Mathematics unravels optimum way of shoe lacing.

The knotty problem of choosing the optimum way of lacing up shoes has been solved by a new mathematical proof.

There are many millions of different possibilities but, reassuringly, the proof shows that centuries of human trial and error has already selected out the strongest lacing patterns. However, the pattern using the least amount of lace possible, the decorative "bow-tie" lacing, is usually only seen in shoe shop displays.

Burkard Polster, of Monash University in Melbourne, Australia, used combinatorial mathematics to come up with his proof. This branch of maths is used to solve a huge variety of problems, including resource allocation and finding the best ways to lay chips in a computer.

Formulas describing the physics of pulleys were used to work out how much force is exerted on the sides of a shoe using different lacing techniques. The most widely used "criss-cross" and "straight" lacing patterns were identified as the strongest.

But criss-cross came out on top for a short, wide set of eyeholes - that is, when the vertical distance between eyeholes is low, and horizontal distance is high. Straight lacing came out tops for a long, skinny set of eyeholes.

Ideal shoe.

Both sets of calculations, which ran to over 30 pages, were based on an idealised shoe. For example, the eyelets are perfectly aligned, and the shoe exists only in one plane. The proofs also ignore certain physical properties, such as the friction exerted by the lace on the eyehole.

Nonetheless, for an idealised shoe, the proofs cover every conceivable lacing pattern. There are 400 million different ways of lacing a shoe with only seven pairs of eyelets. "Even if God wears shoes with 100 eyelets, these proofs will predict the shortest and the strongest lacings," says Polster.

"It's mathematical entertainment - I like it," says Hyam Rubinstein of the University of Melbourne.

In practice, however, the changes in efficiency and strength of different types of lacing are likely to be tiny, says Polster. "Nike probably won't go out and re-lace all their sneakers on the basis of these calculations."

Furthermore, most people, including Polster, opt for criss-cross lacing not because it is stronger, but because it is easy and you do not end up with uneven ends - a big risk with straight lacing. Straight lacing is sometimes used in the army because, if the foot is injured, you can cut the lace with one swipe.