One nail. Hang the picture so it stays up — and falls if the nail is pulled.
nail 1
Ropeno wraps yet — the picture is on the floor
| Nails pulled | Goal | Your rope | |
|---|---|---|---|
| none | hangs | — | |
| 1 | falls | — |
Click a row to pull that set of nails on the board.
Topology you can yank on. Wrap a rope around nails, then pull nails out and see if the picture survives — after Demaine, Demaine, Minsky, Mitchell, Rivest & Pătraşcu, “Picture-Hanging Puzzles”.
One nail. Hang the picture so it stays up — and falls if the nail is pulled.
| Nails pulled | Goal | Your rope | |
|---|---|---|---|
| none | hangs | — | |
| 1 | falls | — |
Click a row to pull that set of nails on the board.
The rope's path is a word in the free group on the nails: passing over nail i left-to-right writes the letter xi, and right-to-left writes its inverse xi−1. Adjacent inverse pairs cancel — a wrap followed by an unwrap is no wrap at all — and the picture hangs precisely when the word doesn't cancel down to nothing.
Pulling a nail out deletes every occurrence of its letter. So the classic two-nail trick is the commutator x1x2x1−1x2−1: intact, nothing cancels and the picture hangs; delete either letter and the survivors collapse in pairs, the rope slides free, and the picture drops. It's the same algebra that makes the Borromean rings hold together three-at-a-time yet fall apart when any one ring is cut.
Nesting the trick — treating a whole commutator as one "virtual nail" inside another commutator — solves 1-out-of-n for any n, and the paper shows any monotone rule ("falls exactly when this family of nail-sets is pulled") is achievable with polynomially much rope. One direction is forced, though: pulling more nails can never re-hang a fallen picture, which is why every puzzle here has a monotone goal.