Discrete Mathematics with Graph Theory (3rd Edition) 150

Discrete Mathematics with Graph Theory (3rd Edition) 150 -...

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148 Solutions to Exercises A B C Initial position 1,2,3 * * Move 1 2,3 1 * Move 2 2,3 * 1 Move 3 3 2 1 Move 4 3 1,2 * MoveS 1,3 2 * Thirteen moves are required. By reversing the Move 6 1,3 * 2 steps, we see that thirteen moves also suffice when Move 7 3 1 2 the tower is initially on the middle peg. Move 8 3 * 1,2 Move 9 * 3 1,2 Move 10 * 1,3 2 Move 11 1 3 2 Move 12 1 2,3 * Move 13 * 1,2,3 * (b) To transfer n discs from peg A to peg B, do the following: 1. Transfer the top n - 1 discs to peg B in an-l moves. 2. Then transfer all n - 1 discs from peg B to peg C in another an-l moves. 3. Transfer the largest disc from A to B in one move. 4. Finally transfer the n - 1 discs from peg C to peg B in another an-l moves. We obtain an = 3an -l + 1. (c) We obtain easily that Pn = -! is a particular solution to the recurrence relation an = 3an -l + 1. The characteristic polynomial associated with an = 3an -l is X2 - 3x with characteristic roots 0,3. Thus, qn = c(3 n ) and Pn + qn = c(3 n ) - !. Since
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