This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

What is known about bounds on solving Rubik's cube?

competition/games/rubiks/rubiks.cube.s

The "official" world record was set by Minh Thai at the 1982 World

Championships in Budapest Hungary, with a time of 22.95 seconds.

Keep in mind mathematicians provided standardized dislocation patterns

for the cubes to be randomized as much as possible.

The fastest cube solvers from 19 different countries had 3 attempts each

to solve the cube as quickly as possible. Minh and several others have

unofficially solved the cube in times between 16 and 19 seconds.

However, Minh averages around 25 to 26 seconds after 10 trials, and my

best average of ten trials is about 27 seconds (although it is usually

higher).

Consider that in the World Championships 19 of the world's fastest cube

solvers each solved the cube 3 times and no one solved the cube in less

than 20 seconds...

God's algorithm is the name given to an as yet (as far as I know)

undiscovered method to solve the rubik's cube in the least number of

moves; as opposed to using 'canned' moves.

The known lower bound is 18 moves. This is established by looking at

things backwards: suppose we can solve a position in N moves. Then by

running the solution backwards, we can also go from the solved position

to the position we started with in N moves. Now we count how many

sequences of N moves there are from the starting position, making

certain that we don't turn the same face twice in a row:

N=0: 1 (empty) sequence;

N=1: 18 sequences (6 faces can be turned, each in 3 different ways)

N=2: 18*15 sequences (take any sequence of length 1, then turn any of the

five faces which is not the last face turned, in any of 3

different ways); N=3: 18*15*15 sequences (take any sequence of

length 2, then turn any of the five faces which is not the last

face turned, in any of 3 different ways); : : N=i: 18*15^(i-1)

sequences.

So there are only 1 + 18 + 18*15 + 18*15^2 + ... + 18*15^(n-1)

sequences of moves of length n or less. This sequence sums to

(18/14)*(15^n - 1) + 1. Trying particular values of n, we find that

there are about 8.4 * 10^18 sequences of length 16 or less, and about

1.3 times 10^20 sequences of length 17 or less.

Since there are 2^10 * 3^7 * 8! * 12!, or about 4.3 * 10^19, possible

positions of the cube, we see that there simply aren't enough sequences

of length 16 or less to reach every position from the starting

position. So not every position can be solved in 16 or less moves -

i.e. some positions require at least 17 moves.

This can be improved to 18 moves by being a bit more careful about

counting sequences which produce the same position. To do this, note

that if you turn one face and then turn the opposite face, you get

exactly the same result as if you'd done the two moves in the opposite

order. When counting the number of essentially different sequences of N

moves, therefore, we can split into two cases:

(a) Last two moves were not of opposite faces. All such sequences can be

obtained by taking a sequence of length N-1, choosing one of the 4 faces

which is neither the face which was last turned nor the face opposite

it, and choosing one of 3 possible ways to turn it. (If N=1, so that the

sequence of length N-1 is empty and doesn't have a last move, we

can choose any of the 6 faces.)

(b) Last two moves were of opposite faces. All such sequences can be

obtained by taking a sequence of length N-2, choosing one of the 2

opposite face pairs that doesn't include the last face turned, and

turning each of the two faces in this pair (3*3 possibilities for how it

was turned). (If N=2, so that the sequence of length N-2 is empty and

doesn't have a last move, we can choose any of the 3 opposite face

pairs.)

This gives us a recurrence relation for the number X_N of sequences of

length N:

N=0: X_0 = 1 (the empty sequence) N=1: X_1 = 18 * X_0 = 18 N=2: X_2 = 12 * X_1 + 27 * X_0 = 243 N=3: X_3 = 12 * X_2 + 18 * X_1 = 3240 : : N=i: X_i = 12 * X_(i-1) + 18 * X_(i-2)

If you do the calculations, you find that X_0 + X_1 + X_2 + ... + X_17 is

about 2.0 * 10^19. So there are fewer essentially different sequences of

moves of length 17 or less than there are positions of the cube, and so some

positions require at least 18 moves.

The upper bound of 50 moves is I believe due to Morwen Thistlethwaite, who

developed a technique to solve the cube in a maximum of 50 moves. It

involved a descent through a chain of subgroups of the full cube group,

starting with the full cube group and ending with the trivial subgroup

(i.e. the one containing the solved position only). Each stage involves a

careful examination of the cube, essentially to work out which coset of the

target subgroup it is in, followed by a table look-up to find a sequence to put

it into that subgroup. Needless to say, it was not a fast technique!

But it was fascinating to watch, because for the first three quarters or

so of the solution, you couldn't really see anything happening - i.e. the

position continued to appear random! If I remember correctly, one of the

final subgroups in the chain was the subgroup generated by all the double

twists of the faces - so near the end of the solution, you would suddenly

notice that each face only had two colours on it. A few moves more and the

solution was complete. Completely different from most cube solutions, in

which you gradually see order return to chaos: with Morwen's solution, the

order only re-appeared in the last 10-15 moves.

* Mark's Update/Comments ----------------------------------------------

* Despite some accounts of Thistlethwaite's method, it is possible to

* follow the progression of his algorithm. Clearly after Stage 2 is

* completed the L and R faces will have L and R colours on them only.

* After Stage 3 is complete the characteristics of the square's group

* is clearly visible on all 6 sides. It is harder to understand what

* is accomplished in Stage 1.

*

* ---------------------------------------------------------------------

With God's algorithm, of course, I would expect this effect to be even more

pronounced: someone solving the cube with God's algorithm would probably

look very much like a film of someone scrambling the cube, run in

reverse!

Finally, something I'd be curious to know in this context: consider the

position in which every cubelet is in the right position, all the corner

cubelets are in the correct orientation, and all the edge cubelets are

"flipped" (i.e. the only change from the solved position is that every edge

is flipped). What is the shortest sequence of moves known to get the cube

into this position, or equivalently to solve it from this position? (I know

of several sequences of 24 moves that do the trick.)

* Mark's Update/Comments ---------------------------------------------- * * This is from my file cmoves.txt which includes the best known * sequences for interesting patterns: * * p3 12 flip R1 L1 D2 B3 L2 F2 R2 U3 D1 R3 D2 F3 B3 D3 F2 D3 (20) * R2 U3 F2 D3 * * ---------------------------------------------------------------------

The reason I'm interested in this particular position: it is the unique

element of the centre of the cube group. As a consequence, I vaguely suspect

(I'd hardly like to call it a conjecture :-) it may lie "opposite" the

solved position in the cube graph - i.e. the graph with a vertex for each

position of the cube and edges connecting positions that can be transformed

into each other with a single move. If this is the case, then it is a good

candidate to require the maximum possible number of moves in God's

algorithm.

-- David Seal dseal@armltd.co.uk

To my knowledge, no one has ever demonstrated a specific cube position

that takes 15 moves to solve. Furthermore, the lower bound is known to

be greater than 15, due to a simple proof.

* --------------------------------------------------------------------- * Mark> Definitely sequences exist in the square's group of length 15 * > e.g. Antipode 2 R2 B2 D2 F2 D2 F2 T2 L2 T2 D2 F2 T2 L2 T2 B2 * ---------------------------------------------------------------------

The way we know the lower bound is by working backwards counting how

many positions we can reach in a small number of moves from the solved

position. If this is less than 43,252,003,274,489,856,000 (the total

number of positions of Rubik's cube) then you need more than that

number of moves to reach the other positions of the cube. Therefore,

those positions will require more moves to solve.

The answer depends on what we consider a move. There are three common

definitions. The most restrictive is the QF metric, in which only a

quarter-turn of a face is allowed as a single move. More common is

the HF metric, in which a half-turn of a face is also counted as a

single move. The most generous is the HS metric, in which a quarter-

turn or half-turn of a central slice is also counted as a single move.

These metrics are sometimes called the 12-generator, 18-generator, and

27-generator metrics, respectively, for the number of primitive moves.

The definition does not affect which positions you can get to, or even

how you get there, only how many moves we count for it.

The answer is that even in the HS metric, the lower bound is 16,

because at most 17,508,850,688,971,332,784 positions can be reached

within 15 HS moves. In the HF metric, the lower bound is 18, because

at most 19,973,266,111,335,481,264 positions can be reached within 17

HF moves. And in the QT metric, the lower bound is 21, because at

most 39,812,499,178,877,773,072 positions can be reached within 20 QT

moves.

-- jjfink@skcla.monsanto.com writes:

Lately in this conference I've noted several messages related to Rubik's

Cube and Square 1. I've been an avid cube fanatic since 1981 and I've

been gathering cube information since.

Around Feb. 1990 I started to produce the Domain of the Cube Newsletter,

which focuses on Rubik's Cube and all the cube variants produced to

date. I include notes on unproduced prototype cubes which don't even

exist, patent information, cube history (and prehistory), computer

simulations of puzzles, etc. I'm up to the 4th issue.

Anyways, if you're interested in other puzzles of the scramble by

rotation type you may be interested in DOTC. It's available free to

anyone interested. I am especially interested in contributing articles

for the newsletter, e.g. ideas for new variants, God's Algorithm.

Anyone ever write a Magic Dodecahedron simulation for a computer?

Drop me a SASE (say empire size) if you're interested in DOTC or if you

would like to exchange notes on Rubik's Cube, Square 1 etc.

I'm also interested in exchanging puzzle simulations, e.g. Rubik's Cube,

Twisty Torus, NxNxN Simulations, etc, for Amiga and IBM computers. I've

written several Rubik's Cube solving programs, and I'm trying to make

the definitive puzzle solving engine. I'm also interested in AI programs

for Rubik's Cube and the like.

Ideal Toy put out the Rubik's Cube Newsletter, starting with

issue #1 on May 1982. There were 4 issues in all.

They are: #1, May 1982 #2, Aug 1982 #3, Winter 1983 #4, Aug 1983

There was another sort of magazine, published in several languages

called Rubik's Logic and Fantasy in space. I believe there were 8

issues in all. Unfortunately I don't have any of these! I'm willing

to buy these off anyone interesting in selling. I would like to get the

originals if at all possible...

I'm also interested in buying any books on the cube or related puzzles.

In particular I am _very_ interested in obtaining the following:

Cube Games Don Taylor, Leanne Rylands Official Solution to Alexander's Star Adam Alexander The Amazing Pyraminx Dr. Ronald Turner-Smith The Winning Solution Minh Thai The Winning Solution to Rubik's Revenge Minh Thai Simple Solutions to Cubic Puzzles James G. Nourse

I'm also interested in buying puzzles of the mechanical type.

I'm still missing the Pyraminx Star (basically a Pyraminx with more tips

on it), Pyraminx Ball, and the Puck.

If anyone out here is a fellow collector I'd like to hear from you.

If you have a cube variant which you think is rare, or an idea for a

cube variant we could swap notes.

I'm in the middle of compiling an exhaustive library for computer

simulations of puzzles. This includes simulations of all Uwe Meffert's

puzzles which he prototyped but _never_ produced. In fact, I'm in the

middle of working on a Pyraminx Hexagon solver. What? Never heard of it?

Meffert did a lot of other puzzles which never were made.

I invented some new "scramble by rotation" puzzles myself. My favourite

creation is the Twisty Torus. It is a torus puzzle with segments (which

slide around 360 degrees) with multiple rings around the circumference.

The computer puzzle simulation library I'm forming will be described

in depth in DOTC #4 (The Domain of the Cube Newsletter). So if you

have any interesting computer puzzle programs please email me and

tell me all about them!

Also to the people interested in obtaining a subscription to DOTC,

who are outside of Canada (which it seems is just about all of you!)

please don't send U.S. or non-Canadian stamps (yeah, I know I said

Self-Addressed Stamped Envelope before). Instead send me an

international money order in Canadian funds for $6. I'll send you

the first 4 issues (issue #4 is almost finished).

Mark Longridge

Address: 259 Thornton Rd N, Oshawa Ontario Canada, L1J 6T2

Email: mark.longridge@canrem.com

One other thing, the six bucks is not for me to make any money. This

is only to cover the cost of producing it and mailing it. I'm

just trying to spread the word about DOTC and to encourage other

mechanical puzzle lovers to share their ideas, books, programs and

puzzles. Most of the programs I've written and/or collected are

shareware for C64, Amiga and IBM. I have source for all my programs

(all in C or Basic) and I am thinking of providing a disk with the

4th issue of DOTC. If the response is favourable I will continue

to provide disks with DOTC.

-- Mark Longridge <mark.longridge@canrem.com> writes:

It may interest people to know that in the latest issue of "Cubism For Fun" %

(# 28 that I just received yesterday) there is an article by Herbert Kociemba

from Darmstadt. He describes a program that solves the cube. He states that

until now he has found no configuration that required more than 21 turns to

solve.

He gives a 20 move manoeuvre to get at the "all edges flipped/ all corners twisted" position: DF^2U'B^2R^2B^2R^2LB'D'FD^2FB^2UF'RLU^2F' or in Varga's parlance: dofitabiribirilobadafodifobitofarolotifa

Other things #28 contains are an analysis of Square 1, an article about

triangular tilings by Martin Gardner, and a number of articles about other

puzzles.

--

% CFF is a newsletter published by the Dutch Cubusts Club NKC.

Secretary:

Anneke Treep

Postbus 8295

6710 AG Ede

The Netherlands

Membership fee for 1992 is DFL 20 (about$ 11).

--

-- dik t. winter <dik@cwi.nl>

References:

Mathematical Papers

-------------------

Rubik's Revenge: The Group Theoretical Solution

Mogens Esrom Larsen A.M.M. Vol. 92 No. 6, June-July 1985, pages

381-390

Rubik's Groups

E. C. Turner & K. F. Gold, American Mathematical Monthly,

vol. 92 No. 9 November 1985, pp. 617-629.

Cubelike Puzzles - What Are They and How Do You Solve Them?

J.A. Eidswick A.M.M. Vol. 93 No. 3 March 1986, pp 157-176

The Slice Group in Rubik's Cube, David Hecker, Ranan Banerji

Mathematics Magazine, Vol. 58 No. 4 Sept 1985

The Group of the Hungarian Magic Cube

Chris Rowley Proceedings of the First Western Austrialian

Conference on Algebra, 1982

Books

-----

Rubik's Cubic Compendium

Erno Rubik, Tamas Varga, et al, Edited by David Singmaster

Oxford University Press, 1987

Enslow Publishers Inc

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