The Turnstile puzzle consists of two overlapping circular disks. Each disk has 12 pieces, 6 are triangular ('corners'), 6 rectangular ('edges') alternating along the rim of the disk around a hexagonal centre. The pieces have curved sides. The disks have three pieces in common, two corners and one edge, so there are altogether 10 corners and 11 edges. Each disk can be rotated. A rotation should be a multiple of 60 degrees since only then can the other disk move again.

There are 5 bright colours used for the corners, a pair of each colour. There is one edge of each of those colours, and the remaining 6 edges are grey. The aim is to put together all pieces of the each colour. The only way to do this is by forming coloured 'lozenges' of two adjacent corners and the edge between them. These lozenges are separated from each other by the grey edges. The rim of the puzzle has small coloured markings, so that the colours of the lozenges should be such that they match the markings.

Turnstile was manufactured by Binary Arts (now called ThinkFun). There is also a version called TwinSpin that was made in the far East. Turnstile was licensed however from Douglas A. Engel who originally invented and produced it. He patented it on 15 November 1983, US4,415,158. He first called it Engel's Enigma, but later marketed it as The Puzzler with three different designs of varying difficulty:

**Novice:**The pieces of one disc are blue, the rest red.**Challenger:**The top half of the discs is green, the bottom half yellow, and the three edge pieces in the middle are black.**Avenger:**Like Turnstile, but with the colours yellow, red, green, blue and black, and using white edges to separate them. There are no colour markings on the puzzle rim.

Note that the Puzzlers sold in the USA were manufactured there and have a hexagonal white base, whereas those sold in the rest of the world were made in Taiwan and have a black oval base.

These puzzles were invented by Douglas A. Engel, who also invented various other puzzles such as:

- Battle Gear
- Farmland Gear puzzles
- Palette 21
- Palette 7
- Palette Mix 4
- Slide Rule Duel (Binary Bisection 5)
- Slide Rule Duel Pentaplenty and Heptalive

If your browser supports it, you can click on the link below to play with a Javascript version of the Turnstile puzzle or any Puzzler variation.

A related puzzle is made in Russia, and called Trio. It has three intersecting disks, arranged in a triangle. The pieces are inside a casing transparent on both sides, and has a slightly different colour scheme on each side. The pieces in the overlapping areas are yellow, the rest of the three disks are red, green, and blue. The other side is the same except that six edges adjacent to the yellow centre are coloured white.

There are 10 corners and 11 edges, so there are at most 11!·10! positions. This limit is not reached on any of these variations because there are many sets of identical pieces. The following table shows the numbers for each variation.

Corners | Edges | Positions | ||
---|---|---|---|---|

Puzzler Novice | 4, 6 | 5, 6 | 10!·11! / (4!·5!·6!^{2}) |
97,020 |

Puzzler Challenger | 5, 5 | 3, 4, 4 | 10!·11! / (3!·4!^{2}·5!^{2}) |
2,910,600 |

Puzzler Avenger /Turnstile | 2, 2, 2, 2, 2 | 1, 1, 1, 1, 1, 6 | 10!·11! / (2!^{5}·6!) |
6,286,896,000 |

Remember however that the Puzzler avenger actually has 5!=120
solutions due to the lack of markings. Equivalently you can consider the positions that
differ by a permutation of colours to be the same, and then it has one solution but only
10!·11! / (2!^{5}·6!·5!)=52,390,800 positions.

I have done a computer analysis of the Puzzler Novice, Puzzler Challenger, and Puzzler Avenger in order to find God's Algorithm for them. The results are in the tables below. Analogous to the Rubik's cube, there are two ways to count the moves. The Face Turn Metric means that a turn of either disk by any amount is a single move. The Sixth Turn Metric means that only 60 degree turns are single moves.

**Puzzler Novice:**

Face turn metric | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | Total | ||

S i x t h t u r n m e t r i c | 0 | 1 | 1 | |||||||||||||

1 | 2 | 2 | ||||||||||||||

2 | 2 | 4 | 6 | |||||||||||||

3 | 1 | 8 | 8 | 17 | ||||||||||||

4 | 8 | 24 | 14 | 46 | ||||||||||||

5 | 4 | 34 | 50 | 24 | 112 | |||||||||||

6 | 1 | 25 | 98 | 116 | 32 | 272 | ||||||||||

7 | 7 | 121 | 294 | 200 | 34 | 656 | ||||||||||

8 | 85 | 424 | 687 | 296 | 18 | 1,510 | ||||||||||

9 | 35 | 397 | 1,327 | 1,265 | 262 | 16 | 3,302 | |||||||||

10 | 6 | 225 | 1,539 | 3,048 | 1,546 | 225 | 11 | 6,600 | ||||||||

11 | 68 | 1,115 | 4,381 | 5,051 | 1,391 | 98 | 2 | 12,106 | ||||||||

12 | 7 | 432 | 3,681 | 9,064 | 5,193 | 557 | 12 | 18,946 | ||||||||

13 | 56 | 1,510 | 8,701 | 10,398 | 2,260 | 72 | 22,997 | |||||||||

14 | 3 | 194 | 3,393 | 9,719 | 4,864 | 372 | 18,545 | |||||||||

15 | 10 | 299 | 3,101 | 4,532 | 947 | 16 | 8,905 | |||||||||

16 | 7 | 192 | 1,376 | 921 | 84 | 6 | 2,586 | |||||||||

17 | 1 | 62 | 209 | 97 | 12 | 381 | ||||||||||

18 | 11 | 16 | 3 | 30 | ||||||||||||

Total | 1 | 5 | 25 | 98 | 409 | 1,555 | 5,391 | 14,419 | 28,341 | 30,236 | 13,760 | 2,546 | 213 | 21 | 97,020 |

The three most difficult positions are shown here:

**Puzzler Challenger:**

Face turn metric | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | Total | ||

S i x t h t u r n m e t r i c | 0 | 1 | 1 | |||||||||||||

1 | 4 | 4 | ||||||||||||||

2 | 4 | 8 | 12 | |||||||||||||

3 | 2 | 16 | 12 | 30 | ||||||||||||

4 | 16 | 36 | 24 | 76 | ||||||||||||

5 | 8 | 56 | 96 | 48 | 208 | |||||||||||

6 | 2 | 52 | 200 | 232 | 82 | 568 | ||||||||||

7 | 32 | 252 | 612 | 516 | 148 | 1,560 | ||||||||||

8 | 12 | 228 | 980 | 1638 | 1132 | 232 | 4,222 | |||||||||

9 | 2 | 140 | 1130 | 3334 | 4350 | 2044 | 200 | 11,200 | ||||||||

10 | 56 | 904 | 4922 | 10475 | 9882 | 2688 | 78 | 29,005 | ||||||||

11 | 12 | 500 | 5198 | 18460 | 29879 | 16130 | 1628 | 16 | 71,823 | |||||||

12 | 2 | 220 | 3658 | 22980 | 62220 | 61987 | 13273 | 368 | 4 | 164,712 | ||||||

13 | 54 | 1530 | 19084 | 88973 | 155887 | 65166 | 4096 | 48 | 334,838 | |||||||

14 | 309 | 9514 | 80520 | 250552 | 204084 | 25234 | 548 | 570,761 | ||||||||

15 | 24 | 2298 | 39498 | 232216 | 376516 | 95187 | 2918 | 8 | 748,665 | |||||||

16 | 176 | 8176 | 99035 | 340627 | 189255 | 10502 | 14 | 647,785 | ||||||||

17 | 448 | 13915 | 107117 | 144389 | 18108 | 30 | 284,007 | |||||||||

18 | 4 | 372 | 6752 | 25210 | 8100 | 28 | 40,466 | |||||||||

19 | 24 | 338 | 293 | 2 | 657 | |||||||||||

Total | 1 | 10 | 50 | 202 | 1,010 | 4,680 | 21,211 | 88,617 | 321,876 | 832,982 | 1,115,265 | 484,093 | 40,521 | 82 | 2,910,600 |

The two most difficult positions are shown here:

**Puzzler Avenger:**

Note that positions that differ by swapping around the colours are considered the same,
so it can be solved with the colours in any order.

Face turn metric | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Total | ||

S i x t h t u r n m e t r i c | 0 | 1 | 1 | |||||||||||||||

1 | 2 | 2 | ||||||||||||||||

2 | 4 | 4 | ||||||||||||||||

3 | 4 | 8 | 12 | |||||||||||||||

4 | 2 | 16 | 16 | 34 | ||||||||||||||

5 | 16 | 52 | 24 | 92 | ||||||||||||||

6 | 8 | 68 | 128 | 48 | 252 | |||||||||||||

7 | 2 | 58 | 252 | 304 | 96 | 712 | ||||||||||||

8 | 30 | 296 | 792 | 668 | 176 | 1,962 | ||||||||||||

9 | 8 | 236 | 1,210 | 2,174 | 1,466 | 304 | 5,398 | |||||||||||

10 | 2 | 125 | 1,244 | 4,216 | 5,638 | 3,054 | 482 | 14,761 | ||||||||||

11 | 46 | 934 | 5,642 | 13,172 | 14,200 | 5,798 | 662 | 40,454 | ||||||||||

12 | 12 | 556 | 5,374 | 21,714 | 39,337 | 32,672 | 9,570 | 540 | 109,775 | |||||||||

13 | 2 | 240 | 3,942 | 25,888 | 77,348 | 108,626 | 66,088 | 10,758 | 168 | 293,060 | ||||||||

14 | 64 | 2,232 | 23,298 | 111,782 | 258,262 | 270,459 | 92,648 | 5,190 | 4 | 763,939 | ||||||||

15 | 8 | 944 | 16,078 | 122,424 | 452,700 | 779,472 | 476,602 | 59,118 | 466 | 1,907,812 | ||||||||

16 | 1 | 226 | 8,405 | 100,270 | 580,475 | 1,630,022 | 1,694,946 | 385,142 | 6,112 | 4,405,599 | ||||||||

17 | 32 | 3,032 | 58,622 | 526,306 | 2,380,099 | 4,225,878 | 1,666,086 | 46,502 | 8,906,557 | |||||||||

18 | 574 | 21,260 | 305,970 | 2,213,807 | 6,766,089 | 4,732,559 | 227,279 | 4 | 14,267,542 | |||||||||

19 | 34 | 3,728 | 95,140 | 1,082,128 | 5,640,167 | 7,221,425 | 641,574 | 64 | 14,684,260 | |||||||||

20 | 248 | 10,712 | 192,953 | 1,638,739 | 3,871,877 | 695,213 | 174 | 6,409,916 | ||||||||||

21 | 240 | 6,398 | 82,914 | 353,130 | 133,180 | 58 | 575,920 | |||||||||||

22 | 4 | 208 | 1,412 | 1,108 | 4 | 2,736 | ||||||||||||

Total | 1 | 2 | 10 | 50 | 234 | 1,121 | 5,401 | 25,546 | 119,475 | 552,577 | 2,377,383 | 8,631,662 | 20,629,489 | 18,296,107 | 1,751,438 | 304 | 52,390,800 |

The four most difficult positions are shown here, but remember that the colours are arbitrary:

The table below summarizes the depth of the worst cases (i.e. diameters) and average depths for both metrics.

Face Turn Metric | Sixth Turn Metric | |||
---|---|---|---|---|

Worst | Average | Worst | Average | |

Puzzler Novice | 13 | 8.3522 | 18 | 12.533 |

Puzzler Challenger | 13 | 9.5556 | 19 | 14.675 |

Puzzler Avenger /Turnstile | 15 | 12.117 | 22 | 18.003 |

PuzzleAtomic is Douglas Engel's site, where he sells many different types of circle puzzle.

Let a clockwise 60 degree rotation of the left disk be denoted by L. Rotations of 120, 180, 240, 300 degrees are then denoted by L2, L3, L4 and L5. Note that L5 can also be considered an anti-clockwise 60 degree turn, and is therefore also denoted by L'. Turns of the right disk are denoted in the same way, but using the letter R.

A solution for any variation of the puzzle is fairly easily constructed by the use of the simple move sequence LR'L'R. This sequence swaps two pairs of corners, and also does a 3-cycle of edges. If you perform it twice then the corners remain unmoved but it is still a 3-cycle of edges. If you perform it three times then the edges remain unmoved but still does a double swap of corners, though if you solve the corners before solving the edges you don't need to do this. Below I will explain a solution to the standard Turnstile and Avenger puzzles which uses some shortcuts instead of applying this move sequence all the time.

**Phase 1:** Pair up the corners.

- First place two corners of the same colour next to each other. This is very easy. If there is not already a pair, turning only one of the disks is nearly always sufficient.
- Rotate the pair to the overlapping region, and then do L2 to place them at the bottom left of the left disk.
- Find the corner matching the colour of the upper left corner. If it does not lie at the top of the left disk (i.e. it is not yet paired up) then it must lie in the right disk. Turn the right disk to bring it to the top and do L R' L'. The left disk must now have two corner pairs.
- If the right disk has one or more pairs then rotate it to bring a pair to the overlapping region, otherwise rotate it so that its top and bottom corners have the same colour.
- Only 4 possibilities remain for the corners other than the three pairs
position.

1. If there are two pairs on the right disk then do R' L R L R.

2. If there is one pair on the right disk then do R3 L4 R3 L2 R.

3. If each pair of corners lie on the opposite sides of the disk then do L2 R2 L2 R2 L.

4. Otherwise do R2 L2 R3 L' R L.

**Phase 2:** Place the edges correct.

- Find a coloured edge which is in a spot where a grey edge belongs (i.e. it does not lie between a matched pair of corners).
- Find the corner pair of the same colour as the edge.
- The following steps should bring the corners to the overlapping region
and the edge adjacent to it.

1. If the edge and its corners lie in the same disk then rotate it to bring the corners to the overlapping region. If the edge is not yet adjacent to the corners then rotate the other disk 120 degrees and go to step 2.

2. If the edge and its corners lie in different disks and are not in the overlapping region then rotate the edge to a position adjacent to the overlap and then turn the other disk to place the corners in the overlap. - The edge can now be placed between the corners by using one of the
following four similar sequences TWICE. Which one you need depends on
the position of the edge piece:

1. Edge is at top left: R'L R L'

2. Edge is at top right: L R'L'R

3. Edge is at bottom left: R L'R'L

4. Edge is at bottom right: L'R L R' - Repeat step a-d for all the edges. If all the grey edges are correct but the coloured ones are not, then you can use the procedure above to insert a grey edge in an incorrectly coloured 'lozenge' to free up a coloured edge for step a.

**Phase 3:** Place the lozenges correct.

- This phase mostly uses only 120 degree turns of the discs, so that the lozenges will not broken be up. First put the correct lozenge in bottom left position which is trivial.
- Find the lozenge belonging at the top left position. If it is not yet in position then rotate it to the top right, and do L2 R4 L4.
- Rotate the right disc to bring the correct lozenge to the overlapping region.
- If the final two colours are not correct then we must break them up and
rebuild them by performing the sequence

L R L' R L R4 L'.

to swap the corners and then solve the remaining edge as in phase 2 by using:

R2 L R' L' R L R' L' R.

1. | L R L R L R L R3 L R' L' R' | |
---|---|---|

2. | L R' L R' L R' L R' L R' | |

3. | L R L2 R' L' R L' R' L R' L R4 | |

4. | L4 R' L R L3 R L' R L' R L' R L2 R L' | |

5. | R L R4 L R L' R2 L |