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 $F_n$. We give explicit, deterministic, polynomial-length constructions for two regimes: $2$-out-of-$n$ with word length at most $\tfrac{8}{3}n^{\log_2 6} - 4n^2$, and $(n-2)$-out-of-$n$ with word length $6n\log_2(n/2)$, both for $n$ a power of two. These improve on W\"astlund'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.