Paper by Erik D. Demaine
- Erik D. Demaine, Martin L. Demaine, Yair N. Minsky, Joseph S. B. Mitchell, Ronald L. Rivest, and Mihai Pǎtraşcu, “Picture-Hanging Puzzles”, in Proceedings of the 6th International Conference on Fun with Algorithms (FUN 2012), Lecture Notes in Computer Science, Venice, Italy, June 4–6, 2012, pages 81–93.
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.
- The full paper is available as arXiv.org:1203.3602 of the Computing Research Repository (CoRR).
- 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.
- Copyright held by the authors.
- The paper is 12 pages.
- The paper is available in PDF (583k).
- See information on file formats.
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- Related papers:
- PictureHanging_TOCS (Picture-Hanging Puzzles)
See also other papers by Erik Demaine.
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