Download Mathematics in Rubik`s cube.

THE MATHEMATICS OF
RUBIK’S CUBES
Sean Rogers
Possibilities
• 43,252,003,274,489,856,000 possible states
• Depends on properties of each face
• That’s a lot!!
• Model each as a set
• Define R_0 as the solved state
• {r_1, r_2, r_4 …, r_9, b_1, b_2, b_3 … b_9, w_1 …}
• So every set has 54 elements
Functions
• Define f: R_x  R_y as this:
• We have a special name for this: L
• Similarly, we have R, U, B, D, R^2, L’, R’, etc.
• These functions are bijections from one set to another
• Obvious- one-to-one correspondence, |R_x|=|R_y|
How to get from A to B
R_6
R_3
R_1
R_0
R_7
R_4
R_2
R_5
Algorithms
• We collect these bijections into algorithms (macros) to get
from one set to another (when you know the properties of
the 2 sets required)
Groups
• A group G is (G, *)
• G is a set of objects, * is an operator acting on them
• 4 axioms:
• Closed (for any group elements a and b, a*b ∈ G)
• Operation * is associative
• For elements a, b, and c, (a*b)*c=a(b*c)
• There exists an identity element e ∈ G s.t. e*g=g*e=g
• Every element in G has an inverse 𝑔−1 relative to * s.t.
• 𝑔 ∗ 𝑔−1 =e
Note that commutatively is not necessarily property
Examples
• Integers are closed under addition
• Identity element is 0, inverse of integer n is -n
• Rational numbers are closed under multiplication
(excluding 0)
• Identity element is 1, inverse of x is
1
𝑥
To Rubik’s Cubes
• Our group will be R, all possible permutations of the
solved state (remember there are ~43 quintillion)
• * will be a rotation of a face (associative so long as order
is preserved)
• Inverse is going the opposite direction
Cycles and Notation
• Cycle- permutation of the elements of some set X which
maps the elements of some subset S set to each other in
a cyclic manner, while fixing all other elements (mapping
them to themselves)
• (1)(2 3 4)
• 1 stays put, 2, 3, and 4 are cycled in some manner
• Ex. {1,2,3,4} {3,4,1,2} is a cycle
• You can’t just switch 2 blocks- permutations are products
of 2-cycles
• Ex. (1 2 3)=(1 2)(1 3)
• Analogue- Prime factorizations
Importance of Cycles
• Parity- amount of 2-cycles that make up a cycle
• Every permutation on the cube has an even parity
• Means you can never exchange just two blocks
• We use at least 3-cycles to reorder blocks in the wrong
place
• Can now quantify the behavior of different blocks on the cube
• Let’s use Ψ
• So Ψ𝑐𝑜𝑟𝑛𝑒𝑟 describes cycle structure of corners
• Ψ𝑒𝑑𝑔𝑒 for edge blocks, and so on
Conjugacy
• Conjugacy ≈ equivalence relations
• Let A be some algorithm (macro) that performs an
operation on the cube, like a cycle of 3 corner pieces.
• Now for some legal cube move M, 𝑀𝐴𝑀−1 is the conjugation of M
by A
• Ex. if M=RUR’U’, then the conjugate of M by F=FRUR’U’F’
• Do something, do something else, undo the first thing
• Conjugacy is an equivalence relation
• Instead of equivalence classes, we have conjugacy classes
• So if we know the conjugacy class of a few blocks, and
how they move (Ψ), we have a way of getting from point A
to point B (or set A to set B, if you prefer)
The Cube
• Several methods to solve
• They make even bigger, harder cubes
• You don’t need this math though- its just a rigorous way of
defining a puzzle
• Invented in 1974 by Ernő Rubik