Introductory Combinatorics

Richard A. Brualdi

Chapter 3 The Pigeonhole Principle - all with Video Answers

Educators

Chapter Questions

Problem 1

Concerning Application 4, show that there is a succession of days during which the chess master will have played exactly $k$ games, for each $k=1,2, \ldots, 21$. (The case $k=21$ is the case treated in Application $4 .$ Is it possible to conclude that there is a succession of days during which the chess master will have played exactly 22 games?

Lauren Long

Numerade Educator

01:23

Problem 2

* Concerning Application 5, show that if 100 integers are chosen from $1,2, \ldots, 200$, and one of the integers chosen is less than 16 , then there are two chosen numbers such that one of them is divisible by the other.

Vysakh M

Numerade Educator

00:54

Problem 3

Generalize Application 5 by choosing (how many?) integers from the set
$$
\{1,2, \ldots, 2 n\}
$$.

Jeyasree R T

Numerade Educator

02:21

Problem 4

Show that if $n+1$ integers are chosen from the set $\{1,2, \ldots, 2 n\}$, then there are always two which differ by 1 .

Nick Johnson

Numerade Educator

11:23

Problem 5

Show that if $n+1$ distinct integers are chosen from the set $\{1,2, \ldots, 3 n\}$, then there are always two which differ by at most $2 .$

Prathan Jarupoonphol

Numerade Educator

00:12

Problem 6

Generalize Exercises 4 and 5 .

Ronald Prasad

Numerade Educator

04:33

Problem 7

. Show that for any given 52 integers there exist two of them whose sum, or else whose difference, is divisible by 100 .

Akash Goyal

Numerade Educator

01:16

Problem 8

Use the pigeonhole principle to prove that the decimal expansion of a rational number $m / n$ eventually is repeating. For example,
$$
\frac{34,478}{99,900}=0.34512512512512512 \cdots
$$

Nick Johnson

Numerade Educator

02:14

Problem 9

In a room there are 10 people, none of whom are older than 60 (ages are given in whole numbers only) but each of whom is at least 1 year old. Prove that we can atways find two groups of people (with no common person) the sum of whose ages is the same. Can 10 be replaced by a smaller number?

Nick Johnson

Numerade Educator

02:35

Problem 10

A child watches TV at least one hour each day for seven weeks but, because of parental rules, never more than 11 hours in any one week. Prove that there is some period of consecutive days in which the child watches exactly 20 hours of TV. (It is assumed that the child watches TV for a whole number of hours each day.)

Nick Johnson

Numerade Educator

02:29

Problem 11

A student has 37 days to prepare for an examination. From past experience she knows that she will require no more than 60 hours of study. She also wishes to study at least 1 hour per day. Show that no matter how she schedules her study time (a whole number of hours per day, however), there is a succession of days during which she will have studied exactly 13 hours.

Tomokazu Switzer

Numerade Educator

02:45

Problem 12

Show by example that the conclusion of the Chinese remainder theorem (Application 6 ) need not hold when $m$ and $n$ are not relatively prime.

Trang Hoang

Numerade Educator

01:19

Problem 13

* Let $S$ be a set of six points in the plane, with no three of the points collinear. Color either red or blue each of the 15 line segments determined by the points of
S. Show that there are at least two triangles determined by points of $S$ which are either red triangles or blue triangles. (Both may be red, or both may be blue, or one may be red and the other blue.)

Erika Bustos

Numerade Educator

03:40

Problem 14

A bag contains 100 apples, 100 bananas, 100 oranges, and 100 pears. If I pick one piece of fruit out of the bag every minute, how tong will it be before I am assured of having picked at least a dozen pieces of fruit of the same kind?

Aparna Shakti

Numerade Educator

01:52

Problem 15

Prove that, for any $n+1$ integers $a_{1}, a_{2}, \ldots, a_{n+1}$, there exist two of the integers $a_{i}$ and $a_{j}$ with $i \neq j$ such that $a_{i}-a_{j}$ is divisible by $n .$

Julian Wong

Numerade Educator

02:14

Problem 16

Prove that in a group of $n>1$ people there are two who have the same number of acquaintances in the group. (It is assumed that no one is acquainted with oneself.)

Nick Johnson

Numerade Educator

02:22

Problem 17

There are 100 people at a party. Each person has an even number (possibly zero) of acquaintances. Prove that there are three people at the party with the same number of acquaintances.

Clarissa Noh

Numerade Educator

02:26

Problem 18

Prove that of any five points chosen within a square of side length 2, there are two whose distance apart is at most $\sqrt{2}$.

Jonathon Brumley

Numerade Educator

06:22

Problem 19

(a) Prove that of any five points chosen within an equilateral triangle of side length 1, there are two whose distance apart is at most $\frac{1}{2}$.
(b) Prove that of any 10 points chosen within an equilateral triangle of side length 1 , there are two whose distance apart is at most $\frac{1}{3}$.
(c) Determine an integer $m_{n}$ such that if $m_{n}$ points are chosen within an equilateral triangle of side length 1, there are two whose distance apart is at most $1 / n$.

Jason Taylor-Pestell

Numerade Educator

00:30

Problem 20

Prove that $r(3,3,3) \leq 17$.

Jimmy Yao

Numerade Educator

05:55

Problem 21

Prove that $r(3,3,3) \geq 17$ by exhibiting a coloring, with colors red, blue, and green, of the line segments joining 16 points with the property that there do not exist three points such that the three line segments joining them are all colored the same.

Khushbu Rani

Numerade Educator

02:11

Problem 22

Prove that
$$
r(\underbrace{3,3, \ldots, 3}_{k+1}) \leq(k+1)(r(\underbrace{3,3, \ldots, 3}_{k})-1)+2 .
$$
Use this result to obtain an upper bound for
$$
r(\underbrace{3,3, \ldots, 3}_{n}) .
$$

Linh Vu

Numerade Educator

02:02

Problem 23

The line segments joining 10 points are arbitrarily colored red or blue. Prove that there must exist three points such that the three line segments joining them are all red, or four points such that the six line segments joining them are all blue (that is, $r(3,4) \leq 10)$.

Donna Densmore

Numerade Educator

00:44

Problem 24

Let $g_{3}$ and $t$ be positive integers with $q_{3} \geq t .$ Determine the Ramsey number $r_{t}\left(t, t, q_{3}\right)$.

Julie Silva

Numerade Educator

02:48

Problem 25

Let $q_{1}, q_{2}, \ldots, q_{k}, t$ be positive integers, where $q_{1} \geq t, q_{2} \geq t, \ldots, q_{k} \geq t .$ Let $m$ be the largest of $q_{1}, q_{2}, \ldots, q_{k} .$ Show that
$$
r_{t}(m, m, \ldots, m) \geq r_{t}\left(q_{1}, q_{2}, \ldots, q_{k}\right)
$$
Conclude that, to prove Ramsey's theorem, it is enough to prove it in the case that $q_{1}=q_{2}=\cdots=q_{k}$.

James Chok

Numerade Educator

03:51

Problem 26

Suppose that the $m n$ people of a marching band are standing in a rectangular forination of $m$ rows and $n$ columns in such a way that in each row each person is taller than the one to his or her left. Suppose that the leader rearranges the people in each column in increasing order of height from front to back. Show that the rows are still arranged in increasing order of height from left to right.

Drake Ellis

Numerade Educator

02:21

Problem 27

A collection of subsets of $\{1,2, \ldots, n\}$ has the property that each pair of subsets has at least one element in common. Prove that there are at most $2^{n-1}$ subsets in the collection.

Nick Johnson

Numerade Educator

05:19

Problem 28

At a dance party there are 100 men and 20 women. For each $i$ from $1,2, \ldots, 100$, the ith man selects a group of $a_{1}$ women as potential dance partners (his "dance list," if you will), but in such a way that given any group of 20 men, it is always possible to pair the 20 men with the 20 women, with each man paired with a woman on his dance list. What is the smallest sum $a_{1}+a_{2}+\cdots+a_{100}$ for which there is a selection of dance lists that will guarantee this?

Ahmad Reda

Numerade Educator

01:56

Problem 29

A number of different objects have been distributed into $n$ boxes $B_{1}, B_{2}, \ldots, B_{n}$. All the objects from these boxes are removed and redistributed into $n+1$ new boxes $B_{1}^{*}, B_{2}^{*}, \ldots, B_{n+1}^{*}$, with no new box empty (so the total number of objects must be at least $n+1)$. Prove that there are two objects each of which has the property that it is in a new box that contains fewer objects than the old box that contained it.

Nick Johnson

Numerade Educator