related to his topic, though he did provide a sample puzzle (Figure 1) before going on to mention, "I'm not at all sure the solution is unique or that it can be found by a purely deductive process. At best this is a jokenken." (A pun on joke and KenKen.) Some questions arise immediately. Is there a deductive process for solving his puz- zle, and is the solution unique? How does changing the clues affect the number of ix -ix 1x ix ix -1x Figure 1. A puzzle from the bit-player blog. MSC: 97A20, 05E18 274 © THE MATHEMATICAL ASSOCIATION OF AMERICA

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