Crafty, Geometry Joys, motor skills, Murderous Maths, Notes on everyday life, Simon's sketch book

A Knot Theory Puzzle

Simon has shown us a curious puzzle: if you hang a poster on a string using two pins, what is the way to arrange the string so that the poster definitely falls once you remove any pin? The math behind the trick involves Knot Theory. Simon has learned the trick from this video by Jade, the creator of the science and phlosophy Up an Atom channel that Simon loves.

It’s relatively easy to solve the puzzle for one particular pin. The picture below shows the solution for removing the right pin:

But the puzzle asks us to think of a configuration that makes the poster fall once ANY pin is removed, doesn’t matter which! And that’s way more difficult. Simon said that we should simplify the problem by removing the poster altogether and replacing the pins with two small loops of string.

What Simon did next was show us the math behind the trick, trying to come up with such a combination of the three loops that would stay connected but, if you remove any of the three, the rest of the construction would fall apart. “Wait, that sound familiar! We’ve actually turned the problem into Borromean rings!”

The letters x and y stand for the ways to intertwine the strings, with x wrapping around 1 and y wrapping around 3. The regular x and y are clockwise (x or y) and the inverse x and y are anticlockwise (x^-1 or y^-1). Obviously, a sequence of clockwise-anticlockwise of the same string should be avoided as it unties itself.
the moment of truth!
performing in front of our guests the day after (in Dutch)
the solution

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