How to ‘magically’ untie a shoelace double knot

Topology is a fascinating branch of mathematics that describes the properties of an object that remain unchanged under continuous “smooth” deformations. Actually, many 3D puzzles are based on topological principles and understanding some very basic principles may help you analyze whether a puzzle is possible or not.

Puzzle-Meister G. Sarcone created this amusing everyday-life topological puzzle to help children to easily take their shoes off.

image knots

As you know, the standard shoelace knot is designed for quick release and easily comes untied when either of the working ends is pulled. Thus, most people think that tying a shoelace into a double knot is an effective method of making the knot “permanent”. But is it true?

So, let’s make a standard shoelace knot following the steps ‘a’ to ‘f’ of the diagram: take a shoelace in each hand (a) and cross one lace over the other (b). Poke the end of the lace through the cross hole and pull both ends tight (c). Form with each lace a loop (d). Then, wrap a loop around the other loop and pass it through the cross hole (e) in order to tie both loops tight in a half hitch (f). To make the double knot, cross again the loop over and wrap it around the other loop (g) and pull both loops until tight (h).

Now try to untie your shoelaces WITHOUT touching the double knot!

show solution