The Mathematics of R ubik’s Cubes

1. The Mathematics of Rubik’s Cubes Sean Rogers

2. 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

3. 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|

4. How to get from A to B R_6 R_3 R_1 R_0 R_7 R_4 R_2 R_5

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)

6. 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

7. 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

8. 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

9. 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

10. 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 useΨ • So describes cycle structure of corners • for edge blocks, and so on

11. 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)

12. 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