Pigeonhole Principle
The pigeonhole principle states that if npigeons are put into mpigeonholes, and if n>m,
then at least one pigeonhole must contain more than one pigeon. Another way of stating this would
be that mholes can hold at most mobjects with one ob ject to a hole; adding another object will
force you to reuse one of the holes.
Although the pigeonhole principle may seem to be a trivial observation, it can be used to
demonstrate unexpected results. For example, there must be at least two people in London with
the same number of hairs on their heads. Demonstration: a typical head of hair has around 150,000
hairs. It is reasonable to assume that no-one has more than 1,000,000 hairs on their head. There
are more than 1,000,000 people in London. If we assign a pigeonhole for each number of hairs on
a head, and assign people to the pigeonhole with their number of hairs on it, there must be two
people with the same number of hairs on their heads.
A generalized version of this principle states that, if ndiscrete objects are to be allocated to m
containers, then at least one container must hold no fewer than dn/meobjects, where d· ··e denotes
the ceiling function.
Dirichlet’s Theorem: Let αbe an irrational number. Then there are infinitely many integer
pairs (h, k) where k > 0 such that
α−h
k
<1
k2.
Examples:
1. Five points are situated inside an equilateral triangle whose side has length one unit. Show
that two of them may be chosen which are less than one half unit apart. What if the equilateral
triangle is replaced by a square whose side has length of one unit?
2. Given any n+2 integers, show that there exist two of them whose sum, or else whose difference,
is divisible by 2n?
3. (Putnam 1978) Let A be any set of 20 distinct integers chosen from the arithmetic progres-
sion 1,4,7,··· ,100. Prove that there must be two distinct integers in A whose sum is 104.
[Actually, 20 can be replaced by 19.]
4. Given any n+ 1 distinct integers between 1 and 2n, show that two of them are relatively
prime. Is this result best possible, i.e., is the conclusion still true for nintegers between 1
and 2n?
5. Given any n+ 1 integers between 1 and 2n, show that one of them is divisible by another. Is
this best possible, i.e., is the conclusion still true for nintegers between 1 and 2n?
6. (Putnam 1980, A4(a)) Prove that there exist integers a, b, c not all zero and each of them
absolute value less than one million, such that
|a+b√2 + c√3|<10−11.
7. Prove that there is some integer power of 2 that begins 2006 · · ·.
8. (Putnam 1980, A4(b)) (not related to pigeon hole principle) Let a, b, c be integers not all zero
and each of them absolute value less than one million. Prove that
|a+b√2 + c√3|>10−21.
9. Given any 2nintegers, show that there are nof them whose sum is divisible by n. (Though
superficially similar to some other pigeonhole problems, this problem is much more difficult
and does not really involve the pigeonhole principle.)
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