# Mathematics of the Rubik's cube by Joyner W.D. PDF

By Joyner W.D.

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Additional info for Mathematics of the Rubik's cube

Example text

Sk = ∪i Si (2) the Si ’s are disjoint: for 1 ≤ i ≤ k, 1 ≤ j ≤ k, if i = j then Si ∩ Sj = ∅. ) (3) the Si ’s exhaust the collection of equivalence classes of R: for each 1 ≤ i ≤ k, there is an si ∈ S such that Si = [si ]. ) Example 28 For real numbers x, y, define x ≡ y if and only if x − y is an integer. }, for x real. Each equivalence class has exactly one representative in the half open interval [0, 1). Remark 2 Conversely, given a partition as in (1), there is an equivalence relation R such that Si = [si ], for some some si ∈ S, where [s] = {x ∈ S | s ∼ x} is the equivalence class of s with respect to R.

A one person game is a sequence of moves following certain rules satisfying • there are finitely many moves at each stage, • there is a finite sequence of moves which yields a solution, • there are no chance or random moves, • there is complete information about each move, • each move depends only on the present position, not on the existence or non-existence of a certain previous move (such as chess, where castling is made illegal if the king has been moved previously). A permutation puzzle is a one person game (solitaire) with the following five properties listed below.

The array corresponds to a rotation of the north pole facets by 3π/4. Notation We use matrix notation to denote the 32 facets of the masterball. The generators for the latitudinal rotations are denoted r1 , r2 , r3 , r4 . 4. RAINBOW MASTERBALL 63 As you look down at the ball from the north pole, this move rotates the ball clockwise. The other moves r2 , r3 , r4 rotate the associated band of the ball in the same direction - clockwise as viewed from the north pole. , f8 . For example, f1 sends 11 21 31 41 12 22 32 42 13 23 33 43 14 24 34 44 15 25 35 45 16 26 36 46 17 27 37 47 18 28 38 48 44 34 24 15 43 33 23 14 42 32 22 13 41 31 21 12 16 25 35 45 17 26 36 46 18 27 37 47 11 28 38 48 to which is pictured as: 64 CHAPTER 4.