The Pigeonhole Principle

Colorado Math CircleGalois Group

November 22, 2008

The Pigeonhole PrincipleIf there are k pigeonholes for n pigeons, and n > k, then at least one pigeonhole must contain more than one pigeon.In general, at least one pigeonhole must contain at least dn/ke pigeons. The pigeonhole principle is also known asDirichlet’s box principle.

EXAMPLE. If there are 8 pigeonholes for 9 pigeons, then at least one hole contains more than one pigeon.EXAMPLE. If there are 4 pigeonholes for 9 pigeons, then at least one hole contains at least 3 pigeons.

Basic Problems1. A sock drawer contains blue, brown, black, and white socks. Without looking, how many socks do you need to

pull out to make sure you have at least one matching pair?

2. There are 75, 000 porcupines in Prairie County. If each porcupine has at most 40, 000 quills, show that at leasttwo porcupines have the same number of quills.

3. Thirty-eight cartons of iPods are delivered to the Apple Store. The iPods come in six different colors. Eachcarton contains iPods of a single color. Show that there must be at least seven cartons containing the same color.

4. Twenty students are evenly spaced around a circle. If more than half are girls, show that there must be two girlswho are directly across from each other.

5. If nine children are seated in a row of twelve chairs, show that at least three children are seated in a row.

Number Theory1. Six integers are chosen from the set {1, 2, . . . , 10}. Show that two of them must have an odd sum.

2. Given a set of eight integers, show that two integers can be chosen whose difference is divisible by 7.

3. One hundred jelly beans are randomly distributed among fifteen students. Show that there are two students whohave the same number of jelly beans.

4. The digits from 1 to 7 are divided into three groups. Show that one of the groups must have a product 18 orgreater.

5. The integers 1, 2, . . . , 10 are written in a circle, in any order. Show that there are three adjacent numbers whosesum is 17 or greater.

6. Each box in a 3 × 3 arrangement of boxes is filled with one of the numbers −1, 0, 1. Show that of the eightpossible sums along the rows, columns, and diagonals, two sums must be equal.

7. The sum of the ages of five students is 87 years. Show that three students can be chosen so that the sum of theirages is at least 53.

8. Show that there are two powers of 3 whose difference is divisible by 17.

9. Given a set of ten integers, show that a subset can be chosen with sum divisible by 10.

10. Show that there exists a positive multiple of 13 that contains only the digits 5 and 0.

Pairing Up1. Show that in a group of five people, there are two who have the same number of friends within the group.

2. Fifty people attended a party, where handshakes were exchanged among some of the guests. Show that therewere at least two guests who shook the same number of hands.

3. In a round robin tennis tournament with eighty participants, each player plays exactly one match against everyother player. Show that at any moment during the tournament, there must be two players who have played anidentical number of games.

Geometry1. Every point on the plane is colored red or blue. Show that there must exist two points, exactly one mile apart,

that are the same color.

2. Five points are placed inside a unit equilateral triangle. Show that there must exist two points no more than 1/2unit apart.

3. Every point in three-dimensional space is colored red, blue, or green. Show that there must exist two points,exactly one mile apart, that are the same color.

4. Five points are placed inside a unit square. Show that there must exist two points no more than√

2/2 units apart.

5. Five lattice points are chosen on an infinite square lattice. Segments are drawn connecting each pair of points.Show that the midpoint of one of the segments is also a lattice point. (A lattice point has integer coordinates.)

6. Every point on the circumference of a circle is colored red or blue. Show that there must exist three equallyspaced points of the same color.

ReferencesA. Bogomolny. “Pigeonhole Principle”. Interactive Mathematics Miscellany and Puzzles <http://www.cut-the-knot.org/>.

D. Fomin, S. Genkin, I. Itenberg. Mathematical Circles (Russian Experience). American Mathematical Society, 1996.

A. Soifer. Mathematics as Problem Solving. Center for Excellence in Mathematical Education, 1987.