2 Trillion Lacing Methods?

On an average shoe with six pairs of eyelets, there are almost 2 Trillion ways to feed a shoelace though those 12 eyelets! Impossible? This page shows the maths behind that extraordinary number.

Shoe Lacing Mathematics

It hardly seems possible that there could be quite that many ways to feed a lace through 12 eyelets! So let's look at the mathematics:
Feed through one of 12 eyelets from either inside or outside. That's 24 possible ways to start.
Continue through one of 11 remaining eyelets from either inside or outside (x 22 more ways).
Then 10 remaining eyelets (x 20 more ways). We've only gone through three eyelets and we're already up to 24 x 22 x 20 = 10,560 ways!
By the time we reach the last eyelet (x 2 more ways), the possible ways have multiplied to 24 x 22 x 20 x 18 x 16 x 14 x 12 x 10 x 8 x 6 x 4 x 2 ways, a staggering total of 1,961,990,553,600.

That's almost 2 TRILLION possibilities!

This number can be halved for those paths that are mirror images of other paths, and halved again for those that follow the identical path from opposite directions. That still results in almost 500 billion ways.

Then again, we can multiply by the many different ways the laces can be crossed or interwoven prior to passing through those eyelets, and multiply again if we allow the laces to either pass through any eyelet more than once or skip any eyelet, and even more if we use two or more laces per shoe. This results in almost infinite possibilities, limited mainly by the length of the shoelaces.
 


In the real world however, we can place some sensible constraints, such as:
The lace should generally start and finish from the top pair of eyelets.
The lace should pass through each eyelet only once.
Each eyelet should contribute to pulling together the sides of the shoe.
The lacing should not be too difficult to tighten or loosen.
Any pattern formed should be relatively stable.
Ignore irrelevant variations (eg. changing the direction through a single eyelet).
Above all, the finished result should be visually pleasing.
So how many possible ways are there to lace a shoe with 12 eyelets if we DO take into account some or all of the above constraints? This requires far more complicated maths than the simple multiplications above. For example:



The above combinatorial equation came from research by Australian mathematician Burkard Polster, who caused a sudden worldwide surge of scientific and academic interest in the mathematics of shoelacing following the publication of an article in the respected journal "Nature" in December 2002.

Although not quoted in the Nature article, Polster's calculation for the number of real-world lacing methods for a typical shoe with 12 eyelets came to 43,200.

I'm therefore sure that the number of lacing methods on this site is destined to grow as I discover more worthwhile methods from the thousands of possibilities that I haven't yet explored.

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