Download The Pigeonhole Principle

The Pigeonhole Principle
Colorado Math Circle
Galois Group
November 2015
Examples
1. The Hogwarts School of Witchcraft and Wizardry contains four
houses: Gryffindor, Slytherin, Ravenclaw, and Hufflepuff. If student
names are drawn randomly from a hat, how many names would you
need to draw to ensure that there are three students from the same
house?
The Pigeonhole Principle
If there are more pigeons than pigeonholes, then there is a pigeonhole that
contains more than one pigeon.
In general, if there are n pigeons and k pigeonholes, then there is a
pigeonhole that contains at least r pigeons where r = n/k, rounded up.
• If there are 9 pigeonholes for 10 pigeons, then at least one hole
contains 2 or more pigeons.
• If there are 3 pigeonholes for 10 pigeons, then at least one hole
contains 4 or more pigeons.
2. Suppose a park has 50,000 porcupines and each porcupine has 40,000
quills or fewer. Show that at least two porcupines have the same
number of quills.
3. Suppose a bag of Skittles contains seven different colors (purple,
yellow, green, orange, red, blue, and pink). If there are 100 Skittles
in the bag, show that there must be at least 15 Skittles of the same
color.
4. Every point on the plane is colored red or blue. Show that there must
be two points, exactly one meter apart, that are the same color.
5. Given a set of five integers, show that two integers can be chosen
whose difference is divisible by 4.
6. Show that in a group of five Gryffindor students, there are two
students who have the same number of friends within the group.
Problems
1. If the Hogwarts School has at least 1000 students, show that at least
three students have the same birthday.
2. Ron purchased 100 Bertie Bott’s Every Flavour Beans in six flavors:
chocolate, peppermint, cranberry, spinach, liver, and dirt. Show that
there must be 17 beans of the same flavor.
3. Every point in space is labeled Harry, Ron or Hermione. Show that
there must be two points, exactly one meter apart, that have the
same name.
4. Thirty-six Hogwarts students are seated evenly spaced around a circle.
If more than half are from the Slytherin house, show that there must
be two Slytherin students who are directly across from each other.
5. The powers of 3 are 3, 9, 27, . . .. Show that there are two powers of
3 whose difference is divisible by 17.
6. In the Care of Magical Creatures class, 100 flobberworms are
randomly distributed among 15 students. Show that there are two
students who have the same number of flobberworms.
11. Each square in a 3 × 3 grid of squares is filled with one of the numbers −1, 0, 1. Show that of the eight possible sums along the rows,
columns, and diagonals, two sums must be equal.
12. Five points are placed inside a unit equilateral triangle. Show that
there must be two points no more than 1/2 unit apart.
13. The numbers 1, 2, . . . , 6 are divided into three groups. Show that one
of the groups must have a product of 9 or greater.
14. Five points are placed inside
√ a unit square. Show that there must be
two points no more than 2/2 units apart.
15. The numbers 1, 2, . . . , 10 are written in a circle, in any order. Show
that there are three adjacent numbers whose sum is 17 or greater.
16. Let A be any set of 19 distinct numbers chosen from the arithmetic
progression 1, 4, 7, . . . , 100. Show that there must be two distinct
numbers in A whose sum is 104.
17. Suppose n is an odd number. Rearrange the numbers {1, 2, . . . , n}
to obtain {a1 , a2 , . . . , an }. Now form the product
P = (1 − a1 )(2 − a2 ) · · · (n − an ).
Show that P is even.
7. Twenty-five numbers are chosen from the set {1, 2, . . . , 48}. Show
that two of them must have an odd sum.
8. If 16 Ravenclaw students are seated in a row of 20 chairs, show that
at least 4 students are seated together.
9. The sum of the ages of five Hufflepuff students is 87 years. Show
that three students can be chosen so that the sum of their ages is at
least 53. (Assume that the ages are integers.)
10. Fifty Hogwarts students signed up for a dueling tournament. The
match-ups were chosen at random. Some students were selected for
a few duels while others were selected for many. No student dueled
the same opponent twice. Show that there were at least two students
who fought the same number of duels.
Challenge Problems
1. Every point on the circumference of a circle is labeled Wizard or
Muggle. Show that there must be three equally spaced points with
the same name. (The middle point will be equidistant from the other
two.)
2. Show that there is a number whose digits are all 1’s which is divisible
by 17.
3. 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.)
4. Given a set of ten integers, show that a subset can be chosen with
sum divisible by 10.
5. Choose any 13 distinct numbers from {1, 2, . . . 20}. Show that there
is at least one pair of numbers that differ by 4, one pair that differ by
5, and one pair that differ by 6.
6. Every point on the plane is labeled Wizard or Muggle. Show that
there is a rectangle that has four vertices of the same name.
References
A. 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.