CS173: Discrete Math Homework 10 - Tower of Hanoi Puzzle & Recurrences, Assignments of Discrete Structures and Graph Theory

Download CS173: Discrete Math Homework 10 - Tower of Hanoi Puzzle & Recurrences and more Assignments Discrete Structures and Graph Theory in PDF only on Docsity! CS173: Discrete Mathematical Structures Fall 2005 Homework #10 Due 11/20/05, 8a 1. In the Tower of Hanoi puzzle, suppose our goal is to transfer all n disks from peg 1 to peg 3, but we cannot move a disk directly between pegs 1 and 3. Each move of a disk must be a move involving peg 2. As usual, we cannot place a disk on top of a smaller disk. a. Find a recurrence relation for the number of moves required to solve the puzzle for n disks with this added restriction. b. Solve this recurrence relation to find a formula for the number of moves required to solve the puzzle for n disks. c. How many different arrangements are there of the n disks on three pegs so that no disk is on top of a smaller disk? d. Use your answers to parts b and c to argue that every allowable arrangement of the n disks occurs in the solution of this variation of the puzzle. 2. Solve the following recurrences exactly: a. ! a n = 2a n"1 + 2n 2 , a 1 = 4 . b. ! a n = "a n"1 + 2an"2 + 2 n"1 , a 0 = a 1 =1. c. ! a n = 5a n 3 " 6a n 9 " 3, a 1 = 0, a 3 =1, a 9 = 2. 3. Given the function ! a n = k 2 k=1 n " a. Prove that the sequence an provides a solution to the linear nonhomogeneous recurrence ! a n = a n"1 + n 2 with a base case of a1=1. b. Solve the recurrence using annihilators. 4. Give an asymptotic solution to the following recurrence (that is, provide a Θ bound for the solution). ! a n = 3a n"1 5a n"2 , a i # 0.