# Puck

The puck is a puzzle in the shape of a hockey puck, i.e. a thick disk. In the centre are two semicircular parts, and around these are 12 segment pieces. The centre can rotate with respect to the segments, and one of its halves can be given a 180 degree turn to change the order of the segments.

There are several puck designs. The most common ones are:
1. Numbers 1 to 12, with 'Hockey Puck Puzzle' on the other side.
3. Rainbow colours, two pieces of each colour.

The Hockey Puck was patented by Zoltan Pataki, Udulo Mszmp, and Geza Csomos, on 7 March 1991, WO 91/02574. The mechanism is virtually identical to that of the earlier Kép Korong.

If your browser supports JavaScript, then you can play the Hockey Puck (the numbered version or the rainbow coloured one) by clicking the link below:

## The number of positions:

There are 12 segments giving a maximum of 12! positions relative to the centre. The orientation of the pieces is not important since it depends totally on their position. Assuming the centre has distinguishing marks then it has 2·2 possible orientations, giving a total of 4·12! positions. Given an arrangement of the outer ring, not all of the centre orientations are possible because each half twist is an odd permutation. This leaves 12!·2 = 958,003,200 positions. If rotations of the centre are not considered different, then there are only 11!·2 = 79,833,600 positions.

The rainbow puck has six pairs of identical pieces, so it has about 26=64 times as few positions. Some positions have 180 degree rotational symmetry, so to get the exact number of positions we can use Burnside's Theorem, which gives (12!/64+ 6!)/12 = 623,760 positions.

## Solution 1:

My original solution was mostly found by using a computer. It is further down the page, as solution 2. More recently I have found a much better solution, one that is longer but a lot easier to remember and understand. Furthermore, a similar solution can be applied to the Saturn puzzle, and the Topspin. This method only uses one move sequence

Move Sequence:
Hold the puzzle so that the seam through the centre is vertical, splitting the puzzle into left and right halves. Then do the following:

1. Shift the outer ring so that the piece you want to move lies at the top of the right half, just to the right of the seam.
2. Turn over the right half.
3. Turn the outer ring one step anti-clockwise.
4. Turn over the right half.
5. Turn the outer ring one step clockwise.
6. Do steps b-e again.
7. Do steps b-e a final time.

The effect of this sequence is that the piece mentioned in step a is shifted clockwise 6 steps, i.e. jumping over 6 pieces. The other pieces do not change their order, though the 6 pieces jumped over will all have turned over.

You can apply this move sequence twice on the same piece, so that it jumps over 12 pieces all together. Since there are only 11 other pieces, the net effect is that the piece has really jumped over only one piece. In other words, the piece has swapped with the one to the right of it. In this case all 10 of the other pieces will have turned over.

With this move sequence, it becomes quite easy to solve the pieces one by one. You can solve half the puzzle intuitively (or for details, see solution 2 below). Then use the move sequence to extend the solved block piece by piece. Throughout all this, the orientation of the pieces was ignored. This might lead to the situation where, at the end, all the pieces are upside down. If you get this, do one twist move and solve it again. If the centre halves have visible orientations then you can avoid this by fixing the centre after you have solved 6 of the pieces.

## Notation:

Consider the 12 segments numbered like a clock face, 12 at the top. We will leave piece 12 fixed, and solve the remaining segments anti- clockwise, from 11 to 1. Therefore hold the puzzle so that the piece at the top is correct. If your puzzle has four blank pieces (e.g. Canadian Puck) then it is best to choose piece 12 so that the blank pieces will end up at positions 1-4.
Piece 12 will never move, so are only 6 possible moves:
1. Turn over segments 1-6.
2. Turn over segments 2-7.
3. Turn over segments 3-8.
4. Turn over segments 4-9.
5. Turn over segments 5-10.
6. Turn over segments 6-11.

You will have to rotate the centre of the puck first to accomplish a move.

I have used a computer to calculate God's algorithm for the puzzle, ignoring the centre pieces completely, and with either all 6 possible moves or restricted to fewer moves. The relevant numbers where the centre orientation is not ignored can easily be deduced from this.

Distance  Moves a-cMoves a-dMoves a-eMoves a-f
01111
13456
26122030
3123578147
423100297708
5442861,1303,408
6807924,22516,177
71422,13515,47275,384
82475,55754,143334,836
941113,783178,2741,400,827
1066231,202508,8055,019,071
111,01961,1251,062,16412,427,541
121,48193,3831,195,53614,323,585
132,05993,939557,1746,051,014
142,74549,86751,373263,992
153,46510,10810373
164,126526
174,63325
184,913
194,777
204,163
213,079
221,612
23488
2494
2525
266
273
281
Total  8! = 40,320  9! = 362,880  10! = 3,628,800  11! = 39,916,800

In Sloane's On-Line Encyclopedia of Integer Sequences these are included as sequences A079735, A079736, A079737, and A079738.

## Solution:

Segment 12: This is already correct.

Segment 11:

Find the piece that belongs in position 11.
If it is at 11, go to the next step obviously.
Otherwise do any move(s) to bring the piece to position 6, and then do move f to put it at 11.

Segments 10-7:

These can be solved in a similar way to piece 11, with the proviso that you must not do any move that destroys any part of what you have done already. Just see what the last move is that you would need to use to put the piece in position, and then do any moves you need to bring the piece to the correct spot for that move to work. I will not explain further as it is easy.

Segment 6:

Find the piece that belongs in position 6. Then do the relevant sequence below:
6. Do nothing since it is already correct.
5. ceacae
4. dbcdcd
2. dcdcbd
1. a

Segment 5:

Find the piece that belongs in position 5. Then do the relevant sequence below.
5. Do nothing since it is already correct.
3. cabcbc
2. cebc dcec
1. cbcbac

The Canadian puck is now done, except possibly for the centre. Go to the final step below.

Segment 4:

Find the piece that belongs in position 4. Then do the relevant sequence below.
4. Do nothing since it is already correct.
3. dfcb abfd
1. bdab cbdb

Segment 3:

Find the piece that belongs in position 3. Then do the relevant sequence below.
3. Do nothing since it is already correct.
2. dbdab dbdab