The Pigeonhole Principle

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The Pigeonhole Principle

• Rosen 4.2

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Pigeonhole Principle
If k1 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 that at least 6 to get the
  same grade? (5 choices for gradesA,B,C,D,F)
• Smallest integer N such that ?N/5? 6, 551 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 to guarantee that there
  are at least 100 who come from the same state?
• 5099 1 4951
• ?4951/50? 100

5
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 E
ach 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 ((xixj)/2, (yiyj)/2,
  (zizj)/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 222 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.

7
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
  positive integers, with 1?aj ?45.
• Day of Month Games Played
• 1 a1
• 2 a2
• 3 a3
• 30 a30

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Moreover, a114, a214, . . ., a3014 is also an
increasing sequence of distinct positive integers
with 15 ? aj 15 ? 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 aj14, j 1 to 30 are all distinct,
there must be indices i and j with ai
aj14. This means that exactly 14 games were
played from day j1 to day i.
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Some Definitions

• Suppose 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 lt i2 lt .
  . . lt im ? n
• A sequence is called strictly increasing if each
  term is larger 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 n21 distinct real
numbers contains a subsequence of (at least)
length n1 that is strictly increasing or
strictly decreasing.

• Example 8, 11, 9, 1, 4, 6, 12, 10, 5, 7
• 10 321 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 n21 distinct real
numbers contains a subsequence of at least length
n1 that is strictly increasing or strictly
decreasing.

• Let a1, a2, , a n21 be a sequence of n21
  distinct numbers. Associate an ordered pair
  (ik,dk) with each term of the sequence where ik
  is the length of the longest increasing
  subsequence starting at ak and dk is length of
  the longest decreasing 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 n1 or greater. Then ik and dk are both
  positive integers ? n, for k1 to n21.

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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 n21 ordered pairs (one
for each element in the sequence) two of them
must be identical. Formally ? terms as and at
in the sequence, with sltt such that is it and
ds dt. We will show that this is impossible.
Because the terms in the sequence are distinct,
either as lt at or as gt at. If as lt at, an
increasing subsequence of length it1(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 gt at, it can be shown that ds must be greater
than dt, which is also a contradiction.