# Baguenaudier

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Disentanglement puzzle

A baguenaudier

Diagrammatic representation of a four-ring baguenaudier

A metal version of the puzzle

**Baguenaudier** (pronounced [\[baɡnodje\]](https://en.wikipedia.org/wiki/Help:IPA/French); [French](/source/French_language) for "time-waster"),[1] also known as the **Chinese rings**, **Cardan's suspension**, **Cardano's rings**, **Devil's needle** or **five pillars** puzzle, is a [disentanglement puzzle](/source/Disentanglement_puzzle) featuring a loop which must be disentangled from a sequence of rings on interlinked pillars.[1] 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](/source/Stewart_Culin) related a tradition attributing the puzzle's invention to the 2nd/3rd century Chinese general [Zhuge Liang](/source/Zhuge_Liang)[2][3] 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*.[4] Luca Pacioli's *De Viribus Quantitatis* of 1509 mentions the puzzle[5] 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.[1]

Variations of this include the *Devil's staircase*, *Devil's Halo*[6] 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](/source/%C3%89douard_Lucas), the inventor of the [Tower of Hanoi](/source/Tower_of_Hanoi) puzzle, was known to have come up with an elegant solution which used [binary](/source/Binary_numeral_system) and [Gray codes](/source/Gray_codes), in the same way that his puzzle can be solved.[2] The minimum number of moves to solve an *n*-ringed problem has been found to be[1]

- a ( n ) = { 2 n + 1 − 2 3 when n is even, 2 n + 1 − 1 3 when n is odd. {\displaystyle 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}}}

For other formulae, see OEIS [sequence A000975](https://oeis.org/A000975).

## See also

- [ABACABA pattern](/source/ABACABA_pattern)

- [Disentanglement puzzle](/source/Disentanglement_puzzle)

- [Towers of Hanoi](/source/Towers_of_Hanoi)

## References

Part of a series on Puzzles Types Guessing Riddle Situation Logic Dissection Induction Logic grid Self-reference Mechanical Combination Disentanglement Lock Go problems Folding Stick Tiling Tour Sliding Chess Maze (Logic maze) Word and Number Crossword Sudoku Puzzle video games Metapuzzles Topics Brain teaser Dilemma Joke Optical illusion Packing problems Paradox Problem solving Puzzlehunt Syllogism Tale Lists Impossible puzzles Maze video games Nikoli puzzle types Puzzle video games Puzzle topics v t e

1. ^ [***a***](#cite_ref-MW_1-0) [***b***](#cite_ref-MW_1-1) [***c***](#cite_ref-MW_1-2) [***d***](#cite_ref-MW_1-3) [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Baguenaudier"](https://mathworld.wolfram.com/Baguenaudier.html). *[MathWorld](/source/MathWorld)*.

1. ^ [***a***](#cite_ref-DD_2-0) [***b***](#cite_ref-DD_2-1) David Darling. ["Chinese rings"](http://www.daviddarling.info/encyclopedia/C/Chinese_rings.html). *Encyclopedia of Science*.

1. **[^](#cite_ref-3)** Hinz, Andreas M.; Klavžar, Sandi; Milutinović, Uroš; Petr, Ciril (2015). [*The Tower of Hanoi – Myths and Maths*](/source/The_Tower_of_Hanoi_%E2%80%93_Myths_and_Maths). Birkhäuser. p. 4. [ISBN](/source/ISBN_(identifier)) [978-3034807692](https://en.wikipedia.org/wiki/Special:BookSources/978-3034807692).

1. **[^](#cite_ref-4)** Hinz, Andreas (2019). *Are the Chinese Rings Chinese?*. Recreational Mathematics Colloquium V. pp. 83–101.

1. **[^](#cite_ref-5)** Heefer, Albrecht; Hinz, Andreas (2017). [""A difficult case": Pacioli and Cardano on the Chinese Rings"](https://scispace.com/pdf/a-difficult-case-pacioli-and-cardano-on-the-chinese-rings-3iqevjx8ep.pdf) (PDF). *Recreational Mathematics Magazine*. **8**: 5–23.

1. **[^](#cite_ref-6)** ["The Devil's Halo"](http://www.puzzlemuseum.com/month/picm05/200501d-halo.htm). *The Puzzle Museum*. 2017.

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Adapted from the Wikipedia article [Baguenaudier](https://en.wikipedia.org/wiki/Baguenaudier) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Baguenaudier?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.