# Cohan Circle / Arusloky

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.

## The number of positions:

There are 19 coloured pieces, so there are at most 19! positions. This limit is not reached because The Cohan Circle has three sets of six indistinguishable pieces. This leaves only 19!/6!3 = 325,909,584 positions. If you consider the three coloured sets to be equivalent, then you can divide by a further 3! to get 54,318,264 positions.
Arusloky has two sets of six and one set of seven indistinguishable pieces. It therefore has 19! / (6!2·7!) = 46,558,512 positions.

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 Sixthturnmetric 0 1 2 3 4 5 6 7 8 9 10 11 12 6 24 24 48 12 96 96 96 288 192 48 432 768 360 12 384 1,536 1,920 672 216 1,920 4,800 4,512 1,344 72 1,656 7,656 13,776 10,368 2,688 6 888 8,472 26,832 37,416 23,292 5,220 288 6,744 37,056 85,908 96,984 51,768 9,120 48 3,816 38,160 141,492 260,832 248,160 105,924 11,448 1,392 28,188 177,636 506,928 778,632 607,140 159,156 4,800 264 15,624 168,576 759,792 1,800,432 2,286,288 1,119,540 86,400 120 48 5,904 122,160 887,700 3,250,272 6,454,830 5,176,068 709,416 2,496 1,560 63,696 789,708 4,544,274 14,003,772 17,464,728 3,783,360 25,164 264 22,056 471,384 4,445,148 21,498,660 41,519,520 13,868,748 160,134 24 3,672 149,352 2,358,984 18,030,234 56,024,250 30,471,336 609,204 24 168 15,528 430,644 5,324,604 27,410,358 26,806,164 1,134,564 48 264 12,372 281,952 2,435,406 4,791,876 509,670 24 564 12,024 52,134 16,266

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 Sixthturnmetric 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 4 4 8 2 16 16 16 48 32 8 72 128 60 2 64 256 320 112 36 320 800 752 224 12 276 1,276 2,296 1,728 448 1 148 1,412 4,472 6,228 3,882 886 48 1,124 6,176 14,274 16,120 8,616 1,692 8 636 6,356 23,458 42,966 40,720 18,478 2,824 232 4,698 29,350 82,436 124,540 99,918 36,040 3,412 44 2,592 27,720 121,962 277,178 351,736 228,020 52,524 2,124 8 976 20,198 140,820 479,742 917,036 949,630 393,298 37,832 452 276 10,922 126,848 659,986 1,881,553 2,944,068 1,957,096 333,158 7,508 40 52 4,156 83,990 705,670 3,058,200 7,133,672 7,192,355 1,946,768 72,652 286 8 916 37,364 535,082 3,650,050 13,005,940 19,870,446 8,252,463 477,295 2,534 4 92 8,816 235,294 2,683,637 15,260,696 37,434,232 24,801,555 2,247,136 16,510 12 724 44,184 914,164 8,785,580 37,350,460 44,032,082 6,982,224 81,080 24 20 2,500 100,576 1,698,982 13,116,080 31,080,478 10,325,245 247,016 90 20 1,884 61,496 884,178 4,605,371 3,867,105 267,436 188 1 92 3,294 47,380 135,162 40,391 109 12 76 160 2

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 Sixthturnmetric 0 1 2 3 4 5 6 7 8 9 10 11 12 2 8 8 16 4 32 32 32 96 64 16 144 256 120 4 120 512 636 224 72 608 1,576 1,488 448 16 496 2,348 4,520 3,472 840 2 232 2,544 8,464 12,300 7,668 1,572 56 1,896 11,220 27,524 32,160 16,136 2,376 8 888 10,716 44,052 85,032 80,864 29,260 2,240 288 7,432 52,244 162,592 256,720 177,912 33,772 592 48 3,824 46,584 235,762 591,050 694,446 249,480 11,392 8 1,144 29,972 253,296 1,023,098 1,953,822 1,173,934 93,470 160 208 11,748 184,548 1,245,360 3,890,792 3,779,374 484,264 1,792 24 2,356 72,648 875,036 4,595,956 7,487,246 1,607,840 9,546 176 11,992 253,348 2,253,812 6,564,972 2,629,030 31,300 8 440 17,044 265,496 1,447,556 1,237,330 35,884 120 3,052 33,622 69,858 6,158 6 8 28 56 16

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 Sixthturnmetric 0 1 2 3 4 5 6 7 8 9 10 11 12 13 1 4 4 8 2 16 16 16 48 32 8 72 128 60 2 60 256 318 112 36 304 788 744 224 8 248 1,174 2,260 1,736 420 1 116 1,272 4,232 6,150 3,834 786 28 948 5,610 13,758 16,072 8,064 1,208 4 444 5,358 22,022 42,442 40,258 14,750 1,348 144 3,716 26,110 80,992 127,024 88,646 19,282 728 24 1,912 23,264 117,171 289,206 338,453 140,389 13,072 196 572 14,974 125,532 495,119 927,789 650,840 111,670 3,338 16 104 5,878 91,974 603,414 1,831,240 2,107,752 614,308 29,032 216 12 1,218 37,564 448,179 2,324,322 4,591,719 2,299,538 180,077 1,962 4 104 7,200 162,248 1,522,560 5,615,744 5,304,343 741,553 11,672 32 4 512 20,669 382,822 2,791,664 5,621,094 1,675,960 47,070 190 8 694 23,010 352,898 1,660,611 1,283,154 92,716 762 4 116 4,714 61,760 158,713 43,429 964 2 102 1,057 1,626 224 2 3

## Notation:

Let a clockwise 60 degree rotation of the left disc 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 disc are denoted in the same way, but using the letter R.

## Terminology:

 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.

## Solution:

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

1. 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.
2. Find any green/yellow piece that is not yet in the left extended circle.
3. 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.
4. 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).
5. Do R3 L2 R L4 R' L2 R to insert the green/yellow piece into the left extended circle.
6. Repeat b-e for the remaining green/yellow pieces.
7. 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.

1. 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.
2. 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.
3. Find any red piece that is not yet in the left extended circle.
4. 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.
5. 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).
6. 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.
7. 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.
8. Repeat c-g for the remaining red pieces.

Phase 3: Make the circles.

1. 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.
2. If necessary, do R2 or R4 so that the left circle is completely red.
3. Find the non-green piece in the right extended circle.
4. 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.

1. If the white piece is in the left disk, then turn the puzzle around so that it lies in the right disk.
2. 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

## Other neat sequences:

 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