The Cohan Circle consists of two overlapping circular discs. The distance between the centres is exactly one radius length. The discs are made up of many pieces with curved sides, which allow them to be rotated any number of 1/6th turns. Each disc has 6 blank triangles, and 12 petal-shaped pieces. As some pieces are shared between the two discs, there are all together 10 triangles and 19 petals.

The 19 petals come in 4 colours - 6 red, 6 green, 6 blue, and 1 white. The aim is to make the 6 peripheral pieces of each disc of a single colour, forming two circles, and also place the white piece in the centre.

This puzzle was invented and patented by Hooshang Cohan, number US 4,580,783 published 8 April 1986. It may have been sold as the 'Magic Circle Puzzle', as the solution booklet has that title covered by a label with the correct name.

Arusloky is a recent version of this puzzle, made in Spain. It has a slightly different colour scheme, with 6 red petals, 6 yellow petals, and everything else blue. Its starting configuration has a red and a yellow circle.

If your browser supports it, you can click on the link below to play with a Javascript version of both the Cohan Circle and Arusloky.

Arusloky has two sets of six and one set of seven indistinguishable pieces. It therefore has 19! / (6!

I have used a computer to calculate God's algorithm for the Cohan Circle. This first table shows the results if the two circles may be any colour. Any position can be solved in at most 14 moves (10.892 on average), or 20 (16.447 on average) if every 1/6 of a turn is counted as a separate move.

Face turn metric | ||||||||||||||||||
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S i x t h t u r n m e t r i c |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | Total | ||

0 | 6 | 6 | ||||||||||||||||

1 | 24 | 24 | ||||||||||||||||

2 | 24 | 48 | 72 | |||||||||||||||

3 | 12 | 96 | 96 | 204 | ||||||||||||||

4 | 96 | 288 | 192 | 576 | ||||||||||||||

5 | 48 | 432 | 768 | 360 | 1,608 | |||||||||||||

6 | 12 | 384 | 1,536 | 1,920 | 672 | 4,524 | ||||||||||||

7 | 216 | 1,920 | 4,800 | 4,512 | 1,344 | 12,792 | ||||||||||||

8 | 72 | 1,656 | 7,656 | 13,776 | 10,368 | 2,688 | 36,216 | |||||||||||

9 | 6 | 888 | 8,472 | 26,832 | 37,416 | 23,292 | 5,220 | 102,126 | ||||||||||

10 | 288 | 6,744 | 37,056 | 85,908 | 96,984 | 51,768 | 9,120 | 287,868 | ||||||||||

11 | 48 | 3,816 | 38,160 | 141,492 | 260,832 | 248,160 | 105,924 | 11,448 | 809,880 | |||||||||

12 | 1,392 | 28,188 | 177,636 | 506,928 | 778,632 | 607,140 | 159,156 | 4,800 | 2,263,872 | |||||||||

13 | 264 | 15,624 | 168,576 | 759,792 | 1,800,432 | 2,286,288 | 1,119,540 | 86,400 | 120 | 6,237,036 | ||||||||

14 | 48 | 5,904 | 122,160 | 887,700 | 3,250,272 | 6,454,830 | 5,176,068 | 709,416 | 2,496 | 16,608,894 | ||||||||

15 | 1,560 | 63,696 | 789,708 | 4,544,274 | 14,003,772 | 17,464,728 | 3,783,360 | 25,164 | 40,676,262 | |||||||||

16 | 264 | 22,056 | 471,384 | 4,445,148 | 21,498,660 | 41,519,520 | 13,868,748 | 160,134 | 81,985,914 | |||||||||

17 | 24 | 3,672 | 149,352 | 2,358,984 | 18,030,234 | 56,024,250 | 30,471,336 | 609,204 | 24 | 107,647,080 | ||||||||

18 | 168 | 15,528 | 430,644 | 5,324,604 | 27,410,358 | 26,806,164 | 1,134,564 | 48 | 61,122,078 | |||||||||

19 | 264 | 12,372 | 281,952 | 2,435,406 | 4,791,876 | 509,670 | 24 | 8,031,564 | ||||||||||

20 | 564 | 12,024 | 52,134 | 16,266 | 80,988 | |||||||||||||

Total | 6 | 60 | 300 | 1,494 | 7,296 | 35,472 | 172,572 | 834,492 | 3,964,452 | 17,925,906 | 68,603,088 | 151,332,498 | 80,574,234 | 2,457,618 | 96 | 325,909,584 |

This table shows the results if the circles must have particular colours. Any position can be solved in at most 17 moves (12.230 on average), or 23 (18.339 on average) if every 1/6 of a turn is counted as a separate move.

Face turn metric | |||||||||||||||||||||
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S i x t h t u r n m e t r i c |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | Total | ||

0 | 1 | 1 | |||||||||||||||||||

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

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

3 | 2 | 16 | 16 | 34 | |||||||||||||||||

4 | 16 | 48 | 32 | 96 | |||||||||||||||||

5 | 8 | 72 | 128 | 60 | 268 | ||||||||||||||||

6 | 2 | 64 | 256 | 320 | 112 | 754 | |||||||||||||||

7 | 36 | 320 | 800 | 752 | 224 | 2,132 | |||||||||||||||

8 | 12 | 276 | 1,276 | 2,296 | 1,728 | 448 | 6,036 | ||||||||||||||

9 | 1 | 148 | 1,412 | 4,472 | 6,228 | 3,882 | 886 | 17,029 | |||||||||||||

10 | 48 | 1,124 | 6,176 | 14,274 | 16,120 | 8,616 | 1,692 | 48,050 | |||||||||||||

11 | 8 | 636 | 6,356 | 23,458 | 42,966 | 40,720 | 18,478 | 2,824 | 135,446 | ||||||||||||

12 | 232 | 4,698 | 29,350 | 82,436 | 124,540 | 99,918 | 36,040 | 3,412 | 380,626 | ||||||||||||

13 | 44 | 2,592 | 27,720 | 121,962 | 277,178 | 351,736 | 228,020 | 52,524 | 2,124 | 1,063,900 | |||||||||||

14 | 8 | 976 | 20,198 | 140,820 | 479,742 | 917,036 | 949,630 | 393,298 | 37,832 | 452 | 2,939,992 | ||||||||||

15 | 276 | 10,922 | 126,848 | 659,986 | 1,881,553 | 2,944,068 | 1,957,096 | 333,158 | 7,508 | 40 | 7,921,455 | ||||||||||

16 | 52 | 4,156 | 83,990 | 705,670 | 3,058,200 | 7,133,672 | 7,192,355 | 1,946,768 | 72,652 | 286 | 20,197,801 | ||||||||||

17 | 8 | 916 | 37,364 | 535,082 | 3,650,050 | 13,005,940 | 19,870,446 | 8,252,463 | 477,295 | 2,534 | 4 | 45,832,102 | |||||||||

18 | 92 | 8,816 | 235,294 | 2,683,637 | 15,260,696 | 37,434,232 | 24,801,555 | 2,247,136 | 16,510 | 12 | 82,687,980 | ||||||||||

19 | 724 | 44,184 | 914,164 | 8,785,580 | 37,350,460 | 44,032,082 | 6,982,224 | 81,080 | 24 | 98,190,522 | |||||||||||

20 | 20 | 2,500 | 100,576 | 1,698,982 | 13,116,080 | 31,080,478 | 10,325,245 | 247,016 | 90 | 56,570,987 | |||||||||||

21 | 20 | 1,884 | 61,496 | 884,178 | 4,605,371 | 3,867,105 | 267,436 | 188 | 1 | 9,687,679 | |||||||||||

22 | 92 | 3,294 | 47,380 | 135,162 | 40,391 | 109 | 226,428 | ||||||||||||||

23 | 12 | 76 | 160 | 2 | 250 | ||||||||||||||||

Total | 1 | 10 | 50 | 249 | 1,216 | 5,912 | 28,766 | 139,266 | 666,396 | 3,114,418 | 13,678,924 | 50,107,040 | 118,257,375 | 115,139,223 | 24,114,855 | 655,453 | 429 | 1 | 325,909,584 |

The single position that takes 17 face turns is L R' L R' L R2 L2 R L' R L' R' L2 R4 L' R4 L, which has the two circle colours swapped.

Here are the results for the Arusloky. This first table shows the results if the two circle colours may be swapped so there are two solutions. Any position can be solved in at most 14 moves (10.556 on average), or 20 (15.763 on average) if every 1/6 of a turn is counted as a separate move.

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

S i x t h t u r n m e t r i c |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | Total | |||

0: | 2 | 2 | |||||||||||||||||

1: | 8 | 8 | |||||||||||||||||

2: | 8 | 16 | 24 | ||||||||||||||||

3: | 4 | 32 | 32 | 68 | |||||||||||||||

4: | 32 | 96 | 64 | 192 | |||||||||||||||

5: | 16 | 144 | 256 | 120 | 536 | ||||||||||||||

6: | 4 | 120 | 512 | 636 | 224 | 1,496 | |||||||||||||

7: | 72 | 608 | 1,576 | 1,488 | 448 | 4,192 | |||||||||||||

8: | 16 | 496 | 2,348 | 4,520 | 3,472 | 840 | 11,692 | ||||||||||||

9: | 2 | 232 | 2,544 | 8,464 | 12,300 | 7,668 | 1,572 | 32,782 | |||||||||||

10: | 56 | 1,896 | 11,220 | 27,524 | 32,160 | 16,136 | 2,376 | 91,368 | |||||||||||

11: | 8 | 888 | 10,716 | 44,052 | 85,032 | 80,864 | 29,260 | 2,240 | 253,060 | ||||||||||

12: | 288 | 7,432 | 52,244 | 162,592 | 256,720 | 177,912 | 33,772 | 592 | 691,552 | ||||||||||

13: | 48 | 3,824 | 46,584 | 235,762 | 591,050 | 694,446 | 249,480 | 11,392 | 8 | 1,832,594 | |||||||||

14: | 1,144 | 29,972 | 253,296 | 1,023,098 | 1,953,822 | 1,173,934 | 93,470 | 160 | 4,528,896 | ||||||||||

15: | 208 | 11,748 | 184,548 | 1,245,360 | 3,890,792 | 3,779,374 | 484,264 | 1,792 | 9,598,086 | ||||||||||

16: | 24 | 2,356 | 72,648 | 875,036 | 4,595,956 | 7,487,246 | 1,607,840 | 9,546 | 14,650,652 | ||||||||||

17: | 176 | 11,992 | 253,348 | 2,253,812 | 6,564,972 | 2,629,030 | 31,300 | 11,744,630 | |||||||||||

18: | 8 | 440 | 17,044 | 265,496 | 1,447,556 | 1,237,330 | 35,884 | 3,003,758 | |||||||||||

19: | 120 | 3,052 | 33,622 | 69,858 | 6,158 | 6 | 112,816 | ||||||||||||

20: | 8 | 28 | 56 | 16 | 108 | ||||||||||||||

Total: | 2 | 20 | 100 | 482 | 2,232 | 10,344 | 49,264 | 230,884 | 1,046,978 | 4,360,348 | 13,866,932 | 20,772,224 | 6,133,832 | 84,864 | 6 | 46,558,512 |

And finally, this table shows the results if the Arusloky has only one solved position. This can be reached from any position in at most 15 moves (11.223 on average), or 22 (16.657 on average) if every 1/6 of a turn is counted as a separate move.

Face turn metric | |||||||||||||||||||
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S i x t h t u r n m e t r i c |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | Total | ||

0: | 1 | 1 | |||||||||||||||||

1: | 4 | 4 | |||||||||||||||||

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

3: | 2 | 16 | 16 | 34 | |||||||||||||||

4: | 16 | 48 | 32 | 96 | |||||||||||||||

5: | 8 | 72 | 128 | 60 | 268 | ||||||||||||||

6: | 2 | 60 | 256 | 318 | 112 | 748 | |||||||||||||

7: | 36 | 304 | 788 | 744 | 224 | 2,096 | |||||||||||||

8: | 8 | 248 | 1,174 | 2,260 | 1,736 | 420 | 5,846 | ||||||||||||

9: | 1 | 116 | 1,272 | 4,232 | 6,150 | 3,834 | 786 | 16,391 | |||||||||||

10: | 28 | 948 | 5,610 | 13,758 | 16,072 | 8,064 | 1,208 | 45,688 | |||||||||||

11: | 4 | 444 | 5,358 | 22,022 | 42,442 | 40,258 | 14,750 | 1,348 | 126,626 | ||||||||||

12: | 144 | 3,716 | 26,110 | 80,992 | 127,024 | 88,646 | 19,282 | 728 | 346,642 | ||||||||||

13: | 24 | 1,912 | 23,264 | 117,171 | 289,206 | 338,453 | 140,389 | 13,072 | 196 | 923,687 | |||||||||

14: | 572 | 14,974 | 125,532 | 495,119 | 927,789 | 650,840 | 111,670 | 3,338 | 16 | 2,329,850 | |||||||||

15: | 104 | 5,878 | 91,974 | 603,414 | 1,831,240 | 2,107,752 | 614,308 | 29,032 | 216 | 5,283,918 | |||||||||

16: | 12 | 1,218 | 37,564 | 448,179 | 2,324,322 | 4,591,719 | 2,299,538 | 180,077 | 1,962 | 4 | 9,884,595 | ||||||||

17: | 104 | 7,200 | 162,248 | 1,522,560 | 5,615,744 | 5,304,343 | 741,553 | 11,672 | 32 | 13,365,456 | |||||||||

18: | 4 | 512 | 20,669 | 382,822 | 2,791,664 | 5,621,094 | 1,675,960 | 47,070 | 190 | 10,539,985 | |||||||||

19: | 8 | 694 | 23,010 | 352,898 | 1,660,611 | 1,283,154 | 92,716 | 762 | 3,413,853 | ||||||||||

20: | 4 | 116 | 4,714 | 61,760 | 158,713 | 43,429 | 964 | 269,700 | |||||||||||

21: | 2 | 102 | 1,057 | 1,626 | 224 | 3,011 | |||||||||||||

22: | 2 | 3 | 5 | ||||||||||||||||

Total: | 1 | 10 | 50 | 241 | 1,116 | 5,172 | 24,632 | 115,442 | 523,721 | 2,195,665 | 7,454,916 | 16,276,352 | 15,687,226 | 4,073,080 | 198,709 | 2,179 | 46,558,512 |

The left circle is the 6 piece locations that lie on the rim of the left
disk. Eventually all the pieces of one colour will be placed in the left circle.
Similarly, the right circle is the 6 locations on the rim of the
right disk, which will be made another colour. |
The left extended circle is the left circle plus the horizontal
piece location at the right hand side. Note that the 7 pieces in the left extended
circle will remain in the left extended circle if you do R2, R4, or L moves.
Similarly the right extended circle is the right circle plus the left
horizontal piece location. |
The spokes of the left disk are the six locations of the left disk that
meet in the centre, i.e. those locations that lie inside the left circle. Similarly
the spokes of the right disk are the six locations that meet in the centre
of the right disk. |
The middle is the central horizontal location where the white piece
should be when the puzzle is solved. Note that this is a left spoke as well as
a right spoke. |

**Phase 1:** Put the green/yellow pieces in the left extended circle.

- Try to put as many green (Cohan Circle) or yellow (Arusloky) pieces in the left extended circle as you can before doing the steps below.
- Find any green/yellow piece that is not yet in the left extended circle.
- By doing one of the following steps, move the green/yellow piece so that it becomes the top right spoke of the right disk:

1. If it is one of the left spokes, turn the left disk so that the piece is in the middle, and then do R2.

2. If it is the bottom right spoke of the right disk, then do R4.

3. If it lies in the right circle, then do R2 or R4 to bring the piece into the left disk, and do step 1. - Turn the left disc so that a non-green/non-yellow piece lies at the top right of the left circle (the top left spoke of the right disk).
- Do R3 L2 R L4 R' L2 R to insert the green/yellow piece into the left extended circle.
- Repeat b-e for the remaining green/yellow pieces.
- The left extended circle now contains six green/yellow pieces and one of another colour. If this seventh piece is red, then use steps b-e to place any blue in the left extended circle in place of the red one.

**Phase 2:** Put the red pieces in their extended circle.

- Turn the puzzle around, so that the green/yellow pieces are now in the right extended circle. In the steps below, the red pieces will be placed in the left extended circle in much the same way as the green/yellow ones were in phase 1.
- Try to put as many red pieces in the left circle as you can without disturbing the green/yellow pieces from the right extended circle. Note that if you use only L2, L4, and moves of the right disk then the green/yellows will be safe.
- Find any red piece that is not yet in the left extended circle.
- By doing one of the following steps, move the red piece so that it becomes the top right spoke of the right disk:

If the red piece is one of the left spokes, turn the left disk so that the piece is in the middle, and then do R2.

If the red piece is the bottom right spoke of the right disk, then do R4 so that it becomes the top right spoke. - If necessary do L2 or L4 so that a non-red piece lies at the top right or bottom right of the left circle (the top-left/bottom-left spoke of the right disk).
- If a non-red piece lies at the top right of the left circle, then do R3 L2 R L4 R' L2 R to insert the red piece there.
- If a non-red piece lies at the bottom right of the left circle, then do L2 R2 L4 followed by R3 L4 R' L2 R L4 R' to insert the red piece. Note that the first three moves put the red piece at the bottom right spoke of the right disk, and the rest of the sequence is the mirror image of the sequence of step f.
- Repeat c-g for the remaining red pieces.

**Phase 3:** Make the circles.

- Find the non-red piece in the left extended circle. If it is not one of the spokes of the right disk, then do L2 or L4 to make it so.
- If necessary, do R2 or R4 so that the left circle is completely red.
- Find the non-green piece in the right extended circle.
- If it is at a left spoke, then a L2 or L4 turn will complete the green circle, otherwise do one of the following, depending on where in the right circle it lies:

Top: R2 L4 R2 L2 R4

Top right: R2 L2 R2 L4 R4

Bottom right: R4 L4 R4 L2 R2

Bottom: R4 L2 R4 L4 R2

**Phase 4:** Place the white petal in the centre.

- If the white piece is in the left disk, then turn the puzzle around so that it lies in the right disk.
- Depending on which spoke the white piece lies at, do one of the following:

Top right: L2 R2 L2 R4 L4 R4

Right: R L4 R L4 R2 L4 R L4 R

Bottom right: L4 R4 L4 R2 L2 R2

Start position. | |||||

1. |
Swap circles. | L R L4 R L R' L R' L' R2 L4 R L' R L' R L' R' | L' R' L R' L R' L2 R4 L2 R2 L' R' L4 R' L | ||

2. |
Swap right circle with spokes. | L R' L R4 L R L' R L R L R L3 R' L' | |||

3. |
Wagonwheel. | R L' R' L R L2 R3 L2 R' L3 R L R | L' R' L4 R3 L2 R L2 R4 L R2 | ||

4. |
Pacman. | R' L R4 L' R L3 R' L R3 L4 R3 L4 | L' R4 L R' L R' L2 R4 L R4 L2 | ||

5. |
Hourglass. | R L R2 L R' L2 R2 L2 R' L R3 L' R L4 | R3 L4 R L' R L4 R4 L R' L R' L R2 L4 | ||

6. |
Fish. | L' R' L R' L R L R L' R' L R2 L' R3 L3 | L R L' R L R' L4 R' L R L R L4 R4 |