Magic Square Products – a New Twist on an Old Puzzle Pretzel

Magic Square 3x3 sum

Most of us are familiar with magic squares. They are square arrays of numbers such as the one shown here. The sum of each row, column, and major diagonal is the same, hence the name. The first time students are posed the challenge of creating one, they invariably say “It’s impossible.” After seeing one such as the one at the right, they are entranced. Magic squares exist for all sizes of squares from 3×3 on up, and there are simple rules to create them.

The new twist arose recently when I read one of my favorite RSS feeds from The Math Nexus: Mathematics Portal – News and Ideas for Teachers and Learners of Mathematics at http://mathnexus.wwu.edu/. I look forward to reading each edition of their online newsletter for great math nuggets.

The problem Math Nexus posed was to find a 3×3 Square that was magic, not using addition, but multiplication. In other words, the product of each row, column, and major diagonal must be the same. I had seen this problem posed before, but had never pursued it. This time I wanted to give it a good try. So I made several 3×3 grids, played around with factors, multiples, and so forth, before I realized that I wasn’t going to be successful right away. So I did what I’ve always suggested to my students: set my intent to solve it and let my subconscious know that this was important, did my best to wrestle this puzzler to the mat, and then left my desk. About two hours later, while sitting in my recliner, the solution just presented itself. And not just one solution, an infinity of them. Those who have played around with magic squares know that there are many different solutions to magic square puzzles, and the Magic Square Product puzzle is no different.

I pose this problem today, and will share at least two different solutions to this product puzzle in seven days. As always, there is much to be gained by studying this puzzle, and teachers can look forward to using many math concepts in the explanation of the puzzle solution.

Related posts:

1. Magic Square Products Solutions – a New Twist on an Old Puzzle Pretzel
2. Reverse the Math Puzzle Using 2, 3, and 4
3. Video – Sum Equals Product Puzzle – Part 1
4. Video – Sum Equals Product Puzzle Part 2 – Solution
5. Video – Reverse the Math Puzzle, Part 2: Using 1, 3, and 5