Jaap's Puzzle Page

Lotica / Turn'Push

Turn'Push

This sliding piece puzzle has 13 pieces which form two intersecting disks. One disk can only do half turns, while the other disk can also do quarter turns. There are two piece shapes; seven triangular ones, and six which are teardrop shaped. Each disk has four of each.

The most interesting fact of this puzzle is that it occurs on the Rubik's cube (and the domino). If you scramble the cube using only the moves U and R2, then the corners of the top and right faces correspond to the teardrops, and the edges to the triangular pieces. A brief discussion of the subgroup <U, R2> can be found in David Singmaster's "Notes on Rubik's 'Magic Cube'", fifth edition, page 57.

Related puzzles are Turnstile, and especially Rashkey.

I got this puzzle as an advertising freebee, and on the back it has the title of the puzzle (TURN'PUSH ®) and "Lotica Patent no. 9503633". Other versions show Lotica as the puzzle's name. That patent number refers to the original French patent, filed 28 March 1995, by Serguei and Alexandre Bagdassarian. The equivalent world patent is WO 96/030097.

If your browser supports JavaScript, then you can play Turn'Push by clicking the link below:

JavaScript Turn'Push

The number of positions:

The top and bottom triangle of one disk - the one that only does half turns - can swap but cannot mix with the other triangles. The pieces therefore fall into three separate groups, of 6, 5, and 2 pieces. This leads to a maximum of 6!·5!·2! = 172,800 positions. This limit is not reached because:

This leaves only 6!·5!/6 = 14,400 positions. For an explanation of the first restriction above, see Singmaster's notes, page 55-57.

A computer search gave the following result:

Face turn metric
Q
u
a
r
t
e
r

t
u
r
n

m
e
t
r
i
c
01234567891011121314151617181920Total
011
133
2145
3268
45813
5181221
62161634
77242455
8112442885
91286432125
1044410040188
1189013652286
122813620464432
1324824826484646
144120352372104952
15201885764841361404
16303827224921681794
17885088605741402170
1821128169123161482306
194222782646294161964
20182107043403281312
211618026848512
222571675
23415
2433
2511
Total14612183653100144252364644898150419342544266219881111116914,400

In Sloane's On-Line Encyclopedia of Integer Sequences this is included as sequence A079865.

This shows that any position can be solved in at most 20 or 25 moves, depending on whether you count a half turn of the disk that can do quarter turns as one move or two. On average it takes 16.587 and 14.569 moves. The hardest position is where the all teardrop pieces in the top halves of the discs are swapped with those in the bottom halves, and only the top triangle of the disk that does quarter turns is swapped with triangle in the bottom half.

Notation:

Hold the puzzle so that the disk that can only do half turns is on the right. The letters L and R will denote the two discs. One clockwise quarter turn of the left disk is L, a half turn is L2, and an anticlockwise turn is L3. The right disk can only do half turns, so then I will use R instead of R2 for a right half turn.

Solution:

  1. Find the triangle that belongs on the far right, and put it into the correct position. This is easy; just bring it to the overlap of the disks, and then turn the right disk.
  2. Find the teardrop that belongs at the top right. If it lies at the bottom right then do R L2 R L3 R L3 R first. To move the teardrop from the left disk to the correct position, then turn it to the bottom left position, and do R L2 R L R L R.
  3. Find the teardrop that belongs at the bottom right. If it lies somewhere on the left disk, then turn it to the top left position, and do R L2 R L3 R L3 R.
  4. Turn the left disk so that all its teardrop pieces are correct. Note that this is always possible because when one is correct, the last three will automatically be in their correct positions too.
  5. Find a triangle on the left disk that is incorrect, and find the position where it belongs. Swapping the two triangles involved will then solve at least one of them. To swap two adjacent triangles of the left disk, turn it to bring them to the top and right of the disk, do R L3 R L R L2 R L2 R L R L R, and then turn the left disk back into its original position. To swap two opposing triangles, turn the disk so that they are at the top and bottom, do R L2 R L2 R L2, and turn the left disk to the correct position.
  6. Repeat step e until the puzzle is solved. Note that the last two triangles of the right disk will automatically be solved.