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

By Joyner W.D.

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Example text

2 There are similar identities for r2 , r3 , r4 . Find them. Identify the facets of the masterball with the entries of the array 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 9 24 23 22 21 20 19 18 17 32 31 30 29 28 27 26 25 (there is a reason for labeling the facets ”backwards” like this but it’s not important).

B) Let f (x) = 2x, let S be the real line, and let R be the corresponding relation as in the first example. Is R an equivalence relation? 3 Let R be the corresponding relation as in the second example. Is R an equivalence relation? Let R be an equivalence relation on a set S. For s ∈ S, we call the subset [s] = {t ∈ S | s ∼ t} the equivalence class of s in S. Example 26 For integers x, y, define x ≡ y if and only if 3 divides x − y. }. 4 Show that for any s1 and s2 in S, we have either (a) [s1 ] = [s2 ], or (b) [s1 ] is disjoint from [s2 ].

Such a permutation is called cyclic. The number r is called the length of the cycle. We call two such cycles (a1 a2 ... ar ) and (b1 b2 ... , bt } are disjoint. Lemma 51 If f and g are disjoint cyclic permutations of T then f g = gf . proof: This is clear since the permutations f and g of T affect disjoint collections of integers, so the permutations may be performed in either order. Lemma 52 The cyclic permutation (a1 a2 ... ar ) has order r. 2. , f r−1 (a1 ) = ar , f r (a1 ) = a1 , by definition of f .

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


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