Hanging a Picture Puzzle
Here's a classic which I first saw on the Futility Closet. If you use a single nail to hang a picture then removing the nail would make the picture fall like the image below. How can you use two nails to hang a picture such that removing either nail would make the picture fall? Answer below.
The answer is non unique but here's the simplest answer:
Removing either nail will make it fall. This puzzle is actually extendable to n nails and the proof is by induction. The easiest way to form the solutions as illustrated below for the n=3 case:
Start with n rings and interlock a loop which if any shape is removed it will disconnect all other shapes. You can see an example of this on the left with that pink shape weaving its way through the three rings. This picture can be stretched out as if the n rings are infinitely big as in the second picture. Finally this can be translated into the equivalent nail problem. If you want a challenge, here's a set of related puzzles that I found when researching this:
Most of these are quite hard, but having a loop of string is quite helpful.