Group Theory and Group Theory and Rubik’s CubeRubik’s Cube

Hayley Poole

““What was lacking in the usual approach, What was lacking in the usual approach, even at its best was any sense of even at its best was any sense of

genuine enquiry, or any stimulus to genuine enquiry, or any stimulus to curiosity, or an appeal to the curiosity, or an appeal to the

imagination. There was little feeling imagination. There was little feeling that one can puzzle out an approach to that one can puzzle out an approach to

fresh problems without having to be fresh problems without having to be given detailed instructions.”given detailed instructions.”

From “Mathematical Puzzling” by A Gardiner - From “Mathematical Puzzling” by A Gardiner - Aspects of Secondary Education, HMSO, 1979Aspects of Secondary Education, HMSO, 1979

This presentation will This presentation will cover…cover…

• The history of the Rubik’s CubeThe history of the Rubik’s Cube

• Introduction to group theory Introduction to group theory

• Ideas behind solving the cubeIdeas behind solving the cube

Erno RubikErno Rubik

• Born 13Born 13thth July 1944 in July 1944 in

Budapest, HungaryBudapest, Hungary

• He is an inventor, sculptor and Professor of He is an inventor, sculptor and Professor of Architecture.Architecture.

History of the Rubik’s CubeHistory of the Rubik’s Cube

• Invented in 1974 Invented in 1974

• Originally called “Buvos Kocka” meaning “magic Originally called “Buvos Kocka” meaning “magic cube”cube”

• Rubik was intrigued by movements and Rubik was intrigued by movements and transformations of shapes in space which lead to transformations of shapes in space which lead to his creation of the cube.his creation of the cube.

• Took him 1 month to solve.Took him 1 month to solve.

• By Autumn of 1974 he had devised full solutionsBy Autumn of 1974 he had devised full solutions

History continuedHistory continued• Applied for it to be patented in January 1975Applied for it to be patented in January 1975

• Cube launched in Hungary in 1977Cube launched in Hungary in 1977

• Launched worldwide in 1980Launched worldwide in 1980

• First world championship took place in 1982 First world championship took place in 1982 in Budapest, winner solving it in 22.95 in Budapest, winner solving it in 22.95 secondsseconds

• TV cartoon created about it in 1983TV cartoon created about it in 1983

““Rubik, the amazing cube”Rubik, the amazing cube”

• Shown in America from 1983-1984Shown in America from 1983-1984

• About four children who discover that their About four children who discover that their Rubik's cube is alive (when the coloured Rubik's cube is alive (when the coloured squares on each of its sides are matched squares on each of its sides are matched up), and has amazing powers. They up), and has amazing powers. They befriend the cube, and they use its powers befriend the cube, and they use its powers to solve mysteries.to solve mysteries.

Number of possible Number of possible orientationsorientations• 8 corner cubes each having 3 possible 8 corner cubes each having 3 possible

orientations.orientations.

• 12 edge pieces each having 2 orientations 12 edge pieces each having 2 orientations

• The centre pieces are fixed.The centre pieces are fixed.

• This will give rise to a maximum number This will give rise to a maximum number of positions in the group being:of positions in the group being:

(8! x 3(8! x 388) x (12! X 2) x (12! X 21212) = ) = 519,024,039,293,878,272,000519,024,039,293,878,272,000

• Some positions in the cube occur from a Some positions in the cube occur from a result of another permutation.result of another permutation.

• Eg, in order to rotate one corner cube, Eg, in order to rotate one corner cube, another must also rotate. Hence, the another must also rotate. Hence, the number of positions is reduced.number of positions is reduced.

• This leaves This leaves

(8!x3(8!x377)x(12!x210) = )x(12!x210) = 43,252,003,274,489,856,000 or 43,252,003,274,489,856,000 or 4.3x104.3x101919

positions.positions.

Other CubesOther Cubes

• Pocket Cube: 2x2x2Pocket Cube: 2x2x2

• Rubik’s Revenge: 4x4x4Rubik’s Revenge: 4x4x4

• Professors Cube: 5x5x5Professors Cube: 5x5x5

• Pyraminx: tetrahedronPyraminx: tetrahedron

• Megaminx: DodecahrdronMegaminx: Dodecahrdron

How do we use maths to solve How do we use maths to solve the cube?the cube?

• Every maths problem is a puzzle.Every maths problem is a puzzle.

• A puzzle is a game, toy or problem A puzzle is a game, toy or problem designed to test ingenuity or knowledge.designed to test ingenuity or knowledge.

• We use group theory in solving the Rubik’s We use group theory in solving the Rubik’s cube.cube.

Introduction to groupsIntroduction to groups

• A Group is a set with a binary operation A Group is a set with a binary operation which obeys the following four axioms:which obeys the following four axioms:

• ClosureClosure

• AssociativityAssociativity

• IdentityIdentity

• InverseInverse

Associativity – The order in which the

operation is carried out doesn’t matter.

For every g1,g2,g3 Є G, we have

g1º (g2º g3)=(g1º g2)º g3

Identity – There must exist an element e in

the group such that for every g Є G, we have

e º g = g º e = g

Closure – If two elements are members of the group (G), then any combination

of them must also be a member of the group.

For every g1,g2 Є G, then g1º g2 Є G

Inverse – Every member of the group

must have an inverse. For every g Є G, there is an element g-1 Є G

such that g º g-1 = g-1

º g = e

Groups

Propositions and ProofsPropositions and Proofs

• The identity element of a group G is The identity element of a group G is unique.unique.

• The inverse of an element gThe inverse of an element gЄЄG is unique.G is unique.

• If g,h,If g,h,ЄЄG and gG and g-1-1 is the inverse of g and h is the inverse of g and h-1-1 is the inverse of h then (gh)is the inverse of h then (gh)-1-1=h=h-1-1gg-1-1..

Basic Group TheoryBasic Group Theory

• Consider the group Consider the group {1,2,3,4} under {1,2,3,4} under multiplication modulo 5.multiplication modulo 5.

• The identity is 1.The identity is 1.• 2 and 3 generate the 2 and 3 generate the

group with having order 4.group with having order 4.• 4 has order 2 (44 has order 2 (422=1).=1).• Elements 1 and 4 form a Elements 1 and 4 form a

group by themselves, group by themselves, called a subgroup.called a subgroup.

XX55 11 22 33 44

11 11 22 33 44

22 22 44 11 33

33 33 11 44 22

44 44 33 22 11

Points about Groups and Points about Groups and subgroupssubgroups• The order of an element The order of an element aa is is nn if a if ann=e.=e.

• All subgroups must contain the identity element.All subgroups must contain the identity element.

• The order of a subgroup is always a factor of the The order of a subgroup is always a factor of the order of the group (Lagrange’s Theorem).order of the group (Lagrange’s Theorem).

• The only element of order 1 is the identity.The only element of order 1 is the identity.

• Any element of order 2 is self inverse.Any element of order 2 is self inverse.

• A group of order n is cyclic iff it contains an A group of order n is cyclic iff it contains an element of order n.element of order n.

So what does this have to So what does this have to do with solving Rubik’s do with solving Rubik’s

cube?cube?

Does Rubik’s Cube form a Does Rubik’s Cube form a group?group?

• Closure – yes, whatever moves are carried Closure – yes, whatever moves are carried out we still have a cube.out we still have a cube.

• Associativity – yes (FR)L=F(RL).Associativity – yes (FR)L=F(RL).

• Identity – yes, by doing nothing.Identity – yes, by doing nothing.

• Inverse – yes, by doing the moves Inverse – yes, by doing the moves backwards you get back to the identity, eg backwards you get back to the identity, eg (FRBL)(L(FRBL)(L-1-1BB-1-1RR-1-1FF-1-1)=e)=e

• Therefore we have a group.Therefore we have a group.

Up (U)

Down (D)

Right (R)Left (L)

Back (B)

Face (F)

The corner 3-cycleThe corner 3-cycle• Consider FRFConsider FRF-1-1LFRLFR--1F1F-1-1LL-1-1

• Three corner pieces out of place – Three corner pieces out of place –

permuted cyclicly.permuted cyclicly.

• Why does a long algorithm have such a Why does a long algorithm have such a simple effect?simple effect?

• g and h are two operations g and h are two operations • Denote [g,h]=ghgDenote [g,h]=ghg-1-1hh-1-1 - Commuter of g and - Commuter of g and

h, as [g,h]=1 iff gh=hg.h, as [g,h]=1 iff gh=hg.

• Proved easily: multiple [g.h] by hg on Proved easily: multiple [g.h] by hg on right:right: ghgghg-1-1hh-1-1hghg =hg=hg

ghgghg-1-1gg =hg=hg ghgh =hg=hg

• g and h commute if gh=hg. The equation g and h commute if gh=hg. The equation [g,h]=1 says that the commuter is trivial [g,h]=1 says that the commuter is trivial iff g and h commute with each other.iff g and h commute with each other.

• g is an operation on the cube, the support of g is an operation on the cube, the support of g denoted supp(g) is the set of pieces which g denoted supp(g) is the set of pieces which are changed by g. Similarly for h. are changed by g. Similarly for h.

• If g and h have disjoint support, ie no If g and h have disjoint support, ie no overlap then they commute. overlap then they commute.

• Consider the R and L movement of the Consider the R and L movement of the cube. The support of R consists of the 9 cube. The support of R consists of the 9 cubes on the right and the support of L cubes on the right and the support of L consists of the 9 cubes on the left. Moving consists of the 9 cubes on the left. Moving R doesn’t affect L. R doesn’t affect L.

• Therefore LR=RLTherefore LR=RL

• Now if g and h are two operations whose Now if g and h are two operations whose supports have only a small amount of supports have only a small amount of overlap, then g and h will almost commute. overlap, then g and h will almost commute.

• This means [g,h] will be an operation This means [g,h] will be an operation affecting only a small number of pieces.affecting only a small number of pieces.

• Going back to the initial sequence of moves:Going back to the initial sequence of moves:

FRFFRF-1-1LFRLFR--1F1F-1-1LL-1-1, let g=FRF, let g=FRF-1-1

• h=L only affects the 9 pieces on the left, h=L only affects the 9 pieces on the left, and of these, the previous diagram shows and of these, the previous diagram shows that g=FRFthat g=FRF-1-1 only affects a single piece. only affects a single piece.

• Since there is little overlap between the Since there is little overlap between the supports of g and h, these operations will supports of g and h, these operations will almost commute so their commuter is almost commute so their commuter is almost trivial.almost trivial.

• Therefore, [g,h]=FRFTherefore, [g,h]=FRF-1-1LFRLFR--1F1F-1-1LL-1 -1 should should only affect a small number of pieces, in only affect a small number of pieces, in fact it affects 3. fact it affects 3.

Brief Application to school Brief Application to school levellevel

• describing properties of shapes describing properties of shapes

• nets and how 3D shapes are made nets and how 3D shapes are made

• Rotation and symmetryRotation and symmetry

• Area and volumeArea and volume

ConclusionsConclusions

• Group Theory is a very versatile area Group Theory is a very versatile area of mathematics.of mathematics.

• It is not only used in maths but also It is not only used in maths but also in chemistry to describe symmetry of in chemistry to describe symmetry of molecules.molecules.

• The theory involved in solving the The theory involved in solving the rubik’s cube is very complicated.rubik’s cube is very complicated.