three multiplications of n-bit integers
are carried out using 3f (n)-bit
operations. Each of the additions,
subtractions, and shifts uses a constant
multiple of n-bit operations, and Cn
represents the total number of bit
operations used by these operation

assigned to the outcome of heads and
the outcome of tails if heads comes up
three times as often as tails? 3. a)
Define the conditional probability of
an event E given an event F. b) Suppose
E is the event that when a die is rolled
it comes up an even num

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) Sh

relation for the number of bit strings of
length n that contain three consecutive
0s. b) What are the initial conditions?
c) How many bit strings of length seven
contain three consecutive 0s? 9. a) Find
a recurrence relation for the number
of bit strings

circuit produces a graph with the same
Hamilton circuit. So as we add edges to
a graph, especially when we make sure
to add edges to each vertex, we make it
WILLIAM ROWAN HAMILTON (1805
1865) William Rowan Hamilton, the
most famous Irish scientist ever to

success? 14. a) What is the variance of
the sum of n independent random
variables? b) What is the variance of
the number of successes when n
independent Bernoulli trials, each with
probability p of success, are carried
out? 15. What does Chebyshevs
inequa

we search is in the list is 1/3, and it is
equally likely that this element is any
of the n elements in the list? 8. a) What
is meant by a Bernoulli trial? b) What
is the probability of k successes in n
independent Bernoulli trials? c) What
is the expecte

g(n) arise in many different situations.
It is possible to derive estimates of the
size of functions that satisfy such
recurrence relations. Suppose that f
satisfies this recurrence relation
whenever n is divisible by b. Let n = bk,
where k is a positive

independent Bernoulli trials are
carried out equals C(n, k)pkqnk,
where p is the probability of success
and q = 1 p is the probability of
failure. Bayes theorem: If E and F are
events from a sample space S such that
p(E) = 0 and p(F ) = 0, then p(F | E) =

numbered 1 through 12. 8. a) What is
the expected value of the number that
comes up when a fair dodecahedral die
is rolled? b) What is the variance of the
number that comes up when a fair
dodecahedral die is rolled? 9. Suppose
that a pair of fair octahedr

dominoes. [Hint: Consider separately
the coverings where the position in the
top right corner of the checkerboard is
covered by a domino positioned
horizontally and where it is covered by
a domino positioned vertically.] b)
What are the initial conditions

the minimum number of integer
multiplications needed to solve a
matrix-chain multiplication problem
has exponential worst-case complexity.
[Hint: Do this by first showing that the
order of multiplication of matrices is
specified by parenthesizing the
prod

recurrence relation for the number of
ternary strings of length n that contain
two consecutive symbols that are the
same. b) What are the initial
conditions? c) How many ternary
strings of length six contain
consecutive symbols that are the
same? 19. Mess

not the sixth number, drawn? c) What
is the probability that a player wins
$150 by matching exactly three of the
first five numbers and the sixth
number or by matching four of the first
five numbers but not the sixth
number? d) What is the probability
tha

first five numbers or the last number.
4. What is the probability that a hand
of 13 cards contains no pairs? 5. What
is the probability that a 13-card bridge
hand contains a) all 13 hearts? b) 13
cards of the same suit? c) seven spades
and six clubs? d) s

card at random and observe only one
side. a) If the side is black, what is the
probability that the other side is also
black? b) What is the probability that
the opposite side is the same color as
the one we observed? 18. What is the
probability that when

is the number of moves used by the
FrameStewart algorithm to solve the
Reves puzzle with n disks, where k is
chosen to be the smallest integer with
n k(k + 1)/2, then R(n) satisfies the
recurrence relation R(n) = 2R(n k) +
2k 1, with R(0) = 0 and R(1) = 1

of people linking these people, where
two people adjacent in the chain know
one another. For example, in Figure 6
in Section 10.1, there is a chain of six
people linking Kamini and Ching. Many
social scientists have conjectured that
almost every pair of p

cases where the older child is a boy
born on a Tuesday and then the case
where the older child is not a boy born
on a Tuesday.] 29. Let X be a random
variable on a sample space S. Show
that V (aX + b) = a2V (X) whenever a
and b are real numbers. 30. Use
C

but one come up the same, the person
whose coin comes up different buys the
refreshments. Otherwise, the people
flip the coins again and continue until
just one coin comes up different from
all the others. a) What is the
probability that the odd person ou

n-bit integers using the fast
multiplication algorithm described in
Example 4. Solution: Example 4 shows
that f (n) = 3f (n/2) + Cn, when n is
even, where f (n) is the number of bit
operations required to multiply two nbit integers using the fast
multipli

where Rn is the number of regions that
a plane is divided into by n lines, if no
two of the lines are parallel and no
three of the lines go through the same
point. b) Find Rn using iteration. 22.
a) Find the recurrence relation
satisfied by Rn, where Rn i

and 28 = 256 dollars if the first head
comes up on or after the eighth flip.
What is the expected value of the
amount of money the person wins?
How much money should a person be
willing to pay to play this game? 22.
Suppose that n balls are tossed into b

only one survives. We denote the
number of the survivor by J (n). 33.
Determine the value of J (n) for each
integer n with 1 n 16. 34. Use the
values you found in Exercise 33 to
conjecture a formula for J (n). [Hint:
Write n = 2m + k, where m is a
nonnega

comes up heads, that the third flip
comes up heads, and that exactly one of
the first flip and third flip come up
heads, respectively, when a fair coin is
flipped three times. Are E1, E2, and E3
pairwise independent? Are they
mutually independent? e) How

circuit. THEOREM 4 ORES THEOREM If
G is a simple graph with n vertices with
n 3 such that deg(u) + deg(v) n for
every pair of nonadjacent vertices u
and v in G, then G has a Hamilton
circuit. The proof of Ores theorem is
outlined in Exercise 65. Diracs
th

there should be an easy way to
determine this, because there is a
simple way to answer the similar
question of whether a graph has an
Euler circuit. Surprisingly, there are no
known simple necessary and sufficient
criteria for the existence of Hamilton
ci

FIGURE 9 A Solution to the A Voyage
Round the World Puzzle. Because the
author cannot supply each reader with
a wooden solid with pegs and string,
we will consider the equivalent
question: Is there a circuit in the graph
shown in Figure 8(b) that passes
t

k1 j = 0 aj c = ak f (1) + c k1 j = 0 aj .
When a = 1 we have f (n) = f (1) + ck .
Because n = bk, we have k = logb n.
Hence, f (n) = f (1) + c logb n . When n is
not a power of b, we have bk <n 1. First
assume that n = bk, where k is a
positive integer.

Erdos (who died in 1996). That is, the
Erdos number of a mathematician is
the length of the shortest chain of
mathematicians that begins with Paul
Erdos and ends with this
mathematician, where each adjacent
pair of mathematicians have written a
joint pape