Pigeonhole Examples. Doug Rall Mathematics 110 Spring /6 Doug Rall Pigeonhole Examples

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1 Pigeonhole Examples Doug Rall Mathematics 110 Spring /6 Doug Rall Pigeonhole Examples

2 Statement of PHP Pigeonhole Principle Suppose that n and m are positive integers with m > n. Regardless of how we distribute m objects into n boxes, there will always be a box that contains at least 2 of the objects. 1/6 Doug Rall Pigeonhole Examples

3 Example A certain elementary school has 1400 students. Each of them writes the initials of their first and last names on a card. 2/6 Doug Rall Pigeonhole Examples

4 Example A certain elementary school has 1400 students. Each of them writes the initials of their first and last names on a card. For example, Billy Bogart would write BB, and Tamera Carlyle would write TC. 2/6 Doug Rall Pigeonhole Examples

5 Example A certain elementary school has 1400 students. Each of them writes the initials of their first and last names on a card. For example, Billy Bogart would write BB, and Tamera Carlyle would write TC. At least two of these cards must be the same. 2/6 Doug Rall Pigeonhole Examples

6 Example A certain elementary school has 1400 students. Each of them writes the initials of their first and last names on a card. For example, Billy Bogart would write BB, and Tamera Carlyle would write TC. At least two of these cards must be the same. Use Pigeonhole Principle 2/6 Doug Rall Pigeonhole Examples

7 Example A certain elementary school has 1400 students. Each of them writes the initials of their first and last names on a card. For example, Billy Bogart would write BB, and Tamera Carlyle would write TC. At least two of these cards must be the same. Use Pigeonhole Principle In fact, at least 3 of these cards must be the same! 2/6 Doug Rall Pigeonhole Examples

8 Example A certain elementary school has 1400 students. Each of them writes the initials of their first and last names on a card. For example, Billy Bogart would write BB, and Tamera Carlyle would write TC. At least two of these cards must be the same. Use Pigeonhole Principle In fact, at least 3 of these cards must be the same! Generalized Pigeonhole Principle Suppose that n and m are positive integers with m > n. Regardless of how we distribute m objects into n boxes, there will always be a box that contains at least m/n of the objects. 2/6 Doug Rall Pigeonhole Examples

9 Examples Prove that if any set S of 21 numbers is chosen from {1, 2, 3,..., 40} there will always be two numbers in S whose sum is 41. 3/6 Doug Rall Pigeonhole Examples

10 Examples Prove that if any set S of 21 numbers is chosen from {1, 2, 3,..., 40} there will always be two numbers in S whose sum is , 450 fans attended a Clemson football game one Saturday. The ages of the fans ranged from 6 to 88 inclusive, and their weights (to the nearest pound) ranged from 48 to 315 pounds. Prove there were at least 4 fans in attendance who were the exact same age and had the exact same weight. 3/6 Doug Rall Pigeonhole Examples

11 B 4/6 Doug Rall Pigeonhole Examples

12 B Imagine all the ways that the puzzle piece B could be placed on this chess board having the same orientation as shown. 4/6 Doug Rall Pigeonhole Examples

13 B Imagine all the ways that the puzzle piece B could be placed on this chess board having the same orientation as shown. Imagine all the different patterns that are possible if we color each of the five squares in B either red or gray. 4/6 Doug Rall Pigeonhole Examples

14 B 5/6 Doug Rall Pigeonhole Examples

15 B Prove that regardless of how the 64 squares of the 8 8 chess board are colored with red and gray there will always be (at least) two copies of B, with the given orientation, somewhere on this colored board that have the same color pattern. 5/6 Doug Rall Pigeonhole Examples

16 A Particular Pattern That Occurs More Than Once B Three occurrences of the indicated pattern. One marked by x, another by y and a third by z.

17 A Particular Pattern That Occurs More Than Once B x x x x x Three occurrences of the indicated pattern. One marked by x, another by y and a third by z.

18 A Particular Pattern That Occurs More Than Once B y y y y y x x x x x Three occurrences of the indicated pattern. One marked by x, another by y and a third by z.

19 A Particular Pattern That Occurs More Than Once z z z z z B y y y y y x x x x x Three occurrences of the indicated pattern. One marked by x, another by y and a third by z.

20 A Particular Pattern That Occurs More Than Once z z z z z B y y y y y x x x x x Three occurrences of the indicated pattern. One marked by x, another by y and a third by z. 6/6 Doug Rall Pigeonhole Examples

Counting: Basics. Four main concepts this week 10/12/2016. Product rule Sum rule Inclusion-exclusion principle Pigeonhole principle

Counting: Basics. Four main concepts this week 10/12/2016. Product rule Sum rule Inclusion-exclusion principle Pigeonhole principle Counting: Basics Rosen, Chapter 5.1-2 Motivation: Counting is useful in CS Application domains such as, security, telecom How many password combinations does a hacker need to crack? How many telephone