Lotica / 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.

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:

The position of three teardrop pieces, determines that of the other three (6)
The permutation of pieces must be even (2)

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:

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	Total
Face turn metric
0	1																					1
1		3																				3
2		1	4																			5
3			2	6																		8
4				5	8																	13
5				1	8	12																21
6					2	16	16															34
7						7	24	24														55
8						1	12	44	28													85
9							1	28	64	32												125
10								4	44	100	40											188
11									8	90	136	52										286
12										28	136	204	64									432
13										2	48	248	264	84								646
14											4	120	352	372	104							952
15												20	188	576	484	136						1404
16													30	382	722	492	168					1794
17														88	508	860	574	140				2170
18														2	112	816	912	316	148			2306
19															4	222	782	646	294	16		1964
20																18	210	704	340	32	8	1312
21																	16	180	268	48		512
22																		2	57	16		75
23																			4	1		5
24																				3		3
25																					1	1
Total	1	4	6	12	18	36	53	100	144	252	364	644	898	1504	1934	2544	2662	1988	1111	116	9	14,400

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:

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.
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.
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.
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.
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.
Repeat step e until the puzzle is solved. Note that the last two triangles of the right disk will automatically be solved.

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Lotica / Turn'Push

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The number of positions:

Notation:

Solution: