Tumbler Puzzle http://www.smartbytes.co.uk/Tumbler/

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Tumbler Puzzle Solution

Contents

Starting Positioni.Completing Row 5ii.Completing Row 2iii.Completing the Last Two Rowsiv.Interpreting the MoveDescriptions

v.

Interpreting the Diagramsvi.Background Informationvii.

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Introduction

Phil Seeley and myself wrote a program to solve this puzzle from Nintendo (yes, the games consolepeople). This puzzle is also known as Tenbirion as it apparently has ten billion combinations, you canread more about it on Wikipedia.

The first four sections guide you through completing the puzzle. 'Interpreting the Move Descriptions' isa reference page that you will need to refer to as you carry out the moves to complete the tumbler puzzle,although you will soon become familiar with the descriptions. 'Interpreting the Diagrams' gives someadditional information if you are finding it difficult to relate my diagrams to the Tumbler Puzzle.'Background Information' explains how we derived the solution and is some further reading for thoseinterested.

If you are accessing the Internet via a dial-up connection you may wish to visit each page in turn andallow it to be downloaded to your browser so that the pages are in your cache, at which point you canhang-up.

Starting Position | Background Information | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999

Tumbler Puzzle - Starting Position http://www.smartbytes.co.uk/Tumbler/tumbler0.htm

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Starting Position

The moves detailed do not affect the top row (row1 shown below) or the bottom row (row 6 shownbelow), which actually doesn't contain any beads when the Tumbler is sat at rest.. The starting positionis, therefore, to shuffle the black beads to the top row under each of the pegs as shown in the diagrambelow.

This diagram represents the starting position with the Tumbler puzzle sat atrest.

For further help in understanding thediagram refer to the 'Interpreting theDiagrams' page.

Having shuffled the black beads to the topof the columns containing the pegs, moveon to the Next Page, 'Completing Row 5'.

Tumbler Puzzle Contents | Next Page | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999

Tumbler Puzzle - Starting Position http://www.smartbytes.co.uk/Tumbler/tumbler1.htm

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Completing Row 5

The next stage is to complete row 5 (as labelled in the diagram below). Note that this row will dictatethe colours that appear in each column, thus now is your opportunity to influence the colour order. Theaim of this stage is to place one bead of each colour throughout row 5. You will find that the movedetailed does not affect the top two rows (labelled 1 and 2 on the diagram), the beads directly above thepegs in row 5 and a couple of outer pieces (the cells marked with a 'X' are affected).

Starting Position

R LL LL D

The important thing to note is the bead in row 4, column D is moved into row 5column B, that bead is moved to row 5 column E and the bead is moved off row5 (actually to row 4 column C).

You will probably find that you'll need to repeat this short sequence several times toachieve the desired result. Note, also thata bead from row 3 is moved down to row 4, this may be useful if you are lackingany beads of the desired colour on row 4.

For further help in understanding thediagram refer to the 'Interpreting theDiagrams' page.

Having arranged row 5 so that it containsone bead of each colour, move on to the Next Page, 'Completing Row 2'.

Note for advanced users

The lower barrel section can be rotated to the right, instead of the left, to perform a mirror image of themove (i.e. beads from column C will drop down a row into column E, with those beads shifting tocolumn B of the same row).

Actually, this short sequence performs a rotation on two groups of 5 beads as follows:

D3 -> B4 -> E4 -> C3 -> A3 -> D3D4 -> B5 -> E5 -> C4 -> A4 -> D4

Tumbler Puzzle Contents | Next Page | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Tumbler Puzzle - Completing Row 2 http://www.smartbytes.co.uk/Tumbler/tumbler2.htm

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Completing Row 2

Ok, now the fun begins. The move sequence that is used to complete 'Row 2' has the following effect:

Note that actually two columns are affected. The important thing is to ensure that row 4contains suitably coloured beads before the move is commenced. Note that this requires abit of memory on your part as when row 4 ispositioned you'll loose row 5 as a reference.

First, identify the two locations in row 2 thatare to be targeted (taking into account youroptions given the current contents of row 4).Position the 'pegs' such that columns A and C will be affected (moving both upper and lowerbarrel sections if necessary).

Confirm the destination colours required byusing row 5 has a reference, now rotate thelower section only to select the target beads incolumns A and C.

At first you may be able to take advantage of the double effect and locate two beads at a time. However,there may be occasions where solving one column dislodges a useful bead in the other. Note the differenteffect that happens in column A (a three-way rotation) and column C (a straight swap). You may be ableto take advantage of this to ensure that you obtain appropriate coloured beads in row 4 to use in a furtherexecution of this move.

Clearly, it is necessary to repeat this move several times to complete the whole of row 2. You may evenfind that row 4 does not contain a bead of the required colour and have to shift it in from row 2 or 3using this same move. When doing this take care to limit the damage to beads already correctly positionedin row 2. It may be possible to affect beads that have yet to be correctly placed, or even swap with anotherbead of the appropriate colour.

Here is the sequence of operations that perform the move illustrated above:

Tumbler Puzzle - Completing Row 2 http://www.smartbytes.co.uk/Tumbler/tumbler2.htm

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R UL D UR LL R LR D UL R UR LL D LR

If you want further details on how to interpret these operations, refer to the 'Move Interpretation' page.

Eventually row 2 should contain the same colours as row 5 all the way around the tumbler, as illustratedby the diagram on the right.

Having achieved this, proceed to the Next Page, 'Completing the Last 2 Rows'.

Notes for advanced users

For more flexibility try doing the reverse of this sequence to move the beads in the opposite direction tothe arrows in the figure at the top of this page. Try doing the mirror image of this move (i.e. Rotating Leftinstead of Right and visa-versa) to affect columns D and A. Also take advantage of some of the lattermoves to juggle the beads in row 3 and 4 into suitable positions for this move.

Tumbler Puzzle Contents | Previous Page | Next Page | Interpreting the Move Descriptions | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999

Tumbler Puzzle - Completing the Last 2 Rows http://www.smartbytes.co.uk/Tumbler/tumbler3.htm

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Completing the Last 2 Rows

Ok, you are nearly there. At this stage you have the benefit of a choice of two moves as shown below:

These two moves are completely different and, therefore, give you plenty of options. As you start to complete somecolumns the swap in column E of the first move will not upset a completed column (although the two barrel sections maybe shifted deliberately to achieve the required effect.

It is important to start by studying the current state of the puzzle. You will probably find that some of the beads in rows 3and 4 are already conveniently placed. Choose from the two moves above one that gives a benefit of one or two (or eventhree!) correctly positioned beads.

Once again ensure that the barrels is correctly positioned use the 'pegs' as guidance for which beads are being affected.

Left Hand Side:

R LL LL D UL UL R LR LR D UR UR

Right Hand Side:

R LL D UL UL R LL D UR R LR LR D UR

Tumbler Puzzle - Completing the Last 2 Rows http://www.smartbytes.co.uk/Tumbler/tumbler3.htm

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Continue to use these two sequences to complete the whole puzzle as shown on the right.

With practice you should be able to remember the movesequences. You should also find that it takes you fewernumbers of move sequences as you are able to be moreadventurous and shift the barrels to place more beads.

Finally, study the 'Notes for advanced users' to get moretips on altering the moves to increase your options.

Notes for advanced users

Once again, more flexibility can be obtained by doing the reverse of the sequences to move the beads in the oppositedirection to the arrows in the figures at the top of this page. Also, the mirror image of these move (i.e. Rotating Leftinstead of Right and visa-versa) could be performed to give you greater diversity.

Tumbler Puzzle Contents | Previous Page | Interpreting the Move Descriptions | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999

Tumbler Puzzle - Interpreting the Move Descriptions http://www.smartbytes.co.uk/Tumbler/tumbler4.htm

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Interpreting the MoveDescriptions

Each move sequence is described by a series of mnemonics. Once you are familiar with their meaningyou will find these mnemonics sufficient to take you through the sequence. Eventually you will learn thewhole sequence off by heart. The table below describes the operation represented by each mnemonic.

Mnemonic Meaning Description

R RaiseRaise the transparent barrel section of the puzzle. (You will have tohold it in this raised position as it will obviously want to drop backdown).

D DropDrop the transparent barrel section of the puzzle. (It is in this loweredposition when placed upright on a flat surface).

UL Upper LeftRotate the upper section of the transparent barrel one click to the left.(i.e. you are moving rows 2 and 3 on the diagrams).

UR Upper RightRotate the upper section of the transparent barrel one click to the right.(i.e. you are moving rows 2 and 3 on the diagrams).

LL Lower LeftRotate the lower section of the transparent barrel one click to the left.(i.e. you are moving rows 4 and 5 on the diagrams).

LR Lower RightRotate the lower section of the transparent barrel one click to the right.(i.e. you are moving rows 4 and 5 on the diagrams).

Tumbler Puzzle Contents | Starting Position | Interpreting the Diagrams | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999

Tumbler Puzzle - Interpreting the Diagrams http://www.smartbytes.co.uk/Tumbler/tumbler5.htm

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Interpreting the Diagrams

Throughout these pages the Tumbler Puzzle is represented as a 'flattened' imageas shown on the right. The rows are numbered 1 to 6 (starting at the top) andthe columns A to E (starting with the 'peg'that has a gap on both sides of it).

This diagram shows the completed puzzleas if it was placed upright on a flatsurface. This is the 'Lowered' position (all moves start from the 'Lowered' position).In this position there aren't actually any beads in row 6 as the 'pegs' are filling theholes. Note also that rows 1 and 6 canonly ever have three beads in them and only one of these rows will contain beadsdepending upon whether the barrel is 'Raised' or 'Dropped'.

Tumbler Puzzle Contents | Starting Position | Interpreting the Move Descriptions | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999

Tumbler Puzzle - Background Information http://www.smartbytes.co.uk/Tumbler/tumbler6.htm

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Background Information

The Tumbler Puzzle is made by Nintendo. It is in the shape of a barrel, the bulk of which is transparent.Around the barrel are five columns of coloured beads (four of each colour) and three black beads. Thebarrel also slides up and down, three 'pegs' shift the beads in the corresponding columns (whilst theother two columns are unaffected). The barrel also has two sections (each two beads high) that arerotated around it, moving 10 of the beads at a time. The solution is to have a different colour in eachcolumn except for the top position where the 'pegs' are, these should contain the three black beads.

I can't remember now how the subject first came up but Phil and myself were discussing how such anapparently simple puzzle had defeated us. Also, in its now muddled state it was virtually impossible tofigure anything out as with only a couple of lifts/drops and twists virtually every bead is affected.

It then struck us that the thing to do was write a program that would sequence through the possiblemoves to search for the solution. This was a relatively trivial exercise in recursion. The program onlyknew four moves; raise the tumbler, lower the tumbler, rotate the top barrel left and rotate the bottombarrel left. We figured that rotating to the right would simply be represented as four rotations to the left.We defined the current state of the beads and set the program running.

The program stored the current position in a file. From this file we could monitor progress. We quicklyrealised that despite a very high number of moves being performed, it was going to take years to find asolution. Also, even if we were presented with the sequence of moves that had to be performed tocomplete the puzzle, we would not have a reusable solution.

We changed the program to restrict the depth of recursion and added move to rotate the barrel sections tothe right. We also changed the bead definition so that every bead was unique (there are only 23 beads soeach was represented by a different letter). We then modified the program to stop if 8 or less beads hadbeen affected. The idea was to look for sequences of moves that had a very isolated effect.

We then ran the program gradually increasing the depth of recursion. It found the three move sequencesdetailed on these pages. Obviously there are many more, however, we can be confident that these are theshortest sequences that have useful effects.

Note that we did not encounter any sequences that had an impact on the top row or the two bottomrows, however, we were confident that will a little trial and error these rows are not that difficult tocomplete. Hence the starting position. If people are having trouble with this then I'll resurrect theprogram and search for some suitable moves.

Tumbler Puzzle Contents | Starting Position | Steve Martin's Home PageSolution © Phil Seeley, Steve Martin 1994

Web Page design © Steve Martin, 1999