In this month’s Math Teachers’ Circle session, run by Dr. Christina Jamroz of the University of St. Francis and Dr. Rita Patel of College of DuPage, we investigated Magic Squares. We began the session by examining a magic square of order five to infer some of its defining properties. Participants concluded that a magic square of order *n* must contain each of the numbers 1 through *n*^2 exactly once. Furthermore, every row, column, and main diagonal in the square must add to the same number, called the magic sum.

The session continued with groups working to construct magic squares of order two, three, and four. We observed that we could not construct a 2 *x* 2 magic square, but found eight magic squares of order three. We also realized that all of these magic squares were related by a rotation or reflection. During this discussion, we made conjectures related to patterns we noticed. One of these conjecture concerned the middle number of an odd order magic square. Is this middle number always going to be the median of the list of numbers used to fill the square? After considering more examples, participants found that this is not always true.

The construction of a magic square of order four led to a discussion about strategies we could use to create magic squares. Participants noted that knowing the magic sum would aid them while working on this problem. We observed that all of the magic squares of order three had the same magic sum. Does this generalize to all magic squares of a fixed order? This led to the discovery that the magic sum of a magic square of order four was always 34. Then, we deduced that the magic sum of a magic square of order *n* was *n(n^2+1)/2*.

At the conclusion of this session, we reflected on how we can use this topic in our classrooms. Furthermore, we realized that we have seen magic squares before, with participants mentioning Sudoku puzzles, art, and math history.