Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Pǎtraşcu, “Picture-Hanging Puzzles”, Theory of Computing Systems, volume 54, number 4, May 2014, pages 531–550.

Abstract:

We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get removed. This construction makes for some fun mathematical magic performances. More generally, we characterize the possible Boolean functions characterizing when the picture falls in terms of which nails get removed as all monotone Boolean functions. This construction requires an exponential number of twists in the worst case, but exponential complexity is almost always necessary for general functions.

Comments:

This paper is also available from SpringerLink, and as arXiv.org:1203.3602 of the Computing Research Repository (CoRR).

Updates:

Open Problem 1 was in fact previously solved by Gartside and Greenwood's paper "Brunnian links" (2007). The length of the shortest solution to the 1-out-of-n puzzle is Θ(n^2); in fact, the exact bound matches the 2002 Chris Lusby Taylor construction we present.

Length:

The paper is 18 pages.

Availability:

The paper is available in PDF (6884k).

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Related papers:

PictureHanging_FUN2012 (Picture-Hanging Puzzles)