THE MATHEMATICS OF RUBIK’S CUBESSean 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_7

R_6

R_5

R_4

R_3

R_2

R_1

R_0

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 relative to * s.t.

• =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

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 describes cycle structure of corners• 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, 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