Rubik's Ufo

The Rubik's Ufo puzzle is in the shape of a flying saucer. It consists of two disks, each with six segments. There are two types of moves - the disks can turn with respect to each other (by multiples of 60 degrees), or any half can be given a 180 degree twist so that three segments of the top disk are swapped with three segments of the bottom. The top disk is light grey, the other disk dark grey. The segments have colours along their sides so that adjacent segments should have matching colours.

The Ufo is based on a simpler early puzzle by Rubik, called Rubik's Cheese. That puzzle has only one circular layer of six pieces, so it is like a Ufo that is restricted to do only twists and not disk turns. The Rubik's Cheese is discussed further on the page about a related puzzle called Kep Korong.

The second picture above shows a puzzle that is based on a two-layer Square-1, but with only corner pieces in each layer so that it has a six-pointed star shape. The pieces come in three different colours of plastic, four of each colour. In the solved position, each third of the puzzle has a single colour.

The number of positions:

There are 12 segments giving a maximum of 12! positions. Since rotations of the puzzle do not matter, we can consider any single piece as already solved and just arrange the other 11, so there really are only at most 11! = 39,916,800 positions. All these positions are indeed possible.

In the three-coloured puzzle there are four pieces of each colour. This means that there are 12! / 4!³ = 34,650 positions if rotations are considered different. There are therefore approximately 34,650/12 = 2,887 positions, but this is a slight underestimate because some positions are symmetric and so do not give 12 distinct patterns when the puzzle is reoriented. To get the exact number, we need to use Burnside's Lemma, which gives 2,940 positions. If you include the orientation of the seam of the centre, then there are 8,731 positions (with three of them being solved positions).

I have used a computer to calculate God's algorithm for the Rubik's Ufo puzzle, and the table below shows how many positions there are for each number of moves. It shows that it can be solved in 14 moves (11.126 on average), or 16 (12.965 on average) if every 1/6th turn is counted as a separate move.

Click to show full table.

Depth	Face turn metric	Sixth turn metric
0	1	1
1	8	5
2	36	20
3	189	79
4	870	306
5	4,182	1,185
6	19,502	4,550
7	89,508	17,240
8	390,114	64,582
9	1,623,294	236,713
10	6,100,151	841,951
11	16,424,349	2,825,790
12	14,501,714	7,998,476
13	762,762	14,330,740
14	120	11,028,854
15		2,546,648
16		19,660
Total	39,916,800	39,916,800

In Sloane's On-Line Encyclopedia of Integer Sequences these are included as sequences A079819 and A079820.

The 3-colour puzzle gives the results shown in the table below. It shows that it can be solved in 10 moves (6.7520 on average), or 15 (10.665 on average) if every 1/6th turn is counted as a separate move.

Click to show full table.

Depth	Sixth turn metric	Face turn metric
0	3	3
1	6	18
2	13	43
3	21	147
4	45	297
5	90	876
6	159	1,716
7	276	2,931
8	474	2,292
9	870	390
10	1,455	18
11	2,034
12	1,974
13	1,098
14	201
15	12
Total	8,731	8,731

		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	Total
0	3											3
1		6										6
2		3	10									13
3		3	9	9								21
4		3	6	21	15							45
5		3	9	18	39	21						90
6			9	36	30	60	24					159
7				39	72	48	99	18				276
8				18	75	156	93	120	12			474
9				6	48	234	276	222	84			870
10					15	219	471	534	204	12		1,455
11					3	114	489	852	522	54		2,034
12						24	219	804	831	96		1,974
13							42	357	555	132	12	1,098
14							3	24	84	84	6	201
15										12		12
Total:	3	18	43	147	297	876	1,716	2,931	2,292	390	18	8,731

Notation:

Mentally label the segments of the disk clockwise from A to F. Any twist of three segments can then simply be denoted by the letter of the middle segment. A single clockwise turn of the top layer will be denoted by a u, and an anti-clockwise turn by u'. Turns of the bottom layer are denoted by d for a clockwise turn (when looking from underneath directly at the bottom half) and d' for an anti-clockwise turn.

Rubik's Ufo Solution 1:

Phase 1: Pair up the pieces of the top layer with those of the bottom layer.
Each piece that belongs in the top layer has a twin piece belonging in the bottom layer which has the same two colours. This phase will match up such pairs.

First do a disk turn to try and match up as many pairs of pieces as possible.
Find any unmatched pair of pieces. If at all possible, find an unmatched pair that lies in adjacent or in opposing segments. If you cannot find such a pair, then hold the puzzle such that segment A has one piece of an unmatched pair and do the sequence uCu'd'Cd.
Do any twists to bring the unmatched pair in opposite segments, and in the same disk.
Assuming the unmatched pair are in the top disk at segments A and D, then do uCu'.
Repeat steps b-d until all segments have matching pairs.

Phase 2: Orient the segments

If more than three segments are flipped, then turn over the Ufo.
If any half of the Ufo has two or three flipped segments, then twist that half.
If there are still segments flipped, for example segment A, then do CDE and turn over the puzzle. If necessary, repeat until no flipped segments are left.

Phase 3: Arrange the segments

Find a segment that is incorrectly positioned. It can be put in place using a single swap, by one of the following move sequences:
1. To swap adjacent segments A and B, do: C uCu'd'Cd C.
2. To swap opposing segments A and D, do: D uCu'd'Cd D.
3. To swap other segments A and C, do: u DC u'u' B uu D u'.
Another useful sequence (which works on the Cheese as well) is CDECDE, which cycles the segments B->D->F.
Repeat step a until the Ufo is solved.

Rubik's Ufo Solution 2:

This method was devised by Bram Cohen, who kindly allowed me to include it here.

Phase 1: Solve the top layer.
This is fairly easy so most details will be omitted.

Solve three adjacent pieces of the top layer, i.e. build a block of three light grey pieces so that their colours match.
Put the three solved pieces on the left of the top layer. Note that you can still turn over the right half of the puzzle and turn the bottom layer without disturbing the three pieces at all.
Without disturbing the three solved pieces, put the other three light-grey pieces together in one block on the bottom layer, so that their colours match.
Bring the two blocks of light-grey pieces together in the top layer. The top layer has now been solved.

Phase 2: Solve the bottom layer.
Remember that we mentally labelled the segments A-F clockwise. During this phase I will assume that B is at the front, so that A is just to its right.

Turn the bottom layer so that as many pieces as possible are matched up with those in the top layer.
The bottom layer pieces that are matched up are now considered to be solved. The other pieces in the bottom layer will be solved relative to these. Note that since in a moment the top layer will be temporarily mixed, these bottom layer pieces will be your guide points. If however two adjacent pieces in the bottom layer seem to be exchanged, then choose one of these to be your guide instead, and consider the other 5 pieces to be incorrectly positioned with respect to that piece. It may seem that this is more work, but this is due to a parity restriction - only even permutations are possible by the steps below, so a 5-cycle is possible while a 2-cycle is not.
If the top layer contains no bottom layer pieces (which is the case the first time you do this step), then turn the bottom layer so that one of its incorrect pieces is in the front (in segment B). If there is a bottom layer piece in the top layer, then find out where in the bottom layer that piece belongs, and turn the bottom layer to bring that to the front.
Do the sequence A u' A u' A u A. This mixes the top layer somewhat, but leaves the bottom layer intact except for the front piece, which has been replaced.
If the bottom layer is now solved, then turn the puzzle upside down, and go back to step a to solve the other layer.
This step is exactly the same as step c: If the top layer contains no bottom layer pieces, then turn the bottom layer so that one of its incorrect pieces is in the front (in segment B). If there is a bottom layer piece in the top layer, then find out where in the bottom layer that piece belongs, and turn the bottom layer to bring that to the front.
Do the sequence A u' A u A u A. This unscrambles most of the top layer again, and only affects the front piece of the bottom layer. Note that this sequence is the inverse of the one in step d.
If the bottom layer is solved, then the top layer will be correct too and you are finished. Otherwise, go back and repeat steps c-g.

Three-colour puzzle Solution:

Notation:

Hold the puzzle so that the seam in the centre goes from front to back. A twist of the right hand half of the puzzle will be denoted by R. A single clockwise turn of the top layer will be denoted by a u, and an anti-clockwise turn by u'. Turns of the bottom layer are denoted by d for a clockwise turn (when looking from underneath directly at the bottom half) and d' for an anti-clockwise turn.

Phase 1: Bring the blue pieces together in one layer.
This is fairly easy so most details will be omitted.

Put the biggest block of blue pieces in the left half of the top layer.
Put a loose blue piece into the bottom layer if it does not already have any.
Turn u however much is needed to bring the solved blue block to the back of the left half, adjacent to the seam.
Turn d however much is needed to bring a loose blue block to the front of the left half, adjacent to the seam.
Do R to bring the blue pieces together.
Repeat a-e until all the blue pieces are together in the top layer.

Phase 2: Solve the rest.

Turn the u so that the two non-blue pieces are at the back, one on each side of the seam.
If the non-blue pieces are of different colours, then
1. Do the moves R d' R.
2. Turn d until the piece at the front to the left of the seam is the same colour as the non-blue piece at the back of the top layer.
3. Do the moves R d R
Turn over the puzzle so that the blue pieces are on the bottom layer, with the two identical non-blue pieces at the back.
In the top layer there are two pieces in the u layer that have a different colour to the others. Turn the top layer so that one of them lies at the back of the left half, and the other lies somewhere in the right half.
Depending on the location of the odd piece on the right, do one of the following:
1. At the back: R u u u
2. In the middle: uuu R u'd'd' R u' R u
3. At the front: R uu R u'u' R u'u' R uuu

Jaap's Puzzle Page

Rubik's Ufo

JavaScript Rubik's Ufo

The number of positions:

Links to other useful pages:

Notation:

Rubik's Ufo Solution 1:

Rubik's Ufo Solution 2:

Three-colour puzzle Solution:

Notation:

	Total	1	8	36	189	870	4,182	19,502	89,508	390,114	1,623,294	6,100,151	16,424,349	14,501,714	762,762	120	39,916,800
		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	1															1
	1		5														5
	2		2	18													20
	3		1	12	66												79
	4			6	66	234											306
	5				45	312	828										1,185
	6				12	234	1,389	2,915									4,550
	7					84	1,230	5,856	10,070								17,240
	8					6	597	5,910	24,151	33,918							64,582
	9						126	3,555	27,103	94,002	111,927						236,713
	10						12	1,098	18,872	117,413	356,388	348,168					841,951
	11							168	7,500	91,511	498,632	1,313,774	914,205				2,825,790
	12								1,692	41,890	423,505	2,065,275	4,160,160	1,305,954			7,998,476
	13								120	10,684	191,394	1,715,544	6,631,890	5,615,412	165,696		14,330,740
	14									696	40,056	606,240	4,112,022	5,889,386	380,382	72	11,028,854
	15										1,392	51,150	604,212	1,678,970	210,900	24	2,546,648
	16												1,860	11,992	5,784	24	19,660