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Baguenaudier

ChineseRings

The baguenaudier is a puzzle involving disentangling a set of rings from a looped double rod, originally used by French peasants to lock chests (Steinhaus 1999). The word "baguenaudier" means "time-waster" in French, and the puzzle is also called the Chinese rings or Devil's needle puzzle. ("Bague" also means "ring," but this appears to be an etymological coincidence. Interestingly, the bladder-senna tree is also known as "baguenaudier" in French.) Culin (1965) attributes the puzzle to Chinese general Hung Ming (A.D. 181-234), who gave it to his wife as a present to occupy her while he was away at the wars.

Seven-ring baguenaudier illustrated by Minguet e Yrol (1733)

Minguet e Yrol (1733, p. 47) illustrates a seven-ring form of the puzzle, with the ring assembly labeled A and the separate loop labeled B, under the heading "Juego del nudo gordiano, ò lazo de las sortijas enredadas."

The solution of the baguenaudier is intimately related to the theory of Gray codes.

The minimum number of moves a(n) needed for n rings is

a(n)=[2/3(2^n-1)]
(1)
={1/3(2^(n+1)-2) n even; 1/3(2^(n+1)-1) n odd,
(2)

where [x] is the ceiling function, giving 1, 2, 5, 10, 21, 42, 85, 170, 341, 682, ... (OEIS A000975). The generating function for these numbers is

1/((1-2x)(1-x^2))=1+2x+5x^2+10x^3+21x^4+....
(3)

They are also given by the recurrence relation

a(n)=a(n-1)+2a(n-2)+1
(4)

with a(1)=1 and a(2)=2.

By simultaneously moving the two end rings, the number of moves for n rings can be reduced to

b(n)={2^(n-1)-1 n even; 2^(n-1) n odd,
(5)

giving 1, 1, 4, 7, 16, 31, 64, 127, 256, 511, ... (OEIS A051049).

Defining the complexity of a solution as the minimal number of times the ring passes through the arc from the last ring to the base of the puzzle, the minimal complexity of a solution is 2^(n-1), as conjectured by Kauffman (1996) and proved by Przytycki and Sikora (2000).

See also

Gray Code, Habiro Move

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References

Culin, S. "Ryou-Kaik-Tjyo--Delay Guest Instrument (Ring Puzzle)." §20 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 31-32, 1965.Dubrovsky, V. "Nesting Puzzles, Part II: Chinese Rings Produce a Chinese Monster." Quantum 6, 61-65 (Mar.) and 58-59 (Apr.), 1996.Elliott Avedon Museum and Archive of Games, University of Waterloo, Ontario, Canada. "Wire and Ring Puzzles." http://www.ahs.uwaterloo.ca/~museum/puzzles/wire/.Gardner, M. "The Binary Gray Code." In Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 15-17, 1986.Kauffman, L. H. "Tangle Complexity and the Topology of the Chinese Rings." In Mathematical Approaches to Biomolecular Structure and Dynamics. New York: Springer-Verlag, pp. 1-10, 1996.Kraitchik, M. "Chinese Rings." §3.12.3 in Mathematical Recreations. New York: W. W. Norton, pp. 89-91, 1942.Minguet e Yrol, P. Engaños a ojos vistas, y diversion de trabajos mundanos, fundada en licitos juegos de manos. Madrid, Spain: Pedro Joseph Alonso y Padilla, p. 47, 1733. https://archive.org/details/enganos-a-ojos-vistas-y-diversion-de-trabajos-mundanos-fundada-en-licitos-juegos/page/n62/mode/1up.Przytycki, J. H. and Sikora, A. S. "Topological Insights from the Chinese Rings." 21 Jul 2000. https://arxiv.org/abs/math/0007134.Sloane, N. J. A. Sequences A000975 and A051049 in "The On-Line Encyclopedia of Integer Sequences."Slocum, J. and Botermans, J. Puzzles Old and New: How to Make and Solve Them. Seattle, WA: University of Washington Press, p. 105, 1988.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 268-269, 1999.

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Baguenaudier

Cite this as:

Weisstein, Eric W. "Baguenaudier." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Baguenaudier.html

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