CS173: Discrete Mathematical Structures
Spring 2006
Homework #11
Due 04/23/05, 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. an=2anโ1+2n2, a1=4 .
b. an= โanโ1+2anโ2+2nโ1, a0=a1=1.
c. an=9an
2
+n2,a1=0.
d. an=9an
4
+n2, a1=0, a2=1.
e. an
=
5an
3
โ
6an
9
โ
3, a1
=
0, a3
=
1, a9
=
2.
3. Given the function an=k3
k=1
n
โ
a. Prove that the sequence an provides a solution to the linear
nonhomogeneous recurrence an=anโ1+n3 with a base case of a1=1.
b. Solve the recurrence using annihilators.
4.
a. Find the general solution for T(n) = 2T(n-1) + 15T(m-2) + 3*2m.
b. Transform b(n) = 2b(n/2) + 15b(n/4) + 3n into the linear recurrence above.
c. Give the general solution for b(n).
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