I have always enjoyed playing with mechanical equipment. As a child I was endlessly fascinated by combination locks on safes, pulley systems, epicyclic gearboxes and the like. Even the humble elastic band can provide hours of entertainment in the right pair of hands. I remember spending days making an huge super-ball that was responsible for many high energy collisions at home, a fly-swatter that could stun flies with pinpoint accuracy from a distance of about half a meter, and a number of puzzle-boxes with internal guillotine-like mechanisms that would delight guests once the pain caused by almost amputating their fingers had subsided. Elastic bands are also a rich source of amenable puzzles.
Two identical elastic bands are each of natural length L. One is looped so that it is doubled, creating a band of natural length L ⁄ 2. The other is tripled, so that its natural length is L ⁄ 3. The bands are knotted together at one end creating a band of overall length L ⁄ 2 + L ⁄ 3 = 5L ⁄ 6. The combined band is stretched over two nails of separation L. Assuming linear behaviour, what position does the knot take?