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Euler Squares - Discourse and Extension

Elaine Young

Comparing solutions and methods between groups led teachers to realize that there were multiple solutions and multiple methods for solving Euler squares. The instructor then challenged groups to determine how many possible solutions there might be using their particular method. Groups using diagonalization speculated on the permutations of their current design, and discussed which were rotations or reflections and whether these patterns counted as unique.

Another group began to extend their exploration by creating a table of Euler squares of increasing order. They soon discovered that an Euler square of order two was impossible to make. However, using only one attribute, they found there were two Latin squares. They moved on to Euler squares of higher order and began counting how many solutions existed. By the end of the class period they were trying to identify a formula to predict the number of solutions as well as wondering why a 6x6 Euler square did not exist.

A simple child’s toy and an intriguing historical problem can be combined to encourage problem solving. Teachers not only learned about Euler squares and related mathematical topics, they learned a lot about themselves and their own thinking abilities. The accompanying Flash module will allow the reader to engage in this historical problem.

Elaine Young, "Euler Squares - Discourse and Extension," Loci (May 2011)