By Joyner W.D.

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K. 5 Let S be the set Z of all integers. , an integer multiple of 2). (1) Show that this is an equivalence relation, (2) Find the sets Si in the above lemma which partition S, (3) Find a representative of each equivalence class Si . 4. 4 35 Counting This section quickly surveys the few basic counting principles we shall use later. , Sn denote disjoint finite sets. Then |S1 ∪ ... ∪ 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 | + ...

8) and the latitudinal lines are at φ = π/4, π/2, 3π/4. 141592... ) 62 CHAPTER 4. PERMUTATION PUZZLES The sphere shall be oriented by the right-hand rule - the thumb of the right hand wrapping along the polar axis points towards the north pole. We assume that one of the longitudinal lines has been fixed once and for all. This fixed line shall be labeled ”1”, the next line (with respect to the orientation above) as ”2”, and so on. Allowed moves: One may rotate the masterball east-to-west by multiples of π/4 along each of the 4 latitudinal bands or by multiples of π along each of the 8 longitudinal lines.

0 .. 0 1 is called the n × n identity matrix and denoted I or In . 4 Muliplication and inverses You can multiply an m × n matrix A by a n × p matrix B and get a m × p matrix AB. The (i, j)th entry of AB is computed as follows: 1. Let k = 1 and c0 = 0. 2. If k = m, you’re done and aij = cm . Otherwise proceed to the next step. 3. Take the k th entry of the ith row of A and multiply it by the k th entry of the j th row of B. Let ck = ck−1 + aik bkj . 4. Increment k by 1 and go to step 2.