## MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

## March 26, 2007

### How to Lace Like an Ace

One of your sneakers has a broken shoelace. You want to replace it, but the only laces you have around are much shorter. Is there a way to lace your shoes so that you can use the shorter laces?

There are three common ways to lace shoes: American (or standard) zigzag, European straight, and quick-action shoe store. The lacing style you happen to use is generally the one you learned as a kid.

Different lacing patterns require different lengths of lace. You might wonder which one of the three common lacing patterns requires the shortest laces. Notice that, in all three cases, the lace passes through each hole (or eyelet) just once, alternately crossing from one side of the shoe to the other.

Here's an experiment you can try. Using the same lace and shoe each time, follow the lacing patterns shown in the diagrams. Measure the total length of shoelace hanging loose above the top eyelets in each pattern. Which lacing saves you the most lace?

Mathematicians have proved that, if your shoe has four or more pairs of eyelets, the American style requires the shortest laces, followed by the European, then the shoe-store styles. If your shoe has only three pairs of eyelets, the American lacing remains shortest, but the European and shoe-store lacings are of equal length. Amazingly, the American style also wins when the eyelets are irregularly spaced instead of being neatly arranged in two neat rows.

Shorter lacings are possible if the lace doesn't have to pass alternately through the holes on the left and right side of the shoe. Here are some alternative lacings you could try. Black means the laces are on top; red means the laces are underneath.

The first two work only if your shoes have an even number of eyelet pairs.

Watch out, though. You might find that by saving shoelace length you end up with shoes that slip off your feet more easily or laces that break more often.

Muse, October 1999, p. 33.