That's almost 2 TRILLION possibilities!
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.
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.
Shoe Lacing Methods
Whilst mathematics tells us that there are more than 2 Trillion Methods of feeding a lace through the six pairs of eyelets on an average shoe, this section presents a (somewhat more realistic) typical cross-section of traditional and alternative lacing methods that have either found or created.
The selection is limited to those methods that are widely used, have a particular feature or benefit. - The list is still growing!
34 Different Ways To Lace Shoes
Diagrams are all drawn as though looking from the front of the shoe, such that the lower part of the lacing is at the bottom of the diagram, and are colour-coded to make it easier to follow where the lace runs. Generally, the start of the lacing is grey, whilst the lace end that starts on the left side is blue and the end that starts on the right side is yellow. For colour-blind visitors, Blue=Dark=Left, Yellow=Light=Right.