Picture-Hanging Puzzles
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...
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| Published in | Theory of computing systems Vol. 54; no. 4; pp. 531 - 550 |
|---|---|
| Main Authors | , , , , , |
| Format | Journal Article |
| Language | English |
| Published |
Boston
Springer US
01.05.2014
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| 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. |
|---|---|
| AbstractList | Issue Title: Fun with Algorithms 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.[PUBLICATION 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. 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. |
| Author | Mitchell, Joseph S. B. Demaine, Erik D. Demaine, Martin L. Rivest, Ronald L. Pǎtraşcu, Mihai Minsky, Yair N. |
| Author_xml | – sequence: 1 givenname: Erik D. surname: Demaine fullname: Demaine, Erik D. email: edemaine@mit.edu organization: MIT Computer Science and Artificial Intelligence Laboratory – sequence: 2 givenname: Martin L. surname: Demaine fullname: Demaine, Martin L. organization: MIT Computer Science and Artificial Intelligence Laboratory – sequence: 3 givenname: Yair N. surname: Minsky fullname: Minsky, Yair N. organization: Department of Mathematics, Yale University – sequence: 4 givenname: Joseph S. B. surname: Mitchell fullname: Mitchell, Joseph S. B. organization: Department of Applied Mathematics and Statistics, State University of New York – sequence: 5 givenname: Ronald L. surname: Rivest fullname: Rivest, Ronald L. organization: MIT Computer Science and Artificial Intelligence Laboratory – sequence: 6 givenname: Mihai surname: Pǎtraşcu fullname: Pǎtraşcu, Mihai organization: AT&T Labs—Research |
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| DOI | 10.1007/s00224-013-9501-0 |
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| Keywords | Algorithms Topology Monotone functions Free group Magic |
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| References | Theodore (CR12) 1999 Ajtai, Komlós, Szemerédi (CR1) 1983; 3 Demaine, Demaine, Uehara (CR3) 2010 Pegg (CR7) 2002 CR14 Paterson (CR6) 1990; 5 Sloane (CR10) 2002 CR11 Ellul, Krawetz, Shallit, Wang (CR4) 2005; 9 Makanin (CR5) 1985; 25 Sillke (CR9) 2001 Brunn (CR2) 1892 Rolfsen (CR8) 1976 Tait (CR13) 1876; 28 H. Brunn (9501_CR2) 1892 9501_CR14 9501_CR11 S. Theodore (9501_CR12) 1999 P.G. Tait (9501_CR13) 1876; 28 M. Ajtai (9501_CR1) 1983; 3 N.J.A. Sloane (9501_CR10) 2002 K. Ellul (9501_CR4) 2005; 9 T. Sillke (9501_CR9) 2001 E.D. Demaine (9501_CR3) 2010 Ed Pegg Jr. (9501_CR7) 2002 M.S. Paterson (9501_CR6) 1990; 5 D. Rolfsen (9501_CR8) 1976 G.S. Makanin (9501_CR5) 1985; 25 |
| References_xml | – volume: 9 start-page: 233 issue: 2–3 year: 2005 end-page: 256 ident: CR4 article-title: Regular expressions: new results and open problems publication-title: J. Autom. Lang. Comb. contributor: fullname: Wang – volume: 25 start-page: 75 issue: 1 year: 1985 end-page: 88 ident: CR5 article-title: Decidability of the universal and positive theories of a free group publication-title: Math. USSR, Izv. doi: 10.1070/IM1985v025n01ABEH001269 contributor: fullname: Makanin – year: 2001 ident: CR9 publication-title: Strange painting contributor: fullname: Sillke – volume: 3 start-page: 1 issue: 1 year: 1983 end-page: 19 ident: CR1 article-title: Sorting in log parallel steps publication-title: Combinatorica doi: 10.1007/BF02579338 contributor: fullname: Szemerédi – start-page: 77 year: 1892 end-page: 99 ident: CR2 article-title: Über Verkettung publication-title: Sitzungsbericht der Bayerischen Akademie der Wissenschaft Mathematisch Naturwissenschaftliche Abteilung contributor: fullname: Brunn – ident: CR14 – year: 1999 ident: CR12 publication-title: Brunnian braids and some of their generalizations contributor: fullname: Theodore – ident: CR11 – volume: 5 start-page: 75 year: 1990 end-page: 92 ident: CR6 article-title: Improved sorting networks with (log ) depth publication-title: Algorithmica doi: 10.1007/BF01840378 contributor: fullname: Paterson – start-page: 139 year: 2010 end-page: 142 ident: CR3 article-title: Any monotone Boolean function can be realized by interlocked polygons publication-title: Proceedings of the 22nd Canadian Conference on Computational Geometry contributor: fullname: Uehara – year: 2002 ident: CR7 publication-title: Picture Hanging contributor: fullname: Pegg – volume: 28 start-page: 145 year: 1876 end-page: 190 ident: CR13 article-title: On knots publication-title: Trans. R. Soc. Edinb. contributor: fullname: Tait – year: 1976 ident: CR8 publication-title: Knots and Links contributor: fullname: Rolfsen – year: 2002 ident: CR10 article-title: Sequence A073121 publication-title: On-Line Encyclopedia of Integer Sequences contributor: fullname: Sloane – start-page: 77 volume-title: Sitzungsbericht der Bayerischen Akademie der Wissenschaft Mathematisch Naturwissenschaftliche Abteilung year: 1892 ident: 9501_CR2 contributor: fullname: H. Brunn – volume-title: Brunnian braids and some of their generalizations year: 1999 ident: 9501_CR12 contributor: fullname: S. Theodore – volume-title: Picture Hanging year: 2002 ident: 9501_CR7 contributor: fullname: Ed Pegg Jr. – volume-title: Strange painting year: 2001 ident: 9501_CR9 contributor: fullname: T. Sillke – volume: 28 start-page: 145 year: 1876 ident: 9501_CR13 publication-title: Trans. R. Soc. Edinb. doi: 10.1017/S0080456800090633 contributor: fullname: P.G. Tait – volume: 3 start-page: 1 issue: 1 year: 1983 ident: 9501_CR1 publication-title: Combinatorica doi: 10.1007/BF02579338 contributor: fullname: M. Ajtai – start-page: 139 volume-title: Proceedings of the 22nd Canadian Conference on Computational Geometry year: 2010 ident: 9501_CR3 contributor: fullname: E.D. Demaine – ident: 9501_CR14 – volume-title: On-Line Encyclopedia of Integer Sequences year: 2002 ident: 9501_CR10 contributor: fullname: N.J.A. Sloane – ident: 9501_CR11 – volume: 9 start-page: 233 issue: 2–3 year: 2005 ident: 9501_CR4 publication-title: J. Autom. Lang. Comb. contributor: fullname: K. Ellul – volume: 25 start-page: 75 issue: 1 year: 1985 ident: 9501_CR5 publication-title: Math. USSR, Izv. doi: 10.1070/IM1985v025n01ABEH001269 contributor: fullname: G.S. Makanin – volume: 5 start-page: 75 year: 1990 ident: 9501_CR6 publication-title: Algorithmica doi: 10.1007/BF01840378 contributor: fullname: M.S. Paterson – volume-title: Knots and Links year: 1976 ident: 9501_CR8 contributor: fullname: D. Rolfsen |
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| Snippet | 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... Issue Title: Fun with Algorithms We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture... 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... |
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| SubjectTerms | Algorithms Analysis Applied mathematics Boolean Boolean functions Computer Science Construction Magic & magicians Mathematical analysis Mathematical models Nails Pictures Polynomials Puzzles Studies Theory of Computation Topological manifolds Wrapping |
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| Title | Picture-Hanging Puzzles |
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