Minimum winding lengths for k-out-of-n picture hanging, and the open case 3-out-of-5

Abstract

A picture hangs from a loop of string wound around n nails. A winding is an element of the free group Fₙ on the nails, removing a set of nails sets the corresponding generators to 1, and the picture falls exactly when the winding becomes trivial. The k-out-of-n puzzle asks for a winding that falls as soon as any k nails are removed and survives every removal of fewer than k; M(k,n) is the least length of such a winding. Verhoeff's recent paper determines M(2,4)=16 by exhaustive search and asks for the exact minima of the next cases, naming 3-out-of-5 as the smallest open one. We prove M(3,5)=22 and exhibit an attaining winding. The engine is a projection principle: restricting a solution to a subset of its nails is again a solution of a smaller puzzle, so every four-nail projection of a 3-out-of-5 solution is a 2-out-of-4 solution. With parity this excludes every length below 20 outright and leaves a single occurrence vector at length 20, which one enumeration of 667 979 nodes rules out. Summing the same principle over all co-singletons gives a density bound M(k,n)/n ≥ M(k-1,n-1)/(n-1), hence lower bounds for whole diagonal families: 5 M(n-2,n) ≥ 22n for n ≥ 5, 5 M(n-3,n) ≥ 24n for n ≥ 5, and M(3,6) ≥ 29. For the next cell we prove 24 ≤ M(2,5) ≤ 30; the upper bound comes from a product of two commutator blocks and improves the best published construction, which has length 54. All statements are machine-checked in Lean 4.

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