Circle Puzzle / Whirligig

The Circle Puzzle is a simple puzzle consisting of four overlapping circular disks in a square arrangement. Each disk overlaps with the two adjacent disks but not with its diagonally opposite disk. Where two disks overlap they form a lozenge shaped piece, and each disk has four such pieces around its rim. All together there are 12 moving pieces. Each disk can be rotated. A rotation should be a multiple of 90 degrees since only then can the other disks move again.

The puzzle has a simple colour scheme where each disk has a single colour - yellow, red, blue, and green. The four shared pieces also have one colour each, so that when solved each circle has three pieces matching its centre colour and one shared piece matching the adjacent disk. The puzzle has two solved positions which differ by the colours of the shared pieces.

The puzzle is related to Turnstile, Rashkey, and also to Gripple, but is much simpler than any of those.

The second puzzle pictured above is the Whirligig. It is a 3 by 3 version of the Circle Puzzle, with a 6 colours and four pieces of each colour. The circle centres are not coloured. Solving it is just as easy, but there are more nice patterns possible.

I have done a computer analysis of the Circle Puzzle in order to find God's Algorithm. 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 any disk by any amount is a single move. The "Quarter" Turn Metric means that only 90 degree turns are single moves. If you want to only use one of the two solved positions, then it can be solved in at most 12 face turns or 12 quarter turns (on average 7.5600 and 8.9442 respectively). If either solved position is acceptable then it may take 11 face turns or 12 quarter turns (7.3238 and 8.6523 on average).

One Solution:

Face turn metric

Q
u
a
r
t
e
r

t
u
r
n

m
e
t
r
i
c 0 1 2 3 4 5 6 7 8 9 10 11 12 Total

0 1 1

1 8 8

2 4 40 44

3 40 171 211

4 10 268 647 925

5 124 1,412 1,963 3,499

6 20 864 5,756 4,904 11,544

7 200 4,458 18,315 9,207 32,180

8 6 1,170 16,894 41,580 11,686 71,336

9 61 4,356 41,313 59,203 8,104 113,037

10 184 8,976 53,209 39,747 2,206 104,322

11 210 6,119 20,092 5,363 52 31,836

12 21 274 347 14 1 657

Total 1 12 90 583 3,129 13,408 44,653 101,286 130,238 68,217 7,916 66 1 369,600

Two Solutions:

Face turn metric

Q
u
a
r
t
e
r

t
u
r
n

m
e
t
r
i
c 0 1 2 3 4 5 6 7 8 9 10 11 Total

0 2 2

1 16 16

2 8 72 80

3 72 288 360

4 20 440 1010 1470

5 208 2144 2882 5234

6 40 1344 8244 6608 16236

7 336 6332 23684 11620 41972

8 12 1692 21080 47886 13660 84330

9 82 5212 44246 59148 8134 116822

10 152 8520 44008 32368 1584 86632

11 124 3514 10132 2558 2 16330

12 12 56 48 116

Total 2 24 164 976 4,846 19,232 56,736 112,396 120,342 50,690 4,190 2 369,600

The hardest positions in the face turn metric are shown below. Every edge piece is moved to the diametrically opposite spot compared to the solved position.

Solution:

Phase 1: Solve the bottom left disk.
This is trivial, so I will not elaborate. Once all three pieces of the matching colour are in the bottom left disk then turn it so that they are on the left, top and bottom, i.e. the non-matching piece is on the right.

Phase 2: Solve the top left disk.

Find a piece of the same colour as the top left disk, but which does not yet lie in that disk.
If that piece lies in the bottom right disk, then turn that disk to bring the piece to the top, so that it is in the top right disk.
Turn the top left disk so that one of its unsolved edges is on the right.
Turn the top right disk so that the piece you are solving lies on the left. This puts it in the top left disk, which is where it belongs.
Turn the top left disk back to its original position, so that the bottom left disk is correct again.
Repeat steps a-e until the top left disk is solved.

Phase 3: Solve the remaining two disks.

Rotate the puzzle as a whole anti-clockwise, so that the two as yet unsolved disks are the top left and top right.
Use the method of phase 2 to solve the top left disk.
The top right disk will automatically be solved as well.

Jaap's Puzzle Page

Circle Puzzle / Whirligig

JavaScript Circle Puzzle

The number of positions:

Solution:

	0	1	2	3	4	5	6	7	8	9	10	11	12	Total
Face turn metric
0	1													1
1		8												8
2		4	40											44
3			40	171										211
4			10	268	647									925
5				124	1,412	1,963								3,499
6				20	864	5,756	4,904							11,544
7					200	4,458	18,315	9,207						32,180
8					6	1,170	16,894	41,580	11,686					71,336
9						61	4,356	41,313	59,203	8,104				113,037
10							184	8,976	53,209	39,747	2,206			104,322
11								210	6,119	20,092	5,363	52		31,836
12									21	274	347	14	1	657
Total	1	12	90	583	3,129	13,408	44,653	101,286	130,238	68,217	7,916	66	1	369,600

	0	1	2	3	4	5	6	7	8	9	10	11	Total
Face turn metric
0	2												2
1		16											16
2		8	72										80
3			72	288									360
4			20	440	1010								1470
5				208	2144	2882							5234
6				40	1344	8244	6608						16236
7					336	6332	23684	11620					41972
8					12	1692	21080	47886	13660				84330
9						82	5212	44246	59148	8134			116822
10							152	8520	44008	32368	1584		86632
11								124	3514	10132	2558	2	16330
12									12	56	48		116
Total	2	24	164	976	4,846	19,232	56,736	112,396	120,342	50,690	4,190	2	369,600