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

By Joyner W.D.

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A permutation of T is a bijection from T to itself. ) For example, on the 3 × 3 Rubik’s cube there are 9 · 6 = 54 facets. , 54 (in any way you like) then any move of the Rubik’s cube corresponds to a permutation of T54 . In this chapter we present some basic notation and properties of permutations. Notation: We may denote a permutation f : T → T by a 2 × n array: f↔ 1 2 ... n f (1) f (2) ... f (n) 33 34 CHAPTER 3. PERMUTATIONS Example 36 The identity permutation, denoted by I, is the permutation which doesn’t do anything: 1 2 ...

N}. , n}. We shall denote each permutation by the second row in its 2 × n array notation. For example, in the case n = 2: 1 2 2 1 are the permutations. 3. AN ALGORITHM TO LIST ALL THE PERMUTATIONS In case n = 4, the idea is to (a) write down each row n = 4 times each as follows: 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 3 3 3 3 1 1 1 1 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 3 3 3 3 47 48 CHAPTER 3. ,(aN , bN ), where N = n! , n} is of the form k f= (ai , bi ), i=1 for some k, 1 ≤ k ≤ N .

Sn | = |S1 | + ... + |Sn |. Example 29 If there are n bowls, each containing some distinguishable marbles and if Si is the set of marbles in the ith bowl then the number of ways to pick a marble from exactly one of the bowls is |S1 | + ... + |Sn |, by the addition principle. Corollary 30 (Pigeonhole principle) If there are n objects (pigeons) which must be placed in m (md < n) boxes (pigeonholes) then there is at least one box with at least d + 1 objects. Example 31 If you are in a room with 9 others then there must be either at least 5 people you know or 5 people you don’t know (not counting yourself ).

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