PIGEONHOLE PRINCIPLE
M AT H C L U B L E S S O N
WHAT IS IT?
•  The pigeonhole principle states that if n items are
put into m containers, with n > m, then at least one
container must contain more than one item
•  Although the concept
is very simple, it can be
applied to solve many
problems
•  The key to solving a
problem using
pigeonhole principle is
determining what the
pigeons are and what
the holes are
EASY PROBLEMS
1.  If a Martian has an infinite number of red, blue,
yellow, and black socks in a drawer, how many
socks must the Martian pull out of the drawer to
guarantee he has a pair?
2.  Six distinct positive integers are randomly chosen
between 1 and 2006, inclusive. What is the
probability that some pair of these integers has a
difference that is a multiple of 5?
3.  Suppose S is a set of n + 1 integers. Prove that
there exist distinct a, b in S such that a - b is a
multiple of n.
A LITTLE HARDER
4.  How many bishops can one put on an 8×8
chessboard such that no two bishops can hit each
other.
5.  Show that in any group of n people, there are two
who have an identical number of friends within
the group.
6.  Show that if we randomly take n + 1 numbers from
the set {1, 2, . . . , 2n}, then some pair of numbers
will have no prime factors in common.
HARDER
7.  Show that if we randomly take n+ 1 numbers from
the set {1, 2, . . . , 2n}, then there will be some pair
in which one number is a multiple of the other
one.
8.  Prove that from any set of one hundred whole
numbers, one can choose either one number
which is divisible by 100, or several numbers whose
sum is divisible by 100.
9.  Every point in a plane is either red, green, or blue.
Prove that there exists a rectangle in the plane
such that all of its vertices are the same color