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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.27617 |
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| _version_ | 1866917538051391488 |
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| author | Verhoeff, Tom |
| author_facet | Verhoeff, Tom |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27617 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $k$-out-of-$n$ picture-hanging puzzle: shorter solutions for small $k$ and $n-k$ Verhoeff, Tom Combinatorics Group Theory 20F12 (Primary), 20E05, 06E30, 97A20, 05A16 (Secondary) 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 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. |
| title | The $k$-out-of-$n$ picture-hanging puzzle: shorter solutions for small $k$ and $n-k$ |
| topic | Combinatorics Group Theory 20F12 (Primary), 20E05, 06E30, 97A20, 05A16 (Secondary) |
| url | https://arxiv.org/abs/2605.27617 |