2. Pigeon-hole Principle
 If n>m
pigeons are put
into m
pigeonholes, th
ere's a hole
with more than
one pigeon.

3. Alternative Forms
• if n objects are to be allocated to m
containers, then at least one container
must hold at least ceil(n/m) objects.
• For any finite set A , there does not exist
a bijection between A and a proper
subset of A .
• Let |A| denote the number of elements in
a finite set A. For two finite sets A and
B, there exists a 1-1 correspondence
f: A->B iff |A| = |B|.

4. History
 The first statement of the
principle is believed to
have been made by
Dirichlet in 1834 under the
name Schubfachprinzip
("drawer principle" or
"shelf principle")
 Also known as Dirichlet's
box (or drawer) principle

5. General Problems
 There 750 students in the a batch at
UOM. Prove that at least 3 of them have
their birthdays on the same date ?
○ 366 * 2= 732 < 750
○ Thus at least 3 students have the birthday
on the same date.

6. Problems on relations
 There are 50 people in a room. some of them are
friends. If A is a friend of B then B is also a friend of
A. prove that there are two persons in the room
who have a same number of friends.
 In league T20 tournament of 16 cricket teams,
every two teams have to meet in a game. prove
that at any time there are two teams which played
equal number of matches.

7. Solution
 Case 1  Case 2
 There exists a person  There does not exists
with 49 friends (he is a a person with 49
friend of all other people) friends
 Then there cannot be a  The no of friends vary
person with 0 friends between 0 – 48 .
 The no of friends vary  There are 50 people
between 1 – 49 . and only 49 values.
 There are 50 people and
only 49 values.

8. Problems On Divisibility
 Prove that there
exists a multiple of
2009 whose decimal
expansion contains
only digits 1 and 0.

9. Answer
 Consider 2010 numbers
- 1,11,111,1111, … ,1111…111.
 Each of these numbers produce one of 2009
remainders
- 0,1,2,3 ,…,2008
 We have 2010 numbers and 2009 remainders
 By pigeon-hole principle some two numbers have the
same remainder .Let those 2 numbers be A and B
(A>B)
 Consider A-B. which is a multiple of 2009.
- In the form of 11…1100…000

10. Problems On Divisibility
 Prove that of any 52 natural numbers one can find
two numbers n and m such that either their sum m+n
or difference m-n is divisible by 100.
 Consider sets {0},{1,99},{2,98}….{49,51},{50}
 There are 51 sets
 By pigeon hole principle at least 1 set should have 2
members
 If we consider any set above if they have 2 members in the
set, m+n or m-n is divisible by 100

11. Problems on geometry
 51 points are placed, in a random way, into a
square of side 1 unit. Can we prove that 3 of these
points can be covered by a circle of radius 1/7 units
?

12. Answer
 To prove the result, we may divide
the square into 25 equal smaller
squares of side 1/5 units each.
Then by the Pigeonhole Principle, at
least one of these small squares
should contain at least 3 points.
Otherwise, each of the small
squares will contain 2 or less points
which will then mean that the total
number of points will be less than
50 , which is a contradiction to the
fact that we have 51 points in the
first case !

13. Answer - continue
 Now the circle
circumvented around the
particular square with the
three points inside should
have
 Radius=Sqrt(1/100+1/100
)
=Sqrt(1/50)
<Sqrt(1/49)=1/7
1/10

14. Applications
 Lossless data compression cannot guarantee
compression for all data input files.
 The pigeonhole principle often arises in
computer science. For example, collisions are
inevitable in a hash table because the number
of possible keys exceeds the number of
indices in the array
 In probability theory, the birthday problem, or
birthday paradox pertains to the probability
that in a set of randomly chosen people some
pair of them will have the same birthday

15. Applications
 The proof of Chinese Remainder Theorem
is based on pigeon-hole principle
Let m and n be relatively prime positive
Integers. Then the system:
x = a (mod m)
x = b (mod n)
has a solution.

16. References
 http://en.wikipedia.org/wiki/Pigeonhole_principle
 http://en.wikipedia.org/wiki/Johann_Peter_Gustav_
Lejeune_Dirichlet
 http://en.wikipedia.org/wiki/Lossless_data_compre
ssion#Limitations
 Article on "What is Pigeonhole Principle?" by
Alexandre V. Borovik, Elena V. Bessonova.
 Article on "Applications of the Pigeonhole
Principle" by Edwin Kwek Swee Hee ,Huang
Meiizhuo ,Koh Chan Swee ,Heng Wee Kuan
, River Valley High School

17. Thank You