MA 490-505 Spring 2014
The Pigeonhole Principle
1
Basic Pigeonhole
The Pigeonhole Principle. If we must put N + 1 or more pigeons into N holes, then at least
one of the holes must contain two or more pigeons.
Problems
Problem 1 Two million pine trees grow in a forest. It is known that no pine tree has more
than 1000000 pine needles on it. Show that two pine trees in the forest must have the same
number of pine needles.
Problem 2 The city of Chicago has six million inhabitants. Show that two of these must
have the same number of hairs on their heads, if it is known that no person has more than one
million hairs on his or her head.
Problem 3 A box contains balls of nine colors. What is the smallest number of balls which
must be drawn from the box, without looking, so that among these balls there are two of the
same color?
Problem 4 Given 11 integers, show that two of them can be chosen whose difference is divisible
by 10.
Problem 5 Given 9 integers, show that two of them can be chosen whose difference is divisible
by 5.
Problem 6 Prove that there exist two power of three whose difference is a multiple of 2006.
Problem 7 Given a unit square, show that√if five points are placed anywhere inside or on this
2
units apart.
square, then two of them must be at most
2
Problem 8 Given a unit square, show that√if ten points are placed anywhere inside or on this
2
square, then two of them must be at most
units apart.
3
Problem 9 Show that an equilateral triangle cannot be covered completely by two smaller
equilateral triangles.
Problem 10 Of 50 people seated at a round table, more than half are men. Prove that there
are two men who are seated diametrically opposite each other.
Problem 11 Each box in a 3 × 3 arrangement of boxes is filled with one of the numbers
−1, 0, 1. Prove that of the eight possible sums along the rows, the column and diagonals, two
sums must be equal.
1
Problem 12 Each box in a 4 × 4 arrangement of boxes is filled with one of the numbers 0, 2.
Prove that of the ten possible sums along the rows, the column and diagonals, two sums must
be equal.
Problem 13 Every point on the plane is colored in 2 colors. Prove that now matter how the
coloring is done, there must exist two points, exactly a mile apart, which are the same color.
Problem 14 Show that in any group of six people, there are two who have an identical number
of friends within the group.
Problem 15 Prove that of any 8 integers, two can always be found such that the difference of
their squares is divisible by 13.
Problem 16 Prove that there exists a power of 7 which ends with the digits 01 (in decimal
notation).
Topics are taken from Problem-Solving Strategies by Arthur Engel, 2007; and Mathematical
Circles: Russian Experience by Dmitri Fomin, Sergey Genkin, Ilia V. Itenberg, 1996
2