The Mathematics of Rubik's Cubes: Modeling and Functions

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 _7 R _6 R _5 R _4 R _3 R _2 R _1 R _0

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

7. Examples

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

11. Conjugacy

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