Download Group Theory and the Rubik`s Cube

Group Theory and
Rubik’s Cube
Hayley Poole
“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
This presentation will cover…
• The history of the Rubik’s Cube
• Introduction to group theory
• Ideas behind solving the cube
Erno Rubik
• Born 13th July 1944 in
Budapest, Hungary
• He is an inventor, sculptor and Professor of
Architecture.
History of the Rubik’s 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
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
“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.
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
• 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.
Other Cubes
• Pocket Cube: 2x2x2
• Rubik’s Revenge: 4x4x4
• Professors Cube: 5x5x5
• Pyraminx: tetrahedron
• Megaminx: Dodecahrdron
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 Rubik’s cube.
Introduction to groups
• A Group is a set with a binary operation which
obeys the following four axioms:
• Closure
• Associativity
• Identity
• Inverse
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
g1 º g2 Є G
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
Groups
Identity – There must
exist an element e in the
group such that for every
g Є G, we have
e º g = g ºe = 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-1º g = e
Propositions and Proofs
• The identity element of a group G is unique.
• The inverse of an element gЄG is unique.
• If g,h,ЄG and g-1 is the inverse of g and h-1 is
the inverse of h then (gh)-1=h-1g-1.
Basic Group Theory
• Consider the group {1,2,3,4}
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
•
•
under multiplication modulo 5.
The identity is 1.
2 and 3 generate the group
with having order 4.
4 has order 2 (42=1).
Elements 1 and 4 form a
group by themselves, called a
subgroup.
Points about Groups and subgroups
• The order of an element a is n if an=e.
• All subgroups must contain the identity element.
• The order of a subgroup is always a factor of the
order of the group (Lagrange’s 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.
So what does this have to do
with solving Rubik’s cube?
Does Rubik’s Cube form a group?
• Closure – yes, whatever moves are carried out
we still have a cube.
• Associativity – yes (FR)L=F(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.
Up (U)
Back (B)
Right (R)
Left (L)
Face (F)
Down (D)
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?
• g and h are two operations
• Denote [g,h]=ghg-1h-1 - Commuter of g and h,
as [g,h]=1 iff gh=hg.
• Proved easily: multiple [g.h] by hg on right:
ghg-1h-1hg =hg
ghg-1g =hg
gh =hg
• g and h commute if gh=hg. The equation
[g,h]=1 says that the commuter is trivial iff g
and h commute with each other.
• 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 doesn’t affect L.
• Therefore LR=RL
• 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 g=FRF-1
• h=L only affects the 9 pieces on the left, and of
these, the previous diagram shows that g=FRF-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.
• Therefore, [g,h]=FRF-1LFR-1F-1L-1 should only
affect a small number of pieces, in fact it affects
3.
Brief Application to school level
• describing properties of shapes
• nets and how 3D shapes are made
• Rotation and symmetry
• Area and volume
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 rubik’s
cube is very complicated.