Mysterious Magic Squares

Here is mathemagician Ethan Brown, a high school student at Phillips Academy Andover in Andover, Massachusetts. Watch him build a magic square before a live audience:

The magic square that Ethan builds in this performance is

In this second video, Ethan reveals a surprisingly simple technique that will allow you to build a basic magic square with any desired row/column total in just a few seconds. Really!

And in this third video, Ethan explains how to build more complex magic squares, like the ones he conjures so deftly in his stage performances.

Taking it Further

Here are some magic square challenge problems:

1. If one were to add together two n × n magic squares, would the sum be another magic square?
2. Can a n × n magic square be created for any n ≥ 3?
3. Can a magic square be created for n = 2 (assuming that not all of the entries are the same number)?
4. A multiplicative magic square is a magic square where the product of the numbers in every row, column, and diagonal are equal. If one were to raise 2 to the power of the corresponding entries of an additive magic square, would this create a multiplicative magic square?

Martin Gardner wrote about magic squares in chapter 17 of his book Time Travel and Other Mathematical Bewilderments (W.H. Freeman, 1988). Here he also discusses the fascinating topic of perfect magic cubes: A cubical array of integers whose rows, columns, pillars, and various diagonals all sum to a fixed number.

Magic cubes have a fascinating history. In 1640, Pierre de Fermat attempted (unsuccessfully) to construct a perfect 4 × 4 × 4 (or “order 4”) magic cube. This is the first known example of the concept. Fermat was aware that a perfect magic cube of order 3 was impossible. The first published example of a perfect magic cube appeared more than 200 years later, in 1866. It was an order 7 magic cube, discovered by Rev. A. H. Frost, M.A. of St. John’s College, Cambridge. It was not known if smaller order magic cubes existed until 1972, when Richard Schroeppel proved that a perfect magic cube of order 4 was impossible. In 2003, Christian Boyer and Walter Trump discovered the first perfect magic cube of order 5, resolving once and for all the issue of whether such an object exists. An order 6 cube was discovered in the same year.

Ed Pegg, Jr. has built an interactive demonstration that shows each of the 64 lines found by Fermat in his attempt at finding a magic cube of order 4, and each of the 72 lines found by Walter Trump in his 2004 example. Neither of these are quite perfect, of course. Find the demonstration here (the free Wolfram CDF Player is required).

The Underlying Mathematics

Arthur Benjamin and Ethan Brown recently coauthored a paper titled “Challenging Magic Squares for Magicians” (The College Mathematics Journal, Vol. 45, No. 2, March 2014, pp. 92–100).

Daryl L. Stephens gives a detailed overview of magic squares, their history, and their properties in his thesis “Matrix properties of magic squares.”

Benjamin previously wrote about “Double Birthday Magic Squares” in M-U-M (August 2006, pp. 68–69). This work inspired Ethan Brown to discover his method for building magic squares.

At the elementary level there is a book called Big Magic Number Puzzles (Scholastic, 1999) that features 36 challenges related to the concept of the magic square (arranging numbers in various shaped arrays to achieve a magic sum). These are stimulating challenges for elementary schoolers, and they are easy for teachers to distribute.

PBS Learning Media have developed some activities relating to magic squares and magic stars that teach a basic method to create odd-ordered magic squares. Creating odd-ordered or 4 × 4 magic squares can be an activity in itself.

Hossein Behforooz has been fascinated with magic squares since he was a child. He has published five papers on this topic in The Journal of Recreational Mathematics.