{{Short description|Disentanglement puzzle}} [[File:Staircasepuzzle-4ringspluspolewithcat.jpg|thumbnail|A baguenaudier]] [[File:Baguenaudier.svg|thumbnail|right|Diagrammatic representation of a four-ring baguenaudier]] [[File:Chinese_ring_full_brightened.jpg|thumbnail|right|A metal version of the puzzle]] '''Baguenaudier''' ({{IPA|fr|baɡnodje|pron}}; [[French language|French]] for "time-waster"),<ref name=MW/> also known as the '''Chinese rings''', '''Cardan's suspension''', '''Cardano's rings''', '''Devil's needle''' or '''five pillars''' puzzle, is a [[disentanglement puzzle]] featuring a loop which must be disentangled from a sequence of rings on interlinked pillars.<ref name=MW/> The loop can be either string or a rigid structure.
The origins are obscure, and it is unknown whether the puzzle originated in East Asia or the West. The American ethnographer [[Stewart Culin]] related a tradition attributing the puzzle's invention to the 2nd/3rd century Chinese general [[Zhuge Liang]]<ref name=DD/><ref>{{Cite book |title=The Tower of Hanoi – Myths and Maths|title-link= The Tower of Hanoi – Myths and Maths |last1= Hinz |first1= Andreas M. |last2= Klavžar |first2= Sandi |last3=Milutinović |first3=Uroš |last4=Petr |first4= Ciril |publisher=Birkhäuser |year=2015 |isbn= 978-3034807692 |page=4}}</ref> but Culin was relying on an unknown informant; the earliest definitive East Asian reference is an early 16th century mention of a "nine linked rings" toy by Yang Shen in his ''Sheng an ji''.<ref>{{Cite conference |last=Hinz |first=Andreas |year=2019 |title=Are the Chinese Rings Chinese? |conference=Recreational Mathematics Colloquium V |pages=83-101}}</ref> Luca Pacioli's ''De Viribus Quantitatis'' of 1509 mentions the puzzle<ref>{{Cite journal |last1=Heefer |first1=Albrecht |last2=Hinz |first2=Andreas |date=2017 |title="A difficult case": Pacioli and Cardano on the Chinese Rings |journal=Recreational Mathematics Magazine |volume=8 |pages=5-23 |url=https://scispace.com/pdf/a-difficult-case-pacioli-and-cardano-on-the-chinese-rings-3iqevjx8ep.pdf }}</ref> and may predate Yang Shen by a few years, but both authors treat the puzzle as something already well known.
The puzzle was used by French peasants as a locking mechanism.<ref name=MW>{{MathWorld | urlname=Baguenaudier | title=Baguenaudier}}</ref>
Variations of this include the ''Devil's staircase'', ''Devil's Halo''<ref>{{cite web |url=http://www.puzzlemuseum.com/month/picm05/200501d-halo.htm |title=The Devil's Halo |website=The Puzzle Museum |date=2017}}</ref> and the ''impossible staircase''. Another similar puzzle is the ''Giant's causeway'' which uses a separate pillar with an embedded ring.
==Mathematical solution== The 19th-century French mathematician [[Édouard Lucas]], the inventor of the [[Tower of Hanoi]] puzzle, was known to have come up with an elegant solution which used [[Binary numeral system|binary]] and [[Gray codes]], in the same way that his puzzle can be solved.<ref name=DD>{{cite encyclopedia |url=http://www.daviddarling.info/encyclopedia/C/Chinese_rings.html |author=David Darling |title=Chinese rings |encyclopedia=Encyclopedia of Science}}</ref> The minimum number of moves to solve an ''n''-ringed problem has been found to be<ref name=MW/>
: <math>a(n) = \begin{cases} \dfrac{2^{n+1} - 2}{3} & \text{when }n\text{ is even,} \\ \dfrac{2^{n+1} - 1}{3} & \text{when }n\text{ is odd.} \end{cases}</math>
For other formulae, see {{OEIS el|A000975}}.
==See also== *[[ABACABA pattern]] *[[Disentanglement puzzle]] *[[Towers of Hanoi]]
==References== {{Puzzles |Types}} {{reflist}}
[[Category:Chinese ancient games]] [[Category:Chinese inventions]] [[Category:Educational toys]] [[Category:Mechanical puzzles]]