Hang a picture on three nails so that removing any two will cause the picture to fall but removing just one will not.
Hint: See the solution for two nails in the graphic below: clockwise around a, clockwise around b, counterclockwise around a, counterclockwise around b, also notated +a+b−a−b.
(Read more about picture-hanging problems here.)
Challenge: Hang the picture on three nails so that removing any one nail will cause the picture to fall.
Here’s one possible solution: +a+b+c–a–b–c. Because the string wraps around each nail the same number of times clockwise and counterclockwise, removing the any two nails will cause all the wraps around the remaining one to cancel out and the picture will fall, and we can see that removing any one nail will leave the painting hanging. This is among the shortest possible solutions because each nail must be wrapped around at least once (or else removing it will not affect the painting) and the counterclockwise wraps must be the same amount as the clockwise one. But there are many other possible solutions, for example: 2c–3a–c+a+b+a–2b–c+b+a
Challenge solution: +c+a+b–a–b–c+b+a–b–a
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