Compendium of Chinese-Rings-Like Puzzles


© 2013 by Dr. Goetz Schwandtner

1 Introduction

There are some puzzles that have been there for a long time and have their roots that far back in the past that their origin is not known exactly. The traditional six piece burrs are one such example, the Chinese Rings puzzle another. For some information about the history of the Chinese Rings (also known as "Nine Linked Rings") see [9].

Chinese Rings
Chinese Rings puzzle

In this compendium, we exhibit a collection of all puzzles known to us that are all related to this type of puzzles, why we will call them "Chinese-Rings-like puzzles", or more formally, "CR recursive puzzles". A common similarity for all these puzzles is that they have an astonishing high number of moves to solve and they have several rings or pieces working in a similar way in the solution.
Our Compendium features a large variety of different puzzles: Disentanglement puzzles, sliding pieces puzzles, puzzle boxes, burrs, sequential movement puzzles, ball maze puzzles, and even puzzle games with certain rules.

SpinOut
SpinOut
Crazy Elephants Dance
Crazy Elephants Dance
Kugellager
Kugellager

The last example above, the "Kugellager", is the puzzle that started our research and lead to the first article about this puzzle and similar puzzles: [1]. In that article, the "Kugellager" puzzle with its 1250 moves is analyzed and later on short analyses of other puzzles like "Die Welle" were added and a table was created showing several puzzles that are now in this compendium. After collecting some of the related puzzles, a group page was set up and the term "n-ary puzzle" was termed for these puzzles: [2]. This term n-ary was also used in a nice article [11] about designing n-ary puzzles which appeared in CFF 15 in November 2014. The most recent exteded version of this article can be found in [12].

1.1 Presentation of the Puzzles

The list of puzzles is the main part of this compendium. It lists the puzzles with several details in entries ordered alphabetically by their names.

Puzzle-ID Name(example entry for explanation of fields)
Image(s) of puzzle.

Click on image or
links for bigger
image versions.
Designer Manufacturer Year
Name of creator of puzzle design Name/Company name of manufacturer Year of first release
Arity No of pieces Piece type Solution length function Number of moves
# of Levels (n) special pieces only of special pieces (m) exact or asymptotic (Θ) function of n and m counted/calculated
RemarksRemarks about special features, similar puzzles.
References
Symbols: §=counting moves of special pieces only; =counting moves until first piece comes out
Record Layout — Structure of each puzzle entry in the list

Hint: Even if the list cannot be ordered for other fields, you can easily search for a certain entry using your web browser's built in search function (usually invoked by keys Ctrl-F or F3).

Hint: For previews of secondary images or the variants (entries with IDs), please hover the mouse pointer over the corresponding link (without clicking). In the top right corner of the browser window, a preview will appear. Unfortunately, this does not work on all browsers; in particular some Internet Explorer versions do not show these previews.

Beside this introduction and the list of puzzles there are also a page with a formal definition of the structure of puzzles that belong into this compendium, including some properties and examples. The compendium is concluded by a page to contribute to this compendium by adding puzzles or new information. Please have a look at the navigation bar on top of all pages or the table of contents below.

1.2 Table of Contents/Sitemap

1   Introduction
1.1   Presentation of the Puzzles
1.2   Table of Contents/Sitemap
2   List of puzzles
3   Definitions, Examples and More
3.1   Definition of CR Recursive Puzzle
3.2   Examples
3.2.1   Classic CR puzzle
3.2.2   SpinOut
3.2.3   Crazy Elephants Dance
3.2.4   Kugellager
3.2.5   Tower of Hanoi
3.3   What does "uniform" mean in the definition?
3.4   Where does the term "recursive" come from?
3.5   Tuple Representation and a Different Notation: n-ary Puzzles
3.6   Puzzle Parameter and Different View: Number of Moves
3.7   CR recursive puzzles with solution length only ~2^m
3.8   Tower of Hanoi - binary or ternary?
3.9   A different definition for n-ary puzzles based on number of moves
4   References
5   Acknowledgements
6   Feedback

2 List of puzzles

Puzzle-ID Name(example entry for explanation of fields)
Image(s) of puzzle.

Click on image or
links for bigger
image versions.
Designer Manufacturer Year
Name of creator of puzzle design Name/Company name of manufacturer Year of first release
Arity No of pieces Piece type Solution length function Number of moves
# of Levels (n) special pieces only of special pieces (m) exact or asymptotic (Θ) function of n and m counted/calculated
RemarksRemarks about special features, similar puzzles.
References
Symbols: §=counting moves of special pieces only; =counting moves until first piece comes out


CR091 NameAlgorithme 9
Designer Manufacturer Year
Patrick Farvacque Patrick Farvacque  
Arity No of pieces Piece type Solution length function Number of moves
3 9 discs    
RemarksThe Algorithme series features different puzzles with different number of discs, different disc heights and post heights. They are all some variation of Tower of Hanoi, which can also be seen in the rules: move one disc at a time, which is on top of its pile; no bigger disc may be put on a smaller one (equal size is OK); piles may only go up to post end, not higher.
References

CR174 NameAlken/Kenal

[1]
[2] [3]
Designer Manufacturer Year
Alfons Eyckmans, Ken Johnson Alfons Eyckmans 2017
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel   321
RemarksVariant of CR160, but not coming apart and only with 5 binary pieces, and one lid piece to be removed by the solution sequence. The lid piece has a different structure, and leads to four puzzles: Alken has lid piece which is binary (135 moves) in one orientation and 6-ary (321 moves) in the other. Kenal has a binary (135 moves) and 4-ary (257 moves) lid piece. The 135 move configurations also allow the solution sequence to run over the point branching into the last few moves before lid removal, and then leading to a dead end. These dead ends can also be reached when re-inserting the lid and trying to close the box. Also at the beginning of the 135 move sequences (box closed) there are some dead ends possible. Second pictures shows the box open and details of the lid pieces, the third piece shows the lid to be slid open without removal, possible for the Kenal 135 configuration just before the end of the sequence.
References

CR127 NameAlles Schiebung
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
2 6 slider   42
RemarksAdditional locking mechanism; AKA: Sternary
References

CR172 NameAquarius Drawer (5 devices)
Designer Manufacturer Year
Hiroshi Iwahara Hiroshi Iwahara 2017
Arity No of pieces Piece type Solution length function Number of moves
3 5 block   61
RemarksFive blocks form a ternary chain of pieces, with two small drawers at the ends. First drawer can be opened after 5 device moves, the other requires 61 moves.
References

CR061 NameA Slide-ly Tricky Tower

[1]
[2] [3]
Designer Manufacturer Year
Yee Dian Lee Yee Dian Lee 1999
Arity No of pieces Piece type Solution length function Number of moves
3 6 sliding piece Θ( 3m ) 485
RemarksDifferent movement schemes by using additional blocking pieces. In the worst case (all blocking pieces), it has 2·3m—1—1 moves. Other cases have just Θ(2m) moves.
References

CR046 NameApricot
  Designer Manufacturer Year
Akio Kamei Akio Kamei 2002
Arity No of pieces Piece type Solution length function Number of moves
2   panel    
Remarks 
References

CR023 NameAuf dem Holzweg
Designer Manufacturer Year
Juergen Reiche Siebenstein 2011
Arity No of pieces Piece type Solution length function Number of moves
3 4 slider    
RemarksTwo arity 3 puzzles in one
References

CR039 NameBarcode Burr

[1]
[2] [3]
Designer Manufacturer Year
Lee Krasnow Lee Krasnow 2004
Arity No of pieces Piece type Solution length function Number of moves
2 6 burr piece 2m+1—4 124
RemarksVariant: CR137; some shortcuts exist
References

CR137 NameBarcode Burr (3D printed)
Designer Manufacturer Year
Lee Krasnow Stephen Miller 2014
Arity No of pieces Piece type Solution length function Number of moves
2 6 burr piece 2m+1—4 124
Remarks3D printed reproduction of CR039 in limited run; some shortcuts exist
References

CR160 NameB-Box

[1]
[2]
Designer Manufacturer Year
Goh Pit Khiam Eric Fuller 2016
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel   135
RemarksThis is a combination of puzzle box and burr. Not only the panels can be opened and removed, but also the frame can be taken apart completely. Inside the box is a second puzzle, the Reactor by Eric Fuller, a small puzzle box.
References

CR115 NameBicomplementary Formation b/b:1/2

[1]
[2] [3]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2014
Arity No of pieces Piece type Solution length function Number of moves
2 11 bars, sticks   70
RemarksV1, N01; Two interlocking binary sequences (one of bars, one a bit hidden of the sticks). Beside the binary moves, this puzzle also contains burr moves without an n-ary scheme and with half-notches. The number of moves contains the binary sequences and some of the burr like moves.
References

CR134 NameBi-Nary

[1]
[2]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 slider-pair+ball   259
RemarksThis puzzle combines the mechanisms of CR064 and CR087. The pictures shows second and first edition. The second is more stable and removes a solution issue of the first.
References

CR006 NameBin Laden
Designer Manufacturer Year
Rik van Grol Rik van Grol 2006
Arity No of pieces Piece type Solution length function Number of moves
2 4 drawer   21
RemarksPartially ternary and modified sequence
References

CR150 NameBin Laden Too
Designer Manufacturer Year
Rik van Grol Rik van Grol 2015
Arity No of pieces Piece type Solution length function Number of moves
2 5 drawer    
RemarksAdditional mechanisms, modified sequence, combination of several binary sequences. The objective to remove the dice modifies the sequence even further, as a die can only be taken out when a drawer is fully extended and the drawer above in its starting position inside the box.
References

CR073 NameBinary Bud

[1]
[2] [3] [4] [5] [6]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2013
Arity No of pieces Piece type Solution length function Number of moves
2 6 leaf Θ( 2m )  
RemarksThe puzzle contains different challenges
References

CR169 NameBinary Ladder Disentanglement
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin  
Arity No of pieces Piece type Solution length function Number of moves
2 6 ring    
RemarksName unknown; this version is of early 1990s or shortly before.
References

CR063 NameBinary Key II
Designer Manufacturer Year
Goh Pit Khiam Cubicdissection 2013
Arity No of pieces Piece type Solution length function Number of moves
2 5 switch 16 [—7·(—1)m+2m+4—9] 85
RemarksVariation of CR058
References

CR163 NameB-Nary
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2016
Arity No of pieces Piece type Solution length function Number of moves
3 4 slider Θ( 3m ) 220
RemarksThe mechanism is hidden and seems to consist of the four sliders, several ball bearings, and sliding pieces. There is also one additional ball bearing that has to travel from start to goal, from where it can be put into the start position via a reset feature. During this time, the ternary sequence is executed twice (forwards, then backwards) with 54 slider moves each. The total number of moves includes these slider moves (2·54), the corresponding tilting moves to move ball bearings/sliding pieces (2·54), tilting moves to move the extra ball bearing inside the puzzle and out (3+1). Once the extra ball bearing has reached the half way position, it can go inside the sliders and cause some lockups that have to be undone by reversed moves before the regular sequence can continue.
References

CR166 NameCast Infinity
Designer Manufacturer Year
Vesa Timonen Hanayama 2016
Arity No of pieces Piece type Solution length function Number of moves
6 2 disc    
RemarksTwo interlocking discs which can rotate between six positions and can move up and down. Objective is to remove the discs.
References

CR059 NameChinese Rings 5

[1]
[2] [3]
Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
2 5 ring [2m+1/3] 21
RemarksVariant: CR042. The pictures 2 and 3 show other versions with 5 rings. Reference [14] shows a 7 ring variant including solution.
References

CR042 NameChinese Rings 9
Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
2 9 ring [2m+1/3] 341
RemarksVariant: CR059
References

CR123 NameComplementary p-arity

[1]
[2] [3] [4] [5]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2014
Arity No of pieces Piece type Solution length function Number of moves
3 14 bars, loops   102
RemarksComplimentary combination of several different sequences: Top 5 bars run in a 3-ary sequence, together with the 5 bottom bars, who run (slower) in a 2-ary sequence. These interact with the 4 loop-pieces, which run accross in a 3-ary sequence. First challenge of the puzzle is to understand these sequences, then the second is to disassemble and correctly reassemble the puzzle, with many other parts, alltogether 29 pieces.
References

CR054 NameComputer Loops
Designer Manufacturer Year
  Mag-Nif 1975
Arity No of pieces Piece type Solution length function Number of moves
2 8 ring [2m+1/3] 170
Remarks 
References

CR031 NameComputer Puzzler No 2

[1]
[2]
Designer Manufacturer Year
  Tenyo  
Arity No of pieces Piece type Solution length function Number of moves
2 4 loop Θ( 2m )  
Remarks 
References

CR034 NameComputer Puzzler No 5

[1]
[2]
Designer Manufacturer Year
  Tenyo  
Arity No of pieces Piece type Solution length function Number of moves
2 4 loop    
Remarkswas No 3 earlier; Variants: CR003, CR013, CR024, CR045, CR081, CR122
References

CR010 NameCrazy Elephant Dance

[1]
[2] [3]
Designer Manufacturer Year
Markus Goetz Peter Knoppers 2005
Arity No of pieces Piece type Solution length function Number of moves
3 5 elephant 3·(2m—1)—2·m 83
RemarksThe second and third pictures show the original prototype of the puzzle.
References

CR175 NameCorn on the Cob I
Designer Manufacturer Year
Aaron King (Wang Yulong) Aaron King (Wang Yulong) 2017
Arity No of pieces Piece type Solution length function Number of moves
2 9 ring pairs    
RemarksThis is mainly a (binary) Chinese Rings puzzle with single rings. The second ring of each pair is dropped from the main bar when the corresponding ring get's off the bar. It will then stay unhooked, while the primary ring follows the usual Chinese Rings sequence.
References

CR047 NameCross and Crown

[1]
[2]
Designer Manufacturer Year
Louis S. Burbank   1913
Arity No of pieces Piece type Solution length function Number of moves
5 4 rivet 2·5m 1250
RemarksVariants: CR120, CR121, CR158
References

CR120 NameCross and Crown 2013
Designer Manufacturer Year
Louis S. Burbank Robrecht Louage 2013
Arity No of pieces Piece type Solution length function Number of moves
5 4 rivet 2·5m 1250
RemarksReproduction based on the original patent; Variants: CR047, CR121, CR158
References

CR121 NameCross and Crown 7
Designer Manufacturer Year
Louis S. Burbank, Michel van Ipenburg Robrecht Louage 2014
Arity No of pieces Piece type Solution length function Number of moves
5 4 rivet 2·7m 4802
RemarksVariants: CR047, CR120, CR158
References

CR140 NameCrossing
Designer Manufacturer Year
Jack Krijnen Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
2 4 sliding piece    
Remarks 
References

CR056 NameCUBI
  Designer Manufacturer Year
Akio Kamei Akio Kamei 1985
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel 2m—1 32
RemarksVariant: CR048, CR162
References

CR092 NameDelirium

[1]
[2] [3] [4]
Designer Manufacturer Year
Stéphane Chomine Claus Wenicker 2013
Arity No of pieces Piece type Solution length function Number of moves
2 28 burr pieces (2m+2—1 — (m mod 2))/3 357913941
RemarksSimplified version of: CR012, CR076, pictures show versions with 28, 5, 6, and 38 special pieces. Reference 1 shows form for arbitrary many special pieces. Variant: CR154
References

CR154 NameDelirium 13
Designer Manufacturer Year
Stéphane Chomine Johan Heyns 2016
Arity No of pieces Piece type Solution length function Number of moves
2 12 burr pieces (2m+2—1 — (m mod 2))/3 5461
RemarksSimplified version of: CR012, CR076. Reference 1 shows form for arbitrary many special pieces. Variant: CR092
References

CR075 NameDevil's Cradle
Designer Manufacturer Year
  Rick Irby 2000
Arity No of pieces Piece type Solution length function Number of moves
3 4 pair of loops    
Remarks Variants: CR014, CR030, CR067, CR068, CR069, CR070
References

CR060 NameDie Welle
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2010
Arity No of pieces Piece type Solution length function Number of moves
5 3 ball 5m—1 124
Remarks 
References

CR124 NameDigi Fork Lock

[1]
[2]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2014
Arity No of pieces Piece type Solution length function Number of moves
4 5 slider    
RemarksEnhancement of CR093
References

CR159 NameDigits' Compressor

[1]
[2]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2016
Arity No of pieces Piece type Solution length function Number of moves
2 8 disc   49
RemarksGoal is to compress the digit stack to minimal height by rotating discs and moving them verticalle, and additionally to line up the red markings with the four red markings on top and bottom parts. There are several different-length dead end sequences. The five gray discs move in a binary symmetric Gray code sequence, unlike the black ones. Each disc has an orange pin, which can interock with two different holes in the disc below, i.e. two differet positions for each disc.
References

CR158 NameDisc & Crown CFF 100 Jubilee Edition Puzzle

[1]
[2]
Designer Manufacturer Year
Robrecht Louage, Michel van Ipenburg Robrecht Louage 1916
Arity No of pieces Piece type Solution length function Number of moves
3 3 rivet 2·3m 54
RemarksLimited edition of 500 that was a gift with CFF issue 100. Variants: CR047, CR120, CR121
References

CR074 NameDispersed GC Lock

[1]
[2] [3] [4]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2013
Arity No of pieces Piece type Solution length function Number of moves
2 9 switch   184
RemarksCode corresponds to setting 1100 of CR020
References

CR143 NameDITWIBIN

[1]
[2] [3] [4]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2014
Arity No of pieces Piece type Solution length function Number of moves
2 6 slider Θ( 2m )  
RemarksOne of the simplest designs of a whole puzzle family, with different number of sliders, disks, and arities. This design was devised fist for higher order variants in August 2014, about a month before this puzzle. One of the higher order variants is CR149.
References

CR170 NameDouble Helix
Designer Manufacturer Year
Goh Pit Khiam Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 pairs of burr sticks   73
RemarksThis puzzle consists of 20 pieces, of which 8 are the special pieces (middle layers). A pair of pieces makes up one level, as outlined in the article referenced below.
References

CR135 NameDouble Loop
Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
3 2 loop pairs    
RemarksVariant: CR067 (other variants see there)
References

CR101 NameDragonfly

[1]
[2]
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 4 rings   23
Remarksextra rings for symmetry; second picture shows Airplane puzzle
References

CR146 NameDrunter & Drueber
Designer Manufacturer Year
Juergen Reiche Siebenstein 2015
Arity No of pieces Piece type Solution length function Number of moves
2 4 loop Θ( 2m )  
Remarks 
References

CR014 NameElectro 1

[1]
[2] [3]
Designer Manufacturer Year
  Tenyo  
Arity No of pieces Piece type Solution length function Number of moves
3 3 pair of loops   26
RemarksThe second picture shows an unknown variant. Variants: CR030, CR067, CR068, CR069, CR070, CR075
References

CR015 NameElectro 2
Designer Manufacturer Year
Gabor Vizelyi Tenyo 1981
Arity No of pieces Piece type Solution length function Number of moves
2 7 loop   25
Remarks 
References

CR153 NameElephant Wire Puzzle
Designer Manufacturer Year
  Beijing Oriental Top Science Trading Ltd 2003
Arity No of pieces Piece type Solution length function Number of moves
2 7 loop    
RemarksVariant of: CR031 with a more irregular shape. The instructions lists 11 different starting positions as challenges.
References

CR071 NameExpansion V

[1]
[2]
Designer Manufacturer Year
Akio Kamei Akio Kamei 2013
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel 2m—1+3 35
RemarksSimpler variant of: CR094
References

CR094 NameExpansion VI

[1]
[2] [3]
Designer Manufacturer Year
Akio Kamei Akio Kamei 2013
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel Θ( 2m ) 83
RemarksMore complicated variant of: CR071
References

CR157 NameExtended Chinese Rings
Designer Manufacturer Year
Ruan Liuqi Bob Easter  
Arity No of pieces Piece type Solution length function Number of moves
2   rings Θ( 2m )  
RemarksVarious extended Chinese Rings, based on the designs from the book given in the references section. These designs are e.g.: CR108, CR110, CR111
References

CR072 NameFerris Wheel
Designer Manufacturer Year
Jean Carle Eureka 3D Puzzles
Arity No of pieces Piece type Solution length function Number of moves
2 3 loop    
RemarksVariant: CR144
References

CR147 NameFibula Puzzle
Designer Manufacturer Year
Lord Minimal Monkeys Cage 2015
Arity No of pieces Piece type Solution length function Number of moves
2 3 loop    
Remarks 
References

CR052 NameFidgety Rabbits
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2012
Arity No of pieces Piece type Solution length function Number of moves
2 7 rabbit disc Θ( 2m ) 170
RemarksVariant: CR018
References

CR018 NameFidgety Rabbits ternary
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2012
Arity No of pieces Piece type Solution length function Number of moves
3 6 rabbit disc Θ( 3m ) 230
RemarksVariant: CR052
References

CR129 NameFind my Hole
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 3 discs    
RemarksThis puzzle contains three disks, but from a mathematical view, the top and bottom disc act as one and have to be moved simultaneously in different directions. Additional locking mechanism.
References

CR098 NameFish

[1]
[2]
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 4 rings   27
RemarksThe first picture shows the "Wicked Wire" version by Professorpuzzle.
References

CR161 NameFishing Hook Chain 9-Ring
Designer Manufacturer Year
Aaron King (Wang Yulong) Aaron King (Wang Yulong) 2016
Arity No of pieces Piece type Solution length function Number of moves
3 9 rings+loops    
RemarksThe three states of each ring+loop pair are: main bar through the loop (or "fishing hook", initial configuration), through the ring, and off both. When reassembling the puzzle, an additional challenge arises: it may easily happen that some hooks end up on the main loop in wrong orientation. As this can only be seen after many (up to 1000s) of moves, careful planning is advised and analaysis of smallers problem of the first few hooks only. One feasible approach is to arrange the loops in an alternating pattern above and below the the backbone while running through the sequence.
References

CR114 NameFootball
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings   21
Remarks 
References

CR110 NameFortune
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 12 rings   993
Remarks 
References

CR081 NameFrame & Loop Octet

[1]
[2]
Designer Manufacturer Year
Abraham Jacob Abraham Jacob 2013
Arity No of pieces Piece type Solution length function Number of moves
2 8 loop    
Remarks Variants: CR003, CR013, CR024, CR034, CR045, CR122
References

CR024 NameFrame & Loop Quartet
Designer Manufacturer Year
Abraham Jacob Abraham Jacob 2009
Arity No of pieces Piece type Solution length function Number of moves
2 4 loop    
RemarksVariant of Computer Puzzler No 5; Variants: CR003, CR013, CR034, CR045, CR081, CR122
References

CR003 NameFrame & Loop Quintet
Designer Manufacturer Year
Abraham Jacob Abraham Jacob 2009
Arity No of pieces Piece type Solution length function Number of moves
2 5 loop    
Remarks Variants: CR013, CR024, CR034, CR045, CR081 , CR122
References

CR122 NameFrame & Loop Septet
Designer Manufacturer Year
Abraham Jacob Abraham Jacob 2014
Arity No of pieces Piece type Solution length function Number of moves
2 7 loop    
Remarks Variants: CR003, CR013, CR024, CR034, CR045, CR081
References

CR045 NameFrame & Loop Sextet
Designer Manufacturer Year
Abraham Jacob Abraham Jacob 2009
Arity No of pieces Piece type Solution length function Number of moves
2 6 loop    
Remarks Re-released for IPP34 Exchange; Variants: CR003, CR013, CR024, CR034, CR081, CR122
References

CR013 NameFrame & Loop Trio
Designer Manufacturer Year
Abraham Jacob Abraham Jacob 2009
Arity No of pieces Piece type Solution length function Number of moves
2 3 loop    
RemarksVariants: CR003, CR024, CR034, CR045, CR081, CR122
References

CR079 NameFrequency Doubler 1

[1]
[2]
Designer Manufacturer Year
Oskar van Deventer Tom Lensch 2012
Arity No of pieces Piece type Solution length function Number of moves
5 4 slider 4·(m2+2·m—1) 28
RemarksVariants: CR080, CR090 older design, but first made in this version in 2012
References

CR080 NameFrequency Doubler 2

[1]
[2]
Designer Manufacturer Year
Oskar van Deventer Tom Lensch 2012
Arity No of pieces Piece type Solution length function Number of moves
5 6 slider 4·(m2+2·m—1) 56
RemarksVariants: CR079, CR090 older design, but first made in this version in 2012
References

CR090 NameFrequency Doubler 3

[1]
[2]
Designer Manufacturer Year
Oskar van Deventer Tom Lensch 2012
Arity No of pieces Piece type Solution length function Number of moves
5 8 slider 4·(m2+2·m—1) 92
RemarksVariants: CR079, CR080, second picture shows solved state, older design, but first made in this version in 2012
References

CR089 NameGC machine ternary

[1]
[2] [3] [4] [5] [6]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2013
Arity No of pieces Piece type Solution length function Number of moves
3 5 switch   141
RemarksGoal is to move all switches to the "far out" position
References

CR065 NameGeneration Lock

[1]
[2]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
15 8 slider 2·15m—1 341718750
Remarkssecond picture shows comparison with CR037; Variant: CR037
References

CR067 NameGordian Knot
Designer Manufacturer Year
  Eureka 3D Puzzles  
Arity No of pieces Piece type Solution length function Number of moves
3 2 pair of loops    
Remarks Variants: CR014, CR030, CR068, CR069, CR070, CR075, CR135; alternative version named "Gekkenwerk" was devised by Jack Botermans, see reference section [13] pp. 76 and 77, including a solution
References

CR068 NameGordian Knot 2

[1]
[2]
Designer Manufacturer Year
  Huso Taso 2013
Arity No of pieces Piece type Solution length function Number of moves
3 1 pair of loops    
Remarks The second picture shows a variant built by Jan Sturm (new in 2014). Variants: CR014, CR030, CR067, CR069, CR070, CR075
References

CR069 NameGordian Knot 4
Designer Manufacturer Year
  Huso Taso 2013
Arity No of pieces Piece type Solution length function Number of moves
3 2 pair of loops    
Remarks Variants: CR014, CR030, CR067, CR068, CR070, CR075
References

CR070 NameGordian Knot 6
Designer Manufacturer Year
  Huso Taso 2013
Arity No of pieces Piece type Solution length function Number of moves
3 3 pair of loops    
Remarks Variants: CR014, CR030, CR067, CR068, CR069, CR075
References

CR112 NameGourd
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 3 rings   5
Remarks 
References

CR118 NameGrydlock

[1]
[2]
Designer Manufacturer Year
Robert Hilchie Robert Hilchie 1993
Arity No of pieces Piece type Solution length function Number of moves
2 10 slider 3m/2—1 242
RemarksThe puzzle can be built with various slider shapes, leading to different mazes. Most of them are not n-ary, like the one shown in the pictures. Several puzzles have been implemented as online version (see reference [2]), an n-ary one has also has been implemented — please see reference [3], and for this the solution length and other details are provided here.
References

CR108 NameHappiness
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 13 rings   364
Remarks 
References

CR043 NameHanui
  Designer Manufacturer Year
Yoshiyuki Kotani   1994
Arity No of pieces Piece type Solution length function Number of moves
3 5 U piece 3m—1·2i—1 242
RemarksPiece i not allowed on middle position; 242 moves for i=biggest piece
References

CR020 NameHexadecimal Puzzle
Designer Manufacturer Year
William Keister Binary Arts 1970
Arity No of pieces Piece type Solution length function Number of moves
2 8 switch Θ( 2m ) 170§
RemarksBinary and 170 move sequence for setting 1110; Variants: CR040, CR066
References

CR040 NameHexadecimal Puzzle Reproduction
Designer Manufacturer Year
William Keister Bill Wylie 2011
Arity No of pieces Piece type Solution length function Number of moves
2 8 switch Θ( 2m ) 170§
RemarksBinary and 170 move sequence for setting 1110; Variants: CR020, CR066
References

CR066 NameHexadecimal Puzzle 2013
Designer Manufacturer Year
William Keister Creative Crafthouse 2013
Arity No of pieces Piece type Solution length function Number of moves
2 8 switch Θ( 2m ) 170§
RemarksBinary and 170 move sequence for setting 1110; Variants: CR020, CR040
References

CR164 NameJUNC

[1]
[2] [3]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2016
Arity No of pieces Piece type Solution length function Number of moves
2 5 slider pair    
RemarksGoal is to move all light sliders down and all dark sliders left. By unlocking and removing the transparent lid, all little square pieces can be reoriented, allowing for 425≅1015 different challenges. Not all of these are possible as can be seen from the second picture, where a partial configuration is shown with the two top-left slider pairs blocking each other, unable to move. While the first picture shows the simple standard configuration of the puzzle, the third one shows one adapted from the N522 puzzle (CR087), with nontrivial solution and polynomial solution length. The letters of the name depict the various configurations of the small squares.
References

CR049 NameJunk's Hanoi

[1]
[2]
Designer Manufacturer Year
Junk Kato    
Arity No of pieces Piece type Solution length function Number of moves
3 5 block Θ( 3m ) 161
RemarksVariation: Israelogi by ThinkinGames / Ili Kaufmann. The image shows a different version created by Dirk Weber.
References

CR053 NameK-323

[1]
[2]
Designer Manufacturer Year
Kim Klobucher Kim Klobucher 2010
Arity No of pieces Piece type Solution length function Number of moves
3 6 block   323
RemarksVariants: CR016, CR038
References

CR038 NameK-419
Designer Manufacturer Year
Kim Klobucher Kim Klobucher 2010
Arity No of pieces Piece type Solution length function Number of moves
4 6 block   419
RemarksVariants: CR016, CR053
References

CR017 NameKing-CUBI
Designer Manufacturer Year
Hiroshi Iwahara Hiroshi Iwahara 2010
Arity No of pieces Piece type Solution length function Number of moves
4 6 panel   1536
Remarks 
References

CR051 NameKugellager
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2009
Arity No of pieces Piece type Solution length function Number of moves
5 4 ball 2·5m 1250
RemarksVariants: CR027, CR028
References

CR028 NameKugellager 7
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2010
Arity No of pieces Piece type Solution length function Number of moves
7 4 rivet 2·7m 4802
RemarksVariants: CR027, CR051
References

CR027 NameKugellager 8
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2010
Arity No of pieces Piece type Solution length function Number of moves
5 4 ball 2·5m 1250
RemarksSmaller Version and upside-down to original Kugellager; AKA: Kugellager 2; Variants: CR028, CR051
References

CR088 NameLabynary

[1]
[2]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
2 4 slider pair+switches    
RemarksBeside the 8 main sliders, the puzzle contains several other smaller sliders for the interaction between the 8 main sliders. Additionally, there is a small ball and a ball maze in this puzzle, and the goal is to get the ball out at one of the three maze exits. The maze is also part of the sliders (see second image) and therefore the binary character only holds for the basic puzzle, without the ball.
References

CR029 NameLeft-Right Chinese Rings
Designer Manufacturer Year
  Jan Sturm  
Arity No of pieces Piece type Solution length function Number of moves
2 9 ring    
Remarks 
References

CR078 NameLego Ternary Gray Code Puzzle

[1]
[2]
Designer Manufacturer Year
Adin Townsend Adin Townsend 2013
Arity No of pieces Piece type Solution length function Number of moves
3 6 lever 3·(2m—1)—2·m 177
RemarksLego variant/implementation of CR010. Second picture shows the three different piece states, with one moved out to the right already. Second reference links to building instructions created by Jeremy Rayner; the puzzle can be built with the pieces of a Mindstorms NXT set, but slight modifications might be necessary depending on the actual piece set.
References

CR100 NameLock
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings   23
Remarksextra ring for symmetry
References

CR130 NameLock 14
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 3 slider+ball    
RemarksFirst shackle part has ternary sequence, second part additional trick. Mechanism is like in CR064; AKA: Alphalock
References

CR037 NameLock 250+
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2010
Arity No of pieces Piece type Solution length function Number of moves
6 4 slider Θ( 5m—1 ) 310
RemarksLowest (4th) ring piece has only 2 position and acts as slider, AKA: Big Sliding Lock, Schloss 250+; Variant: CR065
References

CR102 NameLongevity
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
5 rings   37
Remarksextra rings for symmetry
References

CR030 NameLoony Loop
Designer Manufacturer Year
  Trolbourne Ltd 1975
Arity No of pieces Piece type Solution length function Number of moves
3 2 pair of loops    
Remarks Variants: CR014, CR067, CR068, CR069, CR070, CR075
References

CR008 NameMagnetic Tower of Hanoi
  Designer Manufacturer Year
Uri Levy   2009
Arity No of pieces Piece type Solution length function Number of moves
3 5 disc Θ( 3m ) 83
Remarks 
References

CR111 NameMandarin Duck
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 15 rings   3287
Remarks 
References

CR099 NameMaze

[1]
[2]
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings   24
RemarksThe first picture shows "Rat Race" by Puzzlemaster
References

CR162 NameMechanic CUBI
Designer Manufacturer Year
Akio Kamei Akio Kamei 2005
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel 2m—1 32
RemarksVariants: CR048, CR056. Mechanism is completely made out of wood, no metal (pins) used. Kamei also included a second alternate solution with a shortcut, which will only work at the beginning of the usual sequence, and is a couple of moves only.
References

CR077 NameMeiro Maze Variant
Designer Manufacturer Year
  Fujita  
Arity No of pieces Piece type Solution length function Number of moves
3 3 pair of loops    
RemarksFirst goal is to remove the coin, second the whole thread from the metal part. Both challenges are the same ternary puzzle repeated, but for releasing the coin additional restrictions exist. This seems to be a variant of the Meiro Maze shown in reference 2.
References

CR167 NameMerry-go-round

[1]
[2] [3] [4]
Designer Manufacturer Year
Jack Krijnen, Goh Pit Khiam Jack Krijnen 2016
Arity No of pieces Piece type Solution length function Number of moves
6 6 burr sticks   13432
RemarksVariants: CR126, CR136; This puzzle is a further developed variant of the original Power Tower, and as such it also comes as a whole set of pieces. With these pieces coming as 2-ary, 3-ary, 4-ary, 5-ary, and 6-ary (in the version shown in the pictures), different configurations can be created. There is a special binary piece as a key piece that is part of all configurations as top piece. Therefore, there are 6 slots and 5 of each piece arity (only 2 for 6-ary). Reducing the massive block to a slim tower allows pieces of different length and theoretically in arbitary arity without changing the central tower or other pieces. In the pictures, different examples are shown: 3 binary pieces (solved), 6 binary pieces (solved), one of each kind (mid-solution). While the Power Tower has pairs of mirror-symmetric pieces, here all pieces of same arity are the same and have to be entered in a helical pattern. While the sequences for even and odd arity pieces differ especially at the beginning, they are the same in this puzzle. Goal is to choose a configuration, enter the pieces into the tower, and slide them until they are all flush with the tower side on one end. The maximum number of moves for the puzzle in the picture is 13432, with pieces: (2*, 5, 5, 5, 6, 6).
References

CR139 NameMini Num Lock (binary)

[1]
[2]
Designer Manufacturer Year
Goh Pit Khiam Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
2 4 slider    
RemarksVariant: CR125; the second picture shows three different even and odd base variants: bases 2, 3, and 4. The ternary one is the cross referenced Num Lock in this puzzle list.
References

CR149 NameMixTer-MaxTer

[1]
[2] [3]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2015
Arity No of pieces Piece type Solution length function Number of moves
3 8 slider Θ( 3m ) 342
RemarksOne of a whole puzzle family, with different number of sliders, disks, and arities. CR143 is a simpler variant. The goal of MinTer-MaxTer is to move the sliders from the outer discs with two slots to the outer disc with 8 slots and collect them there.
References

CR016 NameMMMDXLVI
Designer Manufacturer Year
Kim Klobucher Kim Klobucher 2010
Arity No of pieces Piece type Solution length function Number of moves
4 9 block   3546
RemarksVariants: CR038, CR053
References

CR171 NameMountain Trail
Designer Manufacturer Year
Aaron King (Wang Yulong) Aaron King (Wang Yulong) 2017
Arity No of pieces Piece type Solution length function Number of moves
2 9 rings+loops    
RemarksMain binary chinese rings chain, with three additional binary chains of 2 rings each, attached to rings 5, 7, and 9. These are interwoven with the main chain, leading to ternay subsequences, with some quaternary positions, where two subchains meet. Of each of those additional sequences, there is always only one of the two rings on the main loop. This is one of six puzzles in the Chinese 99-ring series.
References

CR033 NameMysterians
Designer Manufacturer Year
Oskar van Deventer George Miller 2002
Arity No of pieces Piece type Solution length function Number of moves
5 3 plate 5m—1 124
Remarks 
References

CR131 NameN5
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 2 slider-pair   20
Remarks Variants: CR087, CR132,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. The number of moves is for putting all sliders up/to the right (calculcated with Burr-Tools, see second reference), with one additional move of the left slider to remove the ball, totalling 21.
References

CR132 NameN52
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 3 slider-pair   68
Remarks Variants: CR087, CR131,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. The number of moves is for putting all sliders up/to the right (calculcated with Burr-Tools, see second reference), with three additional moves of the left slider to remove the ball, totalling 71.
References

CR087 NameN522

[1]
[2]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
3 4 slider-pair   212
RemarksAKA: "522"; Variants: CR131, CR132,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. The number of moves is for putting all sliders up/to the right (calculcated with Burr-Tools, see second reference), with five additional moves of the left slider to remove the ball, totalling 217. This is the first model of the series. Physically built have been all versions from 2+2 to 10+10 sliders, and some are presented on this page.
References

CR133 NameN522222222
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 10 slider-pair    
Remarks Variants: CR087, CR131,CR133; Beside ternary sliders in two orientations, there is a simple ball maze built into the left end of the sliders and the ball is to move from bottom to top as goal, while the moving sliders obstruct and open some of the maze parts. The second picture shows the position in which the maze is usable; only the first slider has to be moved up and down while the ball traverses the maze. Number of moves yet to be determined. This is the biggest of the series actually built.
References

CR116 NameNew Puzzle Rings 3
Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
2 3 ring    
RemarksVariant: CR117
References

CR117 NameNew Puzzle Rings 5
Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
2 5 ring    
RemarksVariant: CR116
References

CR106 NameNine Twists
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings    
Remarks 
References

CR125 NameNum Lock

[1]
[2]
Designer Manufacturer Year
Goh Pit Khiam Tom Lensch 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 sliders 16 (3m—2) —1  143
RemarksVariant: CR139
References

CR151 NameOh Sh*t! Puzzle
Designer Manufacturer Year
  Woodenworks  
Arity No of pieces Piece type Solution length function Number of moves
2 8 loop Θ( 2m )  
RemarksVariants: CR032, CR145
References

CR107 NamePagoda

[1]
[2] [3]
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 9 rings   397
RemarksThe third picture shows the simpler variant "Tree Puzzle" by Puzzlemaster
References

CR009 NamePanex Gold
Designer Manufacturer Year
Toshio Akanuma TRICKS 1983
Arity No of pieces Piece type Solution length function Number of moves
3 20 slider   31537
RemarksVariant: CR035
References

CR035 NamePanex Silver
Designer Manufacturer Year
Toshio Akanuma TRICKS 1983
Arity No of pieces Piece type Solution length function Number of moves
3 20 slider   31537
RemarksVariant: CR009
References

CR128 NamePanex Squared
Designer Manufacturer Year
John Haché John Haché 2014
Arity No of pieces Piece type Solution length function Number of moves
3 12 slider   68
RemarksVariant of CR035 which includes overlapping and interacting Panex instances. As the original design was not solvable (as discovered by Bob Hearn), this puzzle has to be modified by removing the blocking mechanism in the center. Goals are: 1) to swap pieces horizontally (e.g. A and B), and 2) swap pieces vertically (e.g. A and C), obeying the Panex rules, i.e.: in the vertical grooves, no smaller piece can be below (i.e. closer to the center) than a larger piece, same for the horizontal grooves (no larger piece closer to the center, but for both sides of the groove). This modification was proposed by Diniar Namdarian in 2015. The solutions provided have 46 moves for swapping A and B, and 68 moves for swapping A and C.
References

CR113 NamePear
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 7 rings   62
Remarks 
References

CR036 NamePharaoh's Dilemma
Designer Manufacturer Year
  Mag Nif 1970
Arity No of pieces Piece type Solution length function Number of moves
3 6 disc    
RemarksTower of Hanoi variant
References

CR109 NamePhoenix

[1]
[2]
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 16 rings   502
Remarks 
References

CR119 NamePin Burr 2

[1]
[2] [3] [4]
Designer Manufacturer Year
Jerry McFarland Jerry McFarland 2014
Arity No of pieces Piece type Solution length function Number of moves
2 4 burr pieces Θ( 2m ) 38§
RemarksBinary sequence, which is non-GC based and uses a pin-maze-mechanism, a little trick was added corrupting the sequence and making it more interesting for the solver. The third picture shows the prototype, which has a simpler frame but same sequence, the last picture shows both puzzles.
References

CR126 NamePower Tower

[1]
[2] [3]
Designer Manufacturer Year
Jack Krijnen, Goh Pit Khiam Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 burr sticks 3mm—1  76
RemarksVariants: CR136, CR167
References

CR136 NamePower Tower (mixed base — variable stage)

[1]
[2]
Designer Manufacturer Year
Jack Krijnen, Goh Pit Khiam Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
5 6 burr sticks 2·(nm—1)/(n—1)—m 7806
RemarksVariants: CR126, CR167; This Power Tower is a whole set with a block hosting up to 6 stages, a blocker piece to set the number of stages (between 3 and 5, 6 stages without blocker), and a set of pieces for each of the two orientations (two different woods). The pieces come in binary, ternary, and quaternary shape and can be combined arbitrarily, leading to mixed (or uniform) base sequences, which can be quite confusing. There are 1080 different possibilities, with the level varying from 11 to 2724. The solution length is for a uniform n-ary configuration with m pieces. Addition: This now includes an extension set of quinary pieces. The overall entry now contains these pieces and there are now solutions possible up to level 7806.The second picture shows this extension set.
References

CR141 NamePower Box

[1]
[2]
Designer Manufacturer Year
Goh Pit Khiam Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
2 6 panels    
Remarks 
References

CR032 NamePuzzle H
Designer Manufacturer Year
  Eureka 3D Puzzles 1997
Arity No of pieces Piece type Solution length function Number of moves
2 5 loop Θ( 2m )  
RemarksVariant: CR145
References

CR004 NamePyraCircle
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2008
Arity No of pieces Piece type Solution length function Number of moves
3 10 block   116
RemarksVariation of Panex, a non-disjoint union of several such puzzles; 116 is minimum number of moves
References

CR082 NameQuatro
Designer Manufacturer Year
Eric Johansson Eureka 3D Puzzles  
Arity No of pieces Piece type Solution length function Number of moves
2 4 loop Θ( 2m ) 7
RemarksOne of the solutions (reference 2) acts like Chinese Rings, please see reference 3. There are also other solutions.
References

CR148 NameRacktangle
Designer Manufacturer Year
Goh Pit Khiam Tom Lensch 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 plate    
RemarksVariable number of stages (1 to 4, box is built modular) and plates of base 2 and 3 included, which together with the solid plate for the lowest position, can be used to create all mixed base 2 and 3 puzzles for up to 4 stages.
References

CR093 NameRailing with Draining ternary

[1]
[2] [3] [4] [5]
Designer Manufacturer Year
Namick Salakhov Namick Salakhov 2013
Arity No of pieces Piece type Solution length function Number of moves
3 5 slider    
Remarks 
References

CR173 NameReflection
Designer Manufacturer Year
Aaron King (Wang Yulong) Aaron King (Wang Yulong) 2017
Arity No of pieces Piece type Solution length function Number of moves
4 9 pairs of rings    
RemarksVariation directly created from Chinese Rings by attaching a small connector ring and a second bigger ring to each ring, below the first and around the same vertical rod. There are four states for each ring pair: main bar through lower ring (initial position), through upper ring, through both rings ("double ring"), and off the rings. All those appear in the solution, and the double ring configuration is used to mimic the classic binary chinese rings. The configurations with one ring on the loop appear exactly once in the solution sequence, and their transitions interrupt the binary sequence in a regular pattern and increase the number of moves considerably. This is one of six puzzles in the Chinese 99-ring series.
References

CR155 NameReTern Key
Designer Manufacturer Year
Goh Pit Khiam Charlie Rayner 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 slider 8 · 3m—1—8·m+1 185
RemarksThe full name is "The Return of Tern Key" and demonstrates a variant of CR125 without a long synchronizing slider piece. Variant: CR168
References

CR168 NameReTern Key with circular pieces
Designer Manufacturer Year
Fredrik Stridsman Fredrik Stridsman 2017
Arity No of pieces Piece type Solution length function Number of moves
3 4 slider    
RemarksVariant: CR155. The ReTern Key was the base for this puzzle, and the designer replaced the groups of small pieces running on the sides of the puzzle for synchronization by circular pieces.
References 

CR021 NameRings Bottle
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2012
Arity No of pieces Piece type Solution length function Number of moves
2 6 ring    
Remarks 
References

CR041 NameRow to Row
  Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
3 8 disc Θ( 2m )  
Remarks 
References

CR007 NameRudenko Clips
Designer Manufacturer Year
Valery Rudenko Roscreative 2011
Arity No of pieces Piece type Solution length function Number of moves
3 7 clip ( 3m—1 )/2 1093
RemarksTower of Hanoi with move restriction
References

CR019 NameRudenko Disc
Designer Manufacturer Year
Valery Rudenko Roscreative 2011
Arity No of pieces Piece type Solution length function Number of moves
3 7 disc Θ( 2m )  
RemarksTower of Hanoi with simplification
References

CR044 NameRudenko Matryoshka
Designer Manufacturer Year
Valery Rudenko Roscreative 2011
Arity No of pieces Piece type Solution length function Number of moves
3 7 slider Θ( 2m )  
RemarksTower of Hanoi equivalent
References

CR086 NameSeestern
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
11 3 Layers 11m—1 1330
Remarks 
References

CR001 NameShort Big Sliding Lock

[1]
[2]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2009
Arity No of pieces Piece type Solution length function Number of moves
5 3 slider 2·5m 250
RemarksAKA: Kleines dickes Schloss, Voidlock
References

CR064 NameSix Bottles
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
3 6 slider+ball 4·(2m—1) 252
RemarksEach metal ball can be in a top-left, bottom-left, or a bottom-right position, and there are corresponding slider positions middle and top. The bottom slider position occurs only during transition of ball between left and right. A newer circular version replacing balls by switches is: CR083
References

CR011 NameSliding-Block Chinese Rings-style Puzzle

[1]
[2] [3]
Designer Manufacturer Year
Bob Hearn   2008
Arity No of pieces Piece type Solution length function Number of moves
3 3 pair of yellow blocks    
RemarksThe second picture shows the different position of the special pieces, the pairs of yellow blocks in positions 0, 1, and 2. There are other positions not part of the solution. Recently, we found a shorter, non-ternary solution that was not intended, with goal configuration in third picture; under investigation.
References

CR142 NameSlots and Pins (mixed base)
Designer Manufacturer Year
Goh Pit Khiam Jack Krijnen 2014
Arity No of pieces Piece type Solution length function Number of moves
3 5 slider    
RemarksThis version has mixed bases, i.e. binary and ternary pieces/piece parts.
References

CR048 NameSmall CUBI
Designer Manufacturer Year
Akio Kamei Akio Kamei 2010
Arity No of pieces Piece type Solution length function Number of moves
2 6 panel 2m—1 32
RemarksVariants: CR056, CR162. Mechanism is completely made out of wood, no metal (pins) used.
References

CR022 NameSpinOut
Designer Manufacturer Year
William Keister Binary Arts 1970 / 2006
Arity No of pieces Piece type Solution length function Number of moves
2 7 disc [2m+1/3] § 85 §
RemarksVersion with elephants and reset shortcut, green/red/orange. There exists an unintended shortcut solution with 49 moves (see Jaaps's page below). Variants: CR026, CR050
References

CR050 NameSpinOut
Designer Manufacturer Year
William Keister Binary Arts 1970 / 1987
Arity No of pieces Piece type Solution length function Number of moves
2 7 disc [2m+1/3] § 85 §
RemarksThere exists an unintended shortcut solution with 49 moves (see Jaaps's page below); Variants: CR022, CR026
References

CR026 NameSpinOut
Designer Manufacturer Year
William Keister ThinkFun 1970 / 2001
Arity No of pieces Piece type Solution length function Number of moves
2 7 disc [2m+1/3] § 85 §
Remarksreset shortcut. There exists an unintended shortcut solution with 49 moves (see Jaaps's page below); Variants: CR022, CR050
References

CR084 NameSpiralschloss
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
3 4 shackle-layer 2·(nm—1) 160
RemarksMechanism similar to CR085. Goal: Open all shackle-layers completely.
References

CR083 NameSteuerrad
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
3 8 slider+switch 3·2m 768
RemarksRound variant of CR064. Goal: Move all handles to the outer position and reveal hidden message "Nicht durchdrehen", German for "do not get mad" and also referring to turning the steering wheel (German: Steuerrad)
References

CR025 NameSuper-CUBI

[1]
[2] [3]
Designer Manufacturer Year
Hiroshi Iwahara Hiroshi Iwahara 2000
Arity No of pieces Piece type Solution length function Number of moves
3 6 panel   324
RemarksFirst image shows newer version (opposite panels following in solution), second and third the older version (panels following in 90° turn order); Variant: CR165
References

CR165 NameSuper-CUBI (small)

[1]
[2]
Designer Manufacturer Year
Hiroshi Iwahara Hiroshi Iwahara 2016
Arity No of pieces Piece type Solution length function Number of moves
3 6 panel   324
RemarksSmaller version of original Super-CUBI with adjacent panels moving on opposite sites. Comes with a solution leaflet showing all 324 moves, and additionally some instructions on how to calculate and identify the current configuration. Varaiant: CR025
References

CR104 NameTeapot
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings   22
Remarks 
References

CR002 NameTern Key
Designer Manufacturer Year
Goh Pit Khiam Cubicdissection 2009
Arity No of pieces Piece type Solution length function Number of moves
3 4 switch 12·(2m)—12·m—10 134
Remarks 
References

CR005 NameTernary Burr
Designer Manufacturer Year
Goh Pit Khiam Jack Krijnen 2010
Arity No of pieces Piece type Solution length function Number of moves
3 4 burr pieces 6·2m—4·m—5 § 75§
RemarksMove count includes control bar; Variant with only two frame pieces; Variants: CR055, CR095
References

CR055 NameTernary Burr
Designer Manufacturer Year
Goh Pit Khiam Mr. Puzzle 2009
Arity No of pieces Piece type Solution length function Number of moves
3 4 burr pieces 6·2m—4·m—5 § 75§
RemarksMove count includes control bar; 95 moves for complete disassembly; Variants: CR005, CR095
References

CR095 NameTernary Burr
Designer Manufacturer Year
Goh Pit Khiam Eric Fuller 2013
Arity No of pieces Piece type Solution length function Number of moves
3 4 burr pieces 6·2m—4·m—5 § 75§
RemarksMove count includes control bar; 95 moves for complete disassembly; Variants: CR005, CR055
References

CR012 NameThe Binary Burr

[1]
[2]
Designer Manufacturer Year
Bill Cutler Jerry McFarland 2003
Arity No of pieces Piece type Solution length function Number of moves
2 6 burr pieces ( (-1)m+1 + 2m+2 ) / 3  § 85§
RemarksMove count includes control bar; There is also a very rare 10 ring piece version, which is shown in the pictures; Variants: CR076, CR156
References

CR076 NameThe Binary Burr
Designer Manufacturer Year
Bill Cutler Eric Fuller 2013
Arity No of pieces Piece type Solution length function Number of moves
2 6 burr pieces ( (-1)m+1 + 2m+2 ) / 3  § 85§
RemarksMove count includes control bar; Variants: CR012, CR156
References

CR156 NameThe Binary Burr (small)

[1]
[2] [3] [4] [5] [6] [7] [8] [9]
Designer Manufacturer Year
Bill Cutler Maurice Vigouroux 2016
Arity No of pieces Piece type Solution length function Number of moves
2 10 burr pieces ( (-1)m+1 + 2m+2 ) / 3  § 1385§
RemarksMove count includes control bar; the first picture shows the whole group of the Binary Burrs (small) with 3 to 10 special pieces, all with solid cage, the other pictures show the individual puzzles; Variants: CR076, CR012
References

CR057 NameThe Brain

[1]
[2] [3] [4]
Designer Manufacturer Year
Marvin H. Allison, Jr. Mag-Nif 1973
Arity No of pieces Piece type Solution length function Number of moves
2 8 switch [2m+1/3] 170
RemarksPictures 2, 3, and 4, and reference 2 show the newer version
References

CR096 NameThe Cat
Designer Manufacturer Year
William Keister Binary Arts 1985
Arity No of pieces Piece type Solution length function Number of moves
2 2 rings    
Remarks 
References

CR097 NameThe Horse
Designer Manufacturer Year
William Keister Binary Arts 1985
Arity No of pieces Piece type Solution length function Number of moves
2 3 rings    
Remarks 
References

CR058 NameThe Key

[1]
[2]
Designer Manufacturer Year
Goh Pit Khiam Walt Hoppe 2004
Arity No of pieces Piece type Solution length function Number of moves
2 4 switch   40
RemarksVariant: CR063
References

CR062 NameTower of Hanoi
Designer Manufacturer Year
Edouard Lucas Philos (and others) 1883
Arity No of pieces Piece type Solution length function Number of moves
2 9 disc Θ( 2m )  
Remarks 
References

CR105 NameTrapeze

[1]
[2] [3]
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings   61
Remarksextra rings for symmetry; third picture shows puzzle Dingo Trap, a variant with the rings separated and held by smaller loops; reference [14] shows this variant including building instructions and solution
References

CR144 NameTricky Frame
Designer Manufacturer Year
  Philos  
Arity No of pieces Piece type Solution length function Number of moves
2 3 loop Θ( 2m )  
RemarksVariant: CR072
References

CR145 NameTricky Mouse
Designer Manufacturer Year
  Philos  
Arity No of pieces Piece type Solution length function Number of moves
2 4 loop Θ( 2m )  
RemarksVariant: CR032
References

CR085 NameUhrwerk

[1]
[2]
Designer Manufacturer Year
Jean-Claude Constantin Jean-Claude Constantin 2013
Arity No of pieces Piece type Solution length function Number of moves
3 4 ball/gear 2·(nm—1) 160
RemarksMechanism similar to CR084. Goal: Move the one ball with the special starting position to its third hole and remove (only) this ball from puzzle. The two pictures show second (more stable) and first edition.
References

CR138 NameUnknown Disentanglement
Designer Manufacturer Year
     
Arity No of pieces Piece type Solution length function Number of moves
2 3 loops    
RemarksChoosing one of the sides, the consecutive loops on that side will act like a Chinese Rings puzzle. All other loops are not part of the solution.
References

CR152 NameViking Box
Designer Manufacturer Year
Sven Baeck, Jean-Claude Constantin Jean-Claude Constantin 2014
Arity No of pieces Piece type Solution length function Number of moves
3 4 switches    
RemarksThe basic mechanism of the box is a ternary mechanism consisting of discs with the switches attached and visible to the puzzler, and some ball bearings. These will move into some cutouts of the discs and block them in various positions, same general concept as in CR064. Additionally, there are two mechanisms interacting with several discs each: One visible as the bottom horizontal slider, the other hidden, but with its state visible through a small hole below the left disc. This mechanism and the ball bearings have to be manipulated via tilting. The lid contains the mechanism and is firmly closed. However, a second variant was released with a transparent top, allowing the puzzler to see most of the mechanism.
References

CR103 NameWheel
Designer Manufacturer Year
Ruan Liuqi Ingenious Rings  
Arity No of pieces Piece type Solution length function Number of moves
2 5 rings   24
Remarks 
References

3 Definitions, Examples and More

This compendium contains a special collection of puzzles which are somehow related to the famous Chinese Rings puzzle. For a definition which puzzles are included in this compendium and for a clear understanding why they are included, we will provide a structural definition of the class of puzzles in this compendium, the CR recursive puzzles. We will provide a definition, with some examples, and with some interesting properties of these puzzles.

3.1 Definition of CR Recursive Puzzle

A CR recursive puzzle is a puzzle that contains m special similar pieces (with m ≥ 1) and
  1. the puzzle can be generalized to other values of m ≥ 1 and
  2. each special piece has n different positions (e.g. 0,...,n—1, with n ≥ 2) and
  3. there is a uniform condition stating that a special piece can only move between some positions if the other special pieces are in certain positions.

Note that beside the special pieces in the definition above, there may be other pieces. For a distinction from the others, the special pieces are sometimes called "ring pieces", in analogy to the classic Chinese Rings puzzle.

This is a short and formal definition related to the structure of the puzzle. Some examples might be useful.

3.2 Examples

3.2.1 Classic CR puzzle

Chinese Rings

In the picture, a typical wooden version of CR with 5 rings is shown. Each ring may be positioned (diagonally) on the horizontal loop or off the loop. These states may be denoted as: 1 - on, 0 - off the loop, so in this example we have:

and the uniform condition for moving a ring is: The k-th ring can be moved between 1 and 0, if and only if ring (k—1) is in position 1 and all rings to the left of (k—1) are in position 0.

To conclude the matching of the particular classic CR puzzle of this example with our definition of a CR recursive puzzle, we note that there exist many different classic CR versions, with various number of rings, that correspond to the different values of m in our definition of CR recursive puzzles.

3.2.2 SpinOut

SpinOut SpinOut

In this puzzle, we have a slider carrying a line of discs. Each disc can be either in a vertical position (denoted by 0) or horizontal position (denoted by 1). This puzzle is equivalent to CR, if we restrict the moves to the ones that are used in the solution: From the vertical position, each disc can be turned left into the horizontal position, or to the right to a different horizontal position. We disregard this (right turned) position, as it is not needed for solving the puzzle. To see the uniform condition for this puzzle, we have a look at the pictures above and note: A disc can be turned between 0 and 1, if the disc immediately to the right is 0 (in vertical position) and all discs further right are 1 (in horizontal position). For the solution, two additional restrictions are implemented: Only a disc at the position with the additional space (second from the right) can be turned. Discs can only be moved out to the right when in position 1 (horizontal).

3.2.3 Crazy Elephants Dance

The Crazy Elephant Dance is a generalized version of SpinOut. Instead of discs, we have a line of 5 elephants on a slider, and each elephant has three possible states: 0 - facing upwards, 1 - facing to the right, and 2 - facing downwards.

The uniform condition is split into two parts in this case: 1. An elephant (the second from the right in the picture) may move between 0 and 1, if the one immediately to the right is in position 2, and (not shown in the picture) all further right are in position 2.

2. An elephant (again the second one) may move between 1 and 2, if the one immediately to the right is in position 0, and (not shown) all further right are in position 2.

Again, as for the SpinOut, the second part of the conditions above arises from the fact that the slider and elephants may only move out to the right when in position 2.

The pictures demonstrating the two parts of the condition also show an example how to move the second elephant from 0 to 2: It first has to be moved from 0 to 1 (first two pictures), then the elephant to the right of it is moved from 2 to 0 (third picture), then the second elephant may finally move from 1 to 2. This gives an idea what is necessary to move the leftmost elephant from 0 to 2, which is a vital step in the solution sequence.

3.2.4 Kugellager

Kugellager

The Kugellager has four balls, which are the special pieces in our definition and one slider containing some mazes for these balls. The four balls move up and down (maze in the slider permitting, green arrows), and the slider moves left and right (blue arrow). Each of the balls has 5 regular positions {0,1,2,3,4} (or {1,2,3,4,5} in the picture), and then there is an additional position 5 (or 6 in the picture), which can only be used if all balls are already in position 4, to remove the slider afterwards. This is why we may disregard this position 5.

The pictures are taken from the article [1] which also describes the movement in more detail and also lists the uniform condition for ball movement. Shortly summarized, a ball may move between positions i and (i+1), if all balls to the left are in a certain set of positions and all balls to the right are in a certain (different) set of positions.

This puzzle can not only be generalized to more balls, as the definition requires, but also to a higher or lower level, as the Kugellager 7 puzzle (n=7), or the Auf dem Holzweg puzzle (n=3) show.

3.2.5 Tower of Hanoi

Tower of Hanoi

At first sight, Tower of Hanoi looks very different from the puzzles we have seen so far, but it is a CR recursive puzzle and complies to our definition: It has m discs (m=9 in the picture), and each of the discs can be on one of the three piles built on the poles, so n=3. There are different variants with different numbers of discs, so the generalization to other values of m (even other values of n with more poles) is easy. All these variants will have to obey simple rules: Move only one disc at a time, and only the top disc of a tower, and a disc may only be laid down on discs that are bigger than itself. The rules deliver the uniform condition we need for our CR recursive definition and can also be translated to: To move a disc, all smaller discs have to be on a pole different from the start and destination positions of the move.

3.3 What does "uniform" mean in the definition?

The uniform definition covers a property that is independent of the actual number of pieces and also allows the same condition to apply to each one of them. This condition states that for all suitable indices i and j, piece number i can move between positions j and j+1 if and only if the "lower" pieces (left of i) and "higher" pieces (right of i) are in certain positions. This works independently of i: No matter if the second piece (i=2) or the fifth piece (i=5) is to be moved.

There are also puzzles for which the "higher" pieces are irrelevant, for example:

3.4 Where does the term "recursive" come from?

All puzzles contained in this compendium allow a recursive description of their solution. As an example, take the Tower of Hanoi. This puzzle has three positions, two of which are initially empty and one carries a stack of discs that are ordered from biggest in the bottom to smallest on top. The aim of the game is to move the stack of discs to the third position (whichever that may be) obeying the following two rules:

  1. Only one disc may be moved at a time
  2. Never place a bigger disc on top of a smaller one

Tower of Hanoi

A typical Algorithm to solve this problem can be described informally as follows:

Move Tower of n discs from startTower to endTower:

  1. 6 - startTower - endTower → auxiliaryTower // the tower that is not one of the two above
  2. if n>1 then
  3. Move Tower of n—1 discs from startTower to auxiliaryTower
  4. Move one disc from startTower to endTower
  5. if n>1 then
  6. Move Tower of n—1 discs from auxiliaryTower to endTower

This algorithm works by moving the n—1 top discs away on the auxiliary tower, then disc n, then the ones on the auxiliary tower to the destination tower.

What algorithm do we use in lines 3 and 6 in order to move an n—1 disc tower? It is our very same algorithm, that calls itself recursively, and now we have our justification to call this puzzle "recursive", as it can be solved by such a recursive algorithm. More details about Tower of Hanoi can be found e.g. here: [4]

While for this puzzle it may seem obvious, the question remains for the other puzzles: Why are the other puzzles in this compendium also recursive?

Well, this drills down to the core of the matter. What do all these puzzles have in common, and how are they related to the "classic" Chinese Rings puzzle? A first insight might be to look at a solution method for Chinese Rings. The goal of this puzzle is to remove all the rings from the loop.

Chinese Rings

This can be established by a recursive algorithm with the following ideas:

  • move i-th ring off the loop:
    1. Move ring i—1 (to the left of ring i) onto the loop
    2. move all rings left of i—1 off the loop
    3. then perform the movement of i-th ring

  • move i-th ring onto the loop:
    1. Move ring i—1 (to the left of ring i) onto the loop
    2. move all rings left of i—1 off the loop
    3. then perform the movement of i-th ring
Here we see that for the movement of the i-th ring, preparation moves must be performed, and for these the algorithm can invoke itself recursively. More details can be found in the book [3] and a more mathematical observations in the book [6].

3.5 Tuple Representation and a Different Notation: n-ary Puzzles

Before the formalization of the recursive puzzles as above, a different notation was used in prior discussions: "n-ary Puzzles". This notation has its roots in well known puzzles: Chinese Rings, The Brain, SpinOut, The Key, and Binary Burr. These are typically called "binary" puzzles, because their "ring pieces" have two different states each, and their solution length function are asymptotically 2m. Ternary Burr, Tern Key, Crazy Elephant Dance are called "ternary" puzzles because they generalize the binary concepts to pieces having three different states. However, interestingly enough, their solution length functions are still asymptotically 2m, which is a justification for our structural definition. Why ternary puzzles can have 2m as solution length function is discussed below.

A mathematical argument for calling these puzzles "n-ary" is that their current state can be represented by an m-tuple of entries ranging in {0,...,n—1}. For example, the SpinOut starting configuration would be: (0,0,0,0,0,0,0) and the goal configuration: (1,1,1,1,1,1,1). A solution could then be described as a sequence of such tuples:

(0,0,0,0,0,0,0) → (0,0,0,0,0,0,1) → (0,0,0,0,0,0,1,1) → ... → (1,1,1,1,1,1,0) → (1,1,1,1,1,1,1)

A ternary example is the Crazy Elephant Dance, with the following encoding:

(0,0,0,0,0) → (0,0,0,0,2) → (0,0,0,1,2) → (0,0,0,1,0) → (0,0,0,2,0) → (0,0,0,2,2) → (0,0,1,2,2) → ... → (2,2,2,2,2)

3.6 Puzzle Parameter and Different View: Number of Moves

You may have noticed that In the definition of CR recursive puzzle we did not refer to the number of moves required to solve the puzzles, which is a central property in a different definition of the class of puzzles in this compendium (see discussion below). The number of moves seems related to the parameters in our definition, but no uniform relation has been determined for all the puzzles of this compendium, and this seems impossible, as we will see: The key count is the number of moves that the special pieces (or "ring pieces") move during the solution process. Similar to approaches in Computer Science, we will sometimes not provide an exact function of the number of moves, but will use an approximate notation. In such cases the exact number of moves might not have been calculated yet, but by analogy one has a strong indication what it could be like.

This notation describes the asymptotic growth and we are only interested in the fastest growing elements of this number of moves function. Following are some examples for this notation:

Exact function Approximation Note
2m+m+3 Θ( 2m ) Linear and constant summands neglectable
2 · 5n Θ( 5n ) Constant factor neglectable
2(m—1) Θ( 2m ) 2(m—1)= (1/2)·2m, constant factor neglectable
3 · (2m—1) — 2·m Θ( 2m ) Combination of examples above

The Θ notation is taken from Computer Science and Mathematics, the formal definition and details are explained in [5].

3.7 CR recursive puzzles with solution length only ~2m

Why do some CR recursive puzzles with n>2 have a solution with only ~2m moves, not ~nm? CR recursive puzzles with solution length ~nm obviously use all different combinations of piece positions in their solution. To see this, just recall that the number of m-tuples over a set with n different values is exactly nm and so all these tuples occur in the description of the solution (see tuple representation above). So the puzzles in question that have solution length ~2m will not use all these tuples in their solution, but leave some out. For example, on the (shortest) solution path for the Ternary Burr, you will never find a configuration corresponding to tuple (1,0,0,0) -- this configuration is not needed. If you have this puzzle or the equivalent Crazy Elephant Dance, just try it out (with the lowest 4 elephants)! When you try the solution for on of these puzzles, you will find that directly before reaching this position, you will have the configuration (1,0,0,1) -- which is not part of the shortest solution either -- and this will be the only successor configuration after (1,0,0,1). Shortly said: only from (1,0,0,1) you reach (1,0,0,0), and your only option is going back to (1,0,0,0), and this is a detour way back from the solution from (0,0,0,0) to (2,0,0,0). Please see picture for these solution steps in the actual puzzle.

Crazy Elephants Dance

This effect occurs in several different puzzles, but what are the reasons for some puzzles to use all possible configurations, and for some others to omit configurations in their solution? Several reasons have been observed so far:

  1. Condition for moves: Puzzles with all configurations (and hence solution length nm) have conditions that involve both lower and higher pieces. For an example see the Kugellager above. Others do only involve a part of the other pieces. The examples in this paragraph are of such nature. For Ternary Burr and Crazy Elephant Dance, it only matters which positions the lower pieces have. Pieces may be moved no matter in which positions the higher pieces currently are. This allows to bring a piece from position 0 to n—1 without touching the higher pieces at all. When solving Kugellager, you will notice that when moving the lowest piece later in the solution, you will have to move higher pieces before.
  2. The second observation deals with Tower of Hanoi and Rudenko Clips. Both have the same general structure (3 positions) and equivalent rule: However, the first one has a solution length ~2m, while the Clips need ~3m moves. Here, the condition is the same, but there is an additional condition in Rudenko Clips: The three positions 0, 1, and 2 are in a row and a clip may only move between positions 0 and 1, if the stack of smaller clips is on 2, and a clip may only move between positions 1 and 2, if the stack of smaller clips is on 0. No direct move from 0 to 2 is possible for all clips except the biggest one. For Towers of Hanoi, we may use positions freely (only obeying the "bigger disc" rule). In the picture below, the red clip is not able to move from position 0 to 2 over the green one, while this would be the canonical next step in Tower of Hanoi. For a discussion on graph representations also leading to 2m and 3m as solution lengths, please see [4].
Rudenko Clips

3.8 Tower of Hanoi - binary or ternary?

In the paragraphs above the relation between binary and ternary regarding the solution length has been discussed. There is also a binary representation that can be used for describing the solution of Tower of Hanoi, for details, please see [4].

Recently Goh Pit Khiam created a nice illustrated example of a variant of Tower of Hanoi: Linear Tower of Hanoi. In this variant, the three poles are in a line and a disc may only move from a pole to an adjacent pole. The obvious implications are that there are no direct moves between pole 0 and 2, and that a bigger disc may not move between poles 0 and 2, whenever there is a smaller one on pole 1. This makes it very similar to a Rudenko Clips (see previous section above). The less obvious implication is that the puzzle follows a ternary gray code. Please see [10] for the illustrated example of this puzzle, which also shows a nice ternary representation of Tower of Hanoi.

3.9 A different definition for n-ary puzzles based on number of moves

Our definition of CR recursive puzzle is based on the structure of the puzzle, which makes it (relatively) easy to spot if a puzzle belongs to this class. However, there is another different and commonly used definition of n-ary puzzles that first requires the puzzle to be analyzed and solved fully before it can be classified. Once one has determined that there are m special pieces and the solution length function is asymptotically equal to nm, it is classified as n-ary (in this solution length based notation). We are using the structural definition provided above (as it seems easier to apply it in most cases), but there might be some confusion. Some ternary puzzles (our definition) may have a solution with (asymptotic) length 2m, while the solution length function based definition would call them "binary" for this reason. One prominent example is the Ternary Burr by Pit Khiam Goh. This ternary (sic!) variation of the Binary Burr by Bill Cutler would be classified as binary following the solution length based definition.

4 References

[1] Goetz Schwandtner. Kugellager.pdf.

[2] Goetz Schwandtner. n-ary Puzzle Group.

[3] Ring of Linked Rings. Sydney N. Afriat. Gerald Duckworth & Co Ltd (November 1982). ISBN 0715616862

[4] Tower of Hanoi (Wikipedia).

[5] Theta Notation (Wikipedia).

[6] The Tower of Hanoi — Myths and Maths Andreas M. Hinz, Sandi Klavzar, Uros Milutinovic, Ciril Petr. Springer Basel 2013.

[7] Ingenious Rings. Yu Chong En and Zhang Wei. Beijing, 1999. The diagrams shown in the puzzle list of this Compendium for the Ingenious Rings Puzzles are taken from this book and were created by Wei Zhang.

[8] ChinesePuzzles.org: Ingenious Rings Wire puzzles

[9] ChinesePuzzle.org: Nine Linked Rings

[10] Goh Pit Khiam. The Linear Tower of Hanoi and the Ternary Gray Code.

[11] Goh Pit Khiam. Design of N-ary Mechanical Puzzles. CFF95. Rijswijk, 2014. The next entry [12] is the most recent updated version

[12] Goh Pit Khiam. Design of N-ary Mechanical Puzzles. (extended version, latest update 2014-12-15; main updates are from page 28 on: The Power Box, Racktangle, Mixed Base Power Tower illustrated overview and simplification)

[13] New Book of Puzzles. Jerry Slocum and Jack Botermans. New York, 1992.

[14] Denkspiele der Welt. Pieter van Delft and Jack Botermans. München, 1977.

[15] Rik van Grol (Ed.). CFF 100 (Cubism For Fun 100). Rijswijk, 2016.

[16] Richard I. Hess. Analysis of Ring Puzzles. Chinese Rings, Pagodas, Zig Zags. For IPP13. Amsterdam, 1993.

5 Acknowledgements

The idea for this compendium dates back a few years, and in 2012 first steps were taken for implementation, starting to collect data, searching for new puzzles and determining a common property to create a formal definition.

I wish to thank Dan Feldman for big support in the creation of this compendium during many discussions, with research on certain puzzles and editorial work on this compendium. My thanks also go to Nick Baxter and Michel van Ipenburg, with whom I had some detail discussions about the compendium and some puzzles included.

Of course I do not own all the puzzles and need pictures of puzzles that I do not have, of course observing the copyright. Thank you for picture contributions or puzzle samples for taking pictures to: Dan Feldman, Jack Krijnen, Namick Salakhov, Rob Stegmann, Dirk Weber, Yvon Pelletier, Claus Wenicker, Allard Walker, Michel van Ipenburg, Nick Baxter, Robert Hilchie, Kevin Sadler, Jerry McFarland, Stephen Miller, Jeremy Rayner, James Dalgety, and Fredrik Stridsman. Thanks to Jan de Ruiter for pointing out the similarity between Quatro and Chinese Rings. Thanks to Pit Khiam Goh for some interesting discussions around BurrTools models of the puzzles and for confirming some puzzle entry details, and also nice illustrations of puzzles and puzzle examples. Many thanks to the designers and craftsmen who provided some of their puzzles for my collection or some detail descriptions of the puzzles.

Ingenious Rings: Many thanks to Wei Zhang, Peter Rasmussen and Nick Baxter for providing me material on these wire puzzles, of which I selected the ones that seem to fit the definition well. Beside the book [7] they also have a nice web site [8] and [9] on this topic. (see references above)

6 Feedback

This compendium is not a static collection of puzzles, but a dynamic overview which will be updated when new puzzles, new details, pictures or references are available. If you would like to send me some feedback on the compendium, submit additional material or information to be added, or update some entries, please send a mail to me:

Your feedback and contributions are welcome, so please do not hesitate. I am always interested to hear some interesting background stories about the puzzles.

If you are sending pictures for publication in the compendium, please clearly indicate that you are the holder of the rights on these pictures and that you allow the use in this compendium.