# Tubie

Tubie is a puzzle consisting of a stack of 6 disks inside a transparent tube. The disks can rotate about the main axis, and they each have two or three small pins around the outside. The tube has a comb-like row of pins too. By turning the tube, the pins engage and the disks are turned with it. The tube can be slid along the axis slightly so that it is in a position where all the pins miss each other, and then the tube can be rotated to a different position without moving any disks. The disk at one end has no pins at all and does not rotate. It is used as a reference point for the other disks.

Apparently there have been many designs of the puzzle, but the Sudoku design is the most common. In this design every disk has the six numbers 1 to 6 on it in some order. The main aim is to arrange the disks so that each column contains all six numbers. You can also try to arrange them so that there is a column containing only one number, repeated six times. The latter is possible with any of the numbers.

## The number of positions:

There are five movable disks, which each have six valid positions. This gives a total of 65 = 7,776 positions. It turns out however that only 7,560 of these are attainable using complete moves. The 216 inaccessible positions can only be reached if you do partial moves, e.g. rotating some disks half a step so as to separate the pins from each other.

The 216 inaccessible positions are due to 18 inaccessible pin arrangements. These can occur in any of the six locations relative to the fixed disk, and the sixth disk can be given a half turn without changing the pin arrangement.

### Solution

Here is the solved Sudoku arrangement.

 o o 2 3 5 4 1 6 6 4 1 5 2 3 4 5 6 2 3 1 3 1 2 6 4 5 1 6 4 3 5 2 5 2 3 1 6 4

## Notation:

Hold the puzzle horizontally, with the fixed disk on the left hand side. Number the moving disks 1 to 5 from left to right. A move will be denoted by the numbers of all the disks that are involved in the move, followed by a U or a D which indicates an upwards or downwards move. To perform such a move you first need to locate a row of pins that exactly matches the list of disk numbers, and then use the comb to push against that row of pins in the right direction a distance of just one step.

Returning to the start position

1. The start position has all the red pins in one column.
2. Put the comb directly below the red pin on disk 5, the right-most disk.
3. Move upwards, until all teeth of the comb have caught a pin
4. If the teeth caught by the comb are not all red, then shift the comb left, move down two blocks, shift right again, and move back up two blocks.
5. Repeat step d if necessary, until all the red pins are lined up.

There are six possible starting positions. I will number them 1 to 6, according to which number shows on the fixed disk just above the row of red pins. You can of course do one or more 12345U or 12345D moves to get to the other starting positions. The table below shows how to reach any of the solved positions starting from any of the other solved position or starting position.
Start Position     Goal Position Sudoku Yellow Orange Red Purple 14U 1234U 124U 1U 15U 23U 3D 123U 123U 2D 45D 24D 14U 1U 25U 124D 14D 1234D 2U 15U 14D 1234D 5U 2U 14D 1234U 25U 134D 134D 25U 124U 1345D 25U 2D 25U 1235D 13D 1245U 2U 1U 123U 15D 2U 2U 13U 123U 15D 5D 1234U 2U 25D 25D 134U 123U 2D 245D 1U 245D 2U 1U 123U 1234U 1U 23D 12U 23D 135D 25D 1245D 25D 2D 135U 25U 25U 25U 14U 1245U 235D 3D 2D 15U 123U 15D 123D 2U 123D 1D 135D 25U 124D 5U 2U 25U 25U 134D 25U 124U 2U 25U 2D 123D 24U 245U 125U 2U 1U - 12U 1235U 245D 245D 2U 13U 14U 245D 245D 123U 123U 2U 14U 135U 135U 1235U 123U 135U 234D 25U 125U 1235U 25U 1245U 1245U 13U 15U 123U 24D 25U 1245U 1245U 13U 15U 1U 5U 245U 13D 4U 2D 12D 135D - 24U 25D 24U 134U 25D 5D 1234U 24U 124U 12345D 1D 124D 1245D 23D 3U 124D 1U 1345D 4D 12U 2U 1U 25U 25U 123D 45U 2D 12D 135D 25U 124D 1U - 1234U 25D 1234U 13D 12U 145U 124U 35D 124U 23U 124D 145U 124U 12D 135D 124U 2U 1U 2U 1234D 135D 25D 123D 1D 135D 123U 1234U 12345U 25U 2U 134U 1234D 1234D 5U 2U - 2U 1234D 1234D 13D 125U 123U 1D 235U 124D 23U 134U 2D 123U 1U 2D 1234U 13U 15U 1U 25U 2U 124U 23D 12U 15U 23U 3D 14D 2D 135U 2D 124D 1234U 12345U 13U 125D 25D 1234U - 2U 14U 1234U 2D 14U 235D 124U 23U 1D 145U 15D 123D 145D 25D 12D 135D 124D 134D 14D 2D 135U 124D 1245D 25U 2D 23U 14D 25D 134U 2D 124D 134D - 124U 1D 123D 1U 145D 3U 123D 145D 25D 12D 135D 12U 124U 23U 124U 15U 12U 124U 14U 1234U 2U 35D 2345D 3D 5U 35U 14D 245D 134D 23U 24D -