The k-out-of-n picture-hanging puzzle: shorter solutions for small k and n-k

The picture-hanging puzzle, popularized by Demaine et al. (2014), asks for a way to wrap a wire around n nails such that the picture hangs as long as fewer than k nails are removed, but falls as soon as any k are removed. Solutions correspond to words in the free group Fₙ. We give explicit, deterministic, polynomial-length constructions for two regimes: 2-out-of-n with word length at most 83n^₂ 6 - 4n², and (n-2) -out-of-n with word length 6n₂ (n/2), both for n a power of two. These improve on Wästlund's quasi-polynomial deterministic construction in their respective regimes. We also report, via exhaustive computer search, the exact minimum length of 16 for the 2-out-of-4 puzzle, attained by two structurally distinct solutions. As an additional contribution, we observe that the natural workshop realization with carabiners on a flat board introduces an over/under ambiguity at every wire crossing; a wrong choice can produce a Whitehead link, which is topologically distinct from the intended commutator.