x 12 b Pigeonhole

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The Pigeonhole Principle

Rosen 4.2


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Pigeonhole Principle

If k+1 or more objects are placed into k boxes, then there is

at least one box containing two or more objects.


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Generalized Pigeonhole Principle

If N objects are placed into k boxes, then there is

at least one box containing at least N/k objects

Examples Among any 100 people there must be at least 100/12

= 9 who were born in the same month.

What is the minimum number of students needed in a

class to be sure thatat least 6 to get the same grade? (5choices for grades:A,B,C,D,F)

Smallest integer N such that N/5 = 6, 5*5+1 = 26


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Example

Whats the minimum number of students,

each of whom comes from one of the 50

states must be enrolled in a university toguarantee that there are at least 100 who

come from the same state?

50*99 + 1 = 49514951/50 = 100


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There are 38 different time periods during which classes at a

university can be scheduled. If there are 677 different classes,

how many different rooms will be needed?

677/38 = 18

A computer network consists of six computers. Each computer is

directly connected to zero or more of the other computers. Show

that there are at least two computers that are directly connected

to the same number of other computers.

Solution:

Each computer can be directly connected to 0,1,2,3,4,5.

But there are really only five choices, not six, since if one computer

is connected to zero other computers, then no computer can be

connected to five others. Six computers, 5 choices. Pigeonhole

principle says that at least two must have the same number of direct

connections.


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Let (xi, yi, zi), i = 1,2,3,..,9 be a set of nine distinct points

with integer coordinates in xyz space. Show that the

midpoint of at least one pair of these points has integer

coordinates.For points (xj, yj, zj) and (xk, yk, zk) we compute the midpoint by ((xi+xj)/2,

(yi+yj)/2, (zi+zj)/2 ).

(1,1,2), (1,2,2), (3,2,7), (10,5,8), (3,1,4), (3,7,2), (2,1,1), (1,2,1), (0,0,0)

The midpoint between (1,1,2) and (3,1,4) = (2,1,3)

Remember from number theory that when we add an odd number to an

odd number, or an even number to an even number, we get an even

number. So the question becomes, does there exist two sets of

coordinates that have the same parity (i.e., their odd/even order is

the same)?

From the product rule there are 2*2*2 = 8 possible parities. There are

nine points, so by the pigeonhole principle two of them must be the

same. Therefore at least one midpoint must have integer

coordinates.


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During a month with 30 days a baseball team plays at least 1

game a day, but no more than 45 games. Show that there must

be a period of some number of consecutive days during which

the team must play exactly 14 games.

Proof: Let aj be the number of games played on or

before the jth day of the month. Then a1, a2, , a30

must be an increasing sequence of distinct positiveintegers, with 1aj45.

Day of Month Games Played

1 a1

2 a23 a3

30 a30


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Moreover, a1+14, a2+14, . . ., a30+14 is also an increasing

sequence of distinct positive integers with 15 aj + 14 59 .

Together the two sequences, each containing 30 integers,contain 60 positive integers, all of which are less than or equal

to 59. By the pigeonhole principle, at least two of these

integers are equal. Since the integers aj, j = 1 to 30, are all

distinct and the integers aj+14, j = 1 to 30 are all distinct, theremust be indices i and j with ai = aj+14.

This means that exactly 14 games were played from day j+1 to

day i.


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Some DefinitionsSuppose that a1,a2, an is a sequence of real numbers.

A subsequence of this sequence is a sequence of the form

ai1, ai2, , aim, where 1 i1 < i2 < . . . < im n

A sequence is called strictly increasing if each term islarger than the term that precedes it.

A sequence is called strictly decreasing if each term is

smaller than the one that precedes it.

Example: {1, 5, 6, 2, 3, 9} is a sequence. {5,6,9} is a subsequence that is strictly increasing


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Theorem: Every sequence of n2+1 distinct real

numbers contains a subsequence of (at least) length

n+1 that is strictly increasing or strictly decreasing.

Example: 8, 11, 9, 1, 4, 6, 12, 10, 5, 7

10 = 32+1 terms so must be a subsequence of length 4 that is

either strictly increasing or strictly decreasing.

1,4,6,12

1,4,6,7

11,9,6,5


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Theorem: Every sequence of n2+1 distinct real

numbers contains a subsequence of at least length

n+1 that is strictly increasing or strictly decreasing.

Let a1, a2, , a n2+1 be a sequence of n2+1 distinct numbers.

Associate an ordered pair (ik,dk) with each term of thesequence where ikis the length of the longest increasing

subsequence starting at akand dkis length of the longestdecreasing subsequence starting at ak.

Example: 8, 11, 9, 1, 4, 6, 12, 10, 5, 7

a2 = 11 , (2,4)

a4 = 1 , (4,1)Proof by contradiction:Now suppose that there are no

increasing or decreasing subsequences of length n+1 orgreater. Then ikand dkare both positive integers n, fork=1 to n2+1.


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By the product rule, there are n2 possible ordered pairs for (ik,dk).

Why? Because each has the range from 1 to n.

By the pigeonhole principle, since we have n2+1 ordered pairs (one

for each element in the sequence) two of them must be identical.

Formally terms as and at in the sequence, with s at.

If as < at, an increasing subsequence of length it+1(or greater) can be

constructed starting at as, by taking as followed by an increasing

subsequence of length it, beginning at at. But we have said that is = it.Thus this is a contradiction.

Similarly, if as > at, it can be shown that ds must be greater than dt,

which is also a contradiction.

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