Rectangles for Mathemagicians

Depending who you ask, mathemagician has at least two different definitions:

1. A person who enjoys both math and magic. (Wikipedia)
2. A person who is so good at math that the answers to math problems seem to come to them magically. (Urban Dictionary)

When professor Art Benjamin told Stephen Colbert that he was a mathemagician, Colbert asked, “What does that mean? Were those two words by itself not nerdy enough?”

Below is a math puzzle involving magic. To be precise, magic rectangles. But first, a little warm-up…

What do you call a quadrilateral with four right angles that’s been in a car accident?

A wrecked angle.

For the last several years, I’ve had the pleasure of creating puzzles for the Daily Puzzle Challenge at the NCTM Annual Meeting. A new set of four or five puzzles appears in each day’s challenge. The following puzzle, which appeared on Friday’s Daily Puzzle Challenge, involves rectangles and is my favorite puzzle from this year’s meeting.

A magic rectangle is an m × n array of the positive integers from 1 to m × n such that the numbers in each row have a constant sum and the numbers in each column have a constant sum (although the row sum need not equal the column sum). Shown below is a 3 × 5 magic rectangle with the integers 1-15.

Below are three arrays that can be filled with the integers 1-24, but only two of them can be filled in such a way as to form a magic rectangle. Construct two magic rectangles below; for the array that cannot be used to construct a magic rectangle, can you explain why not? More generally, can you determine what types of rectangles can be used to construct magic rectangles and which cannot?