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MatheMUSEments
How to Lace Like an Ace
By Ivars Peterson
Muse, October 1999, p. 33.
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.
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