Dual Circle

Dual Circle consists of two tightly linked rings. Each ring has six sections. If a ring is rotated about its axis, the section on the other ring that is located in the hole of the first ring will be rotated too. The rings are coloured so that the orientation of the 12 sections is visible.

This puzzle is produced by Hanayama, and was invented by Bram Cohen and Oskar van Deventer. Bram came up with the idea of how the pieces move and affect each other, but had them arranged in two parallel rows. He called that puzzle Rotabram. Oskar came up with the design using two linked rings.

The number of positions:

There are 12 rotating ring sections, each with 6 possible orientations. This gives a maximum of 6¹² positions. There is however a twist 'parity' restriction, so only 6¹¹ = 362,797,056 positions are possible.

The twist restriction is rather complicated, and not at all obvious. Click the link below for a full explanation.

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The twist constraint

This puzzle has 6¹¹ positions rather than the 6¹² you might expect. I will prove this here by finding an invariant. On most other puzzles invariants are relatively simple. The Rubik's Cube for example has three invariants - the permutation parity, the total edge flip, and the total corner twist (modulo 3). Any legal move on the puzzle leaves these invariants unchanged. For any mixed position these invariants can be calculated, and if they are different than what the solved position would have, then the puzzle is unsolvable. The Dual Circle puzzle also has an invariant - some expression involving the amount of twist of its twelve pieces that remains unchanged (modulo 6) by any move. It is however much more complicated than the total twist.

Lets use the letters a to l to denote the orientations of the twelve pieces. Here a to f are the orientations of the six pieces of the red ring, labelled clockwise looking on the red side, starting from the piece inside the blue ring. Similarly, let g to l be the same for the six blue pieces. These variables have values ranging from 0 to 5, representing the number of steps each piece has been rotated.

Now consider the following five expressions:
S_r = a + b + c + d + e + f
S_b = g + h + i + j + k + l
V_r = 0·a + 1·b + 2·c + 3·d + 4·e + 5·f
V_b = 0·g + 1·h + 2·i + 3·j + 4·k + 5·l
T = V_r + V_b - S_r·S_b
All these are calculated modulo 6, since rotating a piece by 6 steps is equivalent to doing nothing.

If you shift the blue ring, the variables g to l get cyclically permuted, and a gets incremented. The effect of this on the five expressions is this:
S_r gets incremented by one (modulo 6).
S_b remains unchanged.
V_r remains unchanged.
V_b was equal to (0·g + 1·h + 2·i + 3·j + 4·k + 5·l) and becomes (0·l + 1·g + 2·h + 3·i + 4·j + 5·k) so is increased by g + h + i + j + k - 5·l = S_b modulo 6.
Therefore T remains unchanged.

If you shift the red ring instead, T also remains unchanged. This shows that at most 1/6th of the positions are reachable. Since there is a solution method that solves pieces one by one except for the last (which is automatically correct due to that invariant), there are 6¹¹ reachable positions.

I don't know if this kind of invariant has ever been described in the mathematical literature on Wreath products of groups. If it has, then please let me know.

I have calculated God's Algorithm for this puzzle. It turns out that it can be solved in at most 17 turns (12.992 on average), or 36 moves (20.678 on average) if every sixth of a rotation counts as a move.

Click to view full table.

Ring Turns

Sixth Turns

Depth	Positions
0	1
1	10
2	50
3	250
4	1,249
5	6,216
6	30,720
7	144,076
8	629,890
9	2,511,744
10	9,152,378
11	28,273,418
12	69,711,614
13	117,652,183
14	105,615,134
15	27,693,510
16	1,372,965
17	1,648
Total	362,797,056

Depth	Positions
0	1
1	4
2	12
3	34
4	96
5	272
6	770
7	2,152
8	5,898
9	15,794
10	41,726
11	108,152
12	275,073
13	682,370
14	1,642,690
15	3,785,796
16	8,224,011
17	16,436,710
18	29,405,280

Depth	Positions
19	45,583,326
20	59,591,176
21	64,123,266
22	55,809,435
23	38,978,212
24	21,947,610
25	10,142,700
26	3,947,121
27	1,379,198
28	461,742
29	149,240
30	43,514
31	10,916
32	2,332
33	382
34	40
35	4
36	1
Total	362,797,056

The single antipodal position at 36 sixth turns is the position with all pieces twisted 180 degrees.

Solution:

Phase 1: Solve the red ring sections

Rotate the red ring to bring any unsolved section into the middle of the blue ring.
Rotate the blue ring until that red ring section is solved.
Repeat steps a-b until the red ring sections are all solved.

Phase 2: Solve the blue ring sections

Rotate the red ring so as to solve the blue section currently inside the red ring.
Choose a blue ring section that you want to solve.
Rotate blue ring to bring the chosen section into the centre of the red ring.
Rotate the red ring one step.
Rotate the blue ring back to its original position.
Rotate the red ring back one step to its original position.
Repeat steps c-f until the chosen blue section is solved. Note that this will have disturbed one red ring section.
Repeat steps b-f for each unsolved blue section (except for the blue section inside the red ring). Note that in steps d and f you must twist in the same direction as the previous times, so that it is always the same red section that gets disturbed. Once all the blue sections are solved, that disturbed red section should be automatically correct.
Rotate the blue ring so as to solve the red section currently inside the blue ring.

Useful sequences

Phase 2 in the above solution can take quite a number of moves. It may therefore be quite useful to have some further move sequences in your arsenal to speed things up. Below is a complete set of move sequences that only affect one or two pieces in the blue ring. Once you have solved the red ring, you will need to apply no more than four of these sequences to solve the blue ring.

Mentally label the 6 sections of the blue ring with the letters A to F, where A is the one that lies inside the red ring. A letter followed by a + or a - indicates that you have to turn that section one way or the other by one click. Two or three plusses or minuses obviously means it has to move two or three clicks.

Effect	Move sequence	Effect	Move sequence	Effect	Move sequence	Effect	Move sequence
B++, C++	B-- C++ B--	B--, C--	B++ C-- B++
B++, C-	B+ C- B+	B--, C+	B- C+ B-	E-, F++	F+ E- F+	E+, F--	F- E+ F-
B+++, D+	B- C- D+ C+ B--	B+++, D-	B+ C+ D- C- B++	D+, F+++	F- E- D+ E+ F--	D-, F+++	F+ E+ D- E- F++
B+++, D+++	B+++ C+++ D+++ C+++			D+++, F+++	F+++ E+++ D+++ E+++
B++, E+	B++ A- F- E+ F+ A+	B--, E-	B-- A+ F+ E- F- A-	C+, F++	F++ A- B- C+ B+ A+	C-, F--	F-- A+ B+ C- B- A-
B++, E--	B+ E- B+ E-	B--, E++	B- E+ B- E+	C--, F++	F+ C- F+ C-	C++, F--	F- C+ F- C+
B+, F+	F+ A- B+ A+	B-, F-	F- A+ B- A-
B++, F++	F++ A-- B++ A++	B--, F--	F-- A++ B-- A--
B+++, F+++	F+++ A--- B+++ A+++
C+++	B+++ C+++ B+++			E+++	F+++ E+++ F+++
C+++, D++	C- D+ C- D+ C-	C+++, D--	C+ D- C+ D- C+	D++, E+++	E- D+ E- D+ E-	D--, E+++	E+ D- E+ D- E+
C+, E+	E+ A- C+ A+	C-, E-	E- A+ C- A-
C-, E++	E+ C- E+	C+, E--	E- C+ E-	C++, E-	C+ E- C+	C--, E+	C- E+ C-
C++, E++	E-- C++ E--	C--, E--	C++ E-- C++
C+++, E+++	C+++ D+++ E+++ D+++
D++	D+ A- D+ A+	D--	D- A+ D- A-

	Total	1	10	50	250	1,249	6,216	30,720	144,076	629,890	2,511,744	9,152,378	28,273,418	69,711,614	117,652,183	105,615,134	27,693,510	1,372,965	1,648	362,797,056
		Disc turns
S i x t h t u r n s		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	64													272
	6			2	64	256	320	128												770
	7				36	320	800	744	252											2,152
	8				12	272	1,280	2,304	1,604	426										5,898
	9				2	160	1,440	4,424	5,976	3,168	624									15,794
	10					64	1,184	6,216	13,662	13,856	5,736	1,008								41,726
	11					16	720	6,528	23,116	36,672	29,360	10,600	1,140							108,152
	12					1	312	5,164	28,952	73,852	88,412	61,064	16,040	1,276						275,073
	13						84	3,220	28,470	108,004	205,024	204,240	111,156	20,840	1,332					682,370
	14						12	1,472	21,814	125,984	346,764	532,252	422,544	169,948	21,284	616				1,642,690
	15							448	12,824	116,960	473,676	998,172	1,236,072	737,100	200,284	10,212	48			3,785,796
	16							72	5,448	83,302	495,844	1,530,744	2,574,566	2,426,245	979,718	126,448	1,624			8,224,011
	17								1,622	43,952	414,624	1,811,416	4,331,336	5,530,732	3,568,528	711,732	22,768			16,436,710
	18								308	17,456	262,468	1,692,064	5,488,954	10,002,414	8,865,400	2,924,320	151,360	536		29,405,280
	19								28	5,108	127,346	1,217,984	5,531,764	13,357,320	16,855,400	7,771,004	714,044	3,328		45,583,326
	20									1,046	45,838	674,836	4,202,278	13,948,200	23,094,454	15,549,740	2,044,644	30,132	8	59,591,176
	21									96	12,864	289,360	2,547,502	11,078,048	24,364,108	21,473,208	4,294,560	63,512	8	64,123,266
	22									8	2,752	96,920	1,188,200	6,957,471	18,941,266	22,491,518	5,929,168	202,068	64	55,809,435
	23										388	25,402	448,116	3,447,204	11,752,034	16,947,024	6,108,684	249,248	112	38,978,212
	24										24	5,340	131,988	1,399,400	5,644,748	10,114,170	4,305,936	345,748	256	21,947,610
	25											888	32,282	461,028	2,292,854	4,691,996	2,442,452	220,624	576	10,142,700
	26											88	7,520	128,932	762,889	1,863,412	1,041,780	142,364	136	3,947,121
	27												1,664	33,212	225,178	642,344	412,048	64,496	256	1,379,198
	28												272	9,172	61,998	210,012	149,828	30,412	48	461,742
	29												24	2,432	16,112	64,096	53,648	12,832	96	149,240
	30													544	3,728	17,870	15,896	5,444	32	43,514
	31													88	740	4,348	4,076	1,616	48	10,916
	32													8	120	896	812	496		2,332
	33														8	152	118	96	8	382
	34															16	12	12		40
	35																4			4
	36																	1		1

Jaap's Puzzle Page

Dual Circle

JavaScript Dual Circle

The number of positions:

The twist constraint

Solution:

Useful sequences