Group Theory and Rubik

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Group Theory and Rubiks Cube
Hayley Poole
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• What was lacking in the usual approach, even at
  its best was any sense of genuine enquiry, or any
  stimulus to curiosity, or an appeal to the
  imagination. There was little feeling that one
  can puzzle out an approach to fresh problems
  without having to be given detailed
  instructions.
• From Mathematical Puzzling by A Gardiner -
  Aspects of Secondary Education, HMSO, 1979

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This presentation will cover

• The history of the Rubiks Cube
• Introduction to group theory
• Ideas behind solving the cube

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Erno Rubik

• Born 13th July 1944 in
• Budapest, Hungary
• He is an inventor, sculptor and Professor of
  Architecture.

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History of the Rubiks Cube

• Invented in 1974
• Originally called Buvos Kocka meaning magic
  cube
• Rubik was intrigued by movements and
  transformations of shapes in space which lead to
  his creation of the cube.
• Took him 1 month to solve.
• By Autumn of 1974 he had devised full solutions

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History continued

• Applied for it to be patented in January 1975
• Cube launched in Hungary in 1977
• Launched worldwide in 1980
• First world championship took place in 1982 in
  Budapest, winner solving it in 22.95 seconds
• TV cartoon created about it in 1983

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Rubik, the amazing cube

• Shown in America from 1983-1984
• About four children who discover that their
  Rubik's cube is alive (when the coloured squares
  on each of its sides are matched up), and has
  amazing powers. They befriend the cube, and they
  use its powers to solve mysteries.

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Number of possible orientations

• 8 corner cubes each having 3 possible
  orientations.
• 12 edge pieces each having 2 orientations
• The centre pieces are fixed.
• This will give rise to a maximum number of
  positions in the group being
• (8! x 38) x (12! X 212) 519,024,039,293,878,272
  ,000

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• Some positions in the cube occur from a result of
  another permutation.
• Eg, in order to rotate one corner cube, another
  must also rotate. Hence, the number of positions
  is reduced.
• This leaves
• (8!x37)x(12!x210) 43,252,003,274,489,856,
  000 or 4.3x1019
• positions.

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Other Cubes

• Pocket Cube 2x2x2
• Rubiks Revenge 4x4x4
• Professors Cube 5x5x5
• Pyraminx tetrahedron
• Megaminx Dodecahrdron

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How do we use maths to solve the cube?

• Every maths problem is a puzzle.
• A puzzle is a game, toy or problem designed to
  test ingenuity or knowledge.
• We use group theory in solving the Rubiks cube.

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Introduction to groups

• A Group is a set with a binary operation which
  obeys the following four axioms
• Closure
• Associativity
• Identity
• Inverse

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Associativity The order in which the operation
is carried out doesnt matter. For every g1,g2,g3
? G, we have g1º (g2º g3)(g1º g2)º g3
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 ? G,
then g1º g2 ? G
Groups
Identity There must exist an element e in the
group such that for every g ? G, we have e º g
g º e g
Inverse Every member of the group must have an
inverse. For every g ? G, there is an element
g-1 ? G such that g º g-1 g-1º g e
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Propositions and Proofs

• The identity element of a group G is unique.
• The inverse of an element g?G is unique.
• If g,h,?G and g-1 is the inverse of g and h-1 is
  the inverse of h then (gh)-1h-1g-1.

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Basic Group Theory
X5 1 2 3 4
1 1 2 3 4
2 2 4 1 3
3 3 1 4 2
4 4 3 2 1

• Consider the group 1,2,3,4 under multiplication
  modulo 5.
• The identity is 1.
• 2 and 3 generate the group with having order 4.
• 4 has order 2 (421).
• Elements 1 and 4 form a group by themselves,
  called a subgroup.

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Points about Groups and subgroups

• The order of an element a is n if ane.
• All subgroups must contain the identity element.
• The order of a subgroup is always a factor of the
  order of the group (Lagranges Theorem).
• The only element of order 1 is the identity.
• Any element of order 2 is self inverse.
• A group of order n is cyclic iff it contains an
  element of order n.

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So what does this have to do with solving Rubiks
cube?
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Does Rubiks Cube form a group?

• Closure yes, whatever moves are carried out we
  still have a cube.
• Associativity yes (FR)LF(RL).
• Identity yes, by doing nothing.
• Inverse yes, by doing the moves backwards you
  get back to the identity, eg
• (FRBL)(L-1B-1R-1F-1)e
• Therefore we have a group.

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Up (U)
Back (B)
Right (R)
Left (L)
Face (F)
Down (D)
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The corner 3-cycle

• Consider FRF-1LFR-1F-1L-1
• Three corner pieces out of place
• permuted cyclicly.
• Why does a long algorithm have such a simple
  effect?

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• g and h are two operations
• Denote g,hghg-1h-1 - Commuter of g and h, as
  g,h1 iff ghhg.
• Proved easily multiple g.h by hg on right
• ghg-1h-1hg hg
• ghg-1g hg
• gh hg
• g and h commute if ghhg. The equation g,h1
  says that the commuter is trivial iff g and h
  commute with each other.

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• g is an operation on the cube, the support of g
  denoted supp(g) is the set of pieces which are
  changed by g. Similarly for h.
• If g and h have disjoint support, ie no overlap
  then they commute.
• Consider the R and L movement of the cube. The
  support of R consists of the 9 cubes on the right
  and the support of L consists of the 9 cubes on
  the left. Moving R doesnt affect L.
• Therefore LRRL

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• Now if g and h are two operations whose supports
  have only a small amount of overlap, then g and h
  will almost commute.
• This means g,h will be an operation affecting
  only a small number of pieces.
• Going back to the initial sequence of moves
• FRF-1LFR-1F-1L-1, let gFRF-1

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• hL only affects the 9 pieces on the left, and of
  these, the previous diagram shows that gFRF-1
  only affects a single piece.
• Since there is little overlap between the
  supports of g and h, these operations will almost
  commute so their commuter is almost trivial.

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• Therefore, g,hFRF-1LFR-1F-1L-1 should only
  affect a small number of pieces, in fact it
  affects 3.

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Brief Application to school level

• describing properties of shapes
• nets and how 3D shapes are made
• Rotation and symmetry
• Area and volume

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Conclusions

• Group Theory is a very versatile area of
  mathematics.
• It is not only used in maths but also in
  chemistry to describe symmetry of molecules.
• The theory involved in solving the rubiks cube
  is very complicated.