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The pigeonhole principle Marymount Manhattan College April 14, 2010 Outline Introduction 1 (Not So) Magic Squares Pigeonholes Examples 2 Someones been using my initials. Hairs in NYC Triangular dartboard A party problem Birthdays T.

  1. The pigeonhole principle Marymount Manhattan College April 14, 2010

  2. Outline Introduction 1 (Not So) Magic Squares Pigeonholes Examples 2 Someone’s been using my initials. Hairs in NYC Triangular dartboard A party problem Birthdays T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 2 / 21

  3. Introduction 1. Introduction T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 3 / 21

  4. Introduction (Not So) Magic Squares (Not So) Magic Squares The challenge Fill in boxes with 1’s and − 1’s so that columns , rows , and diagonals all have DIFFERENT sums. SURPRISE! It can’t be done! T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 4 / 21

  5. Introduction (Not So) Magic Squares (Not So) Magic Squares 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 1 1 1 1 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 -1 -1 1 T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 5 / 21

  6. Introduction (Not So) Magic Squares (Not So) Magic Squares Why can’t it be done? different sums needed = 2 columns + 2 rows + 2 diagonals = 6 biggest possible sum: 1 + 1 = 2 smallest possible sum: ( − 1) + ( − 1) = − 2. Every possible sum is between (or equal to) − 2 and 2. BUT, only five numbers from − 2 to 2. #(sums needed) > #(sums possible) Therefore at least two of the sums must be the same! This is the Pigeonhole Principle . T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 6 / 21

  7. Introduction Pigeonholes The pigeonhole principle The principle If 6 pigeons have to fit into 5 pigeonholes, then some pigeonhole gets more than one pigeon. More generally, if #(pigeons) > #(pigeonholes), then some pigeonhole gets more than one pigeon. Counting Argument � Combinatorics T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 7 / 21

  8. Introduction Pigeonholes The pigeonhole principle Strategy for using pigeonhole principle Identify the pigeons and pigeonholes . (Want to assign a pigeonhole for each pigeon.) Is #(pigeons) > #(pigeonholes)? If YES, then some pigeonhole has to get more than one pigeon! EXAMPLE: (Not So) Magic Squares = different sums needed (6) pigeons pigeonholes = possible sums ( < 5) Therefore 2 (or more) sums must be the same. T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 8 / 21

  9. Introduction Pigeonholes What about 6 × 6? − 1 1 − 1 1 1 − 1 − 1 1 different sums needed = 6 columns + 6 rows + 2 diagonals = 14 biggest possible sum: 1 + 1 + 1 + 1 + 1 + 1 = 6 smallest possible sum: ( − 1) + ( − 1) + ( − 1) + ( − 1) + ( − 1) + ( − 1) = − 6. pigeons = different sums needed (14) pigeonholes = possible sums ( < 13) Nope! (Actually doesn’t work for any n × n .) T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 9 / 21

  10. Examples 2. Examples T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 10 / 21

  11. Examples Someone’s been using my initials. Someone’s been using my initials. How many first/last name initials are there? 26 possible letters. 26 × 26 = 676 possible pairs of initials. CLAIM: At least 2 students at Marymount Manhattan College have the same first/last initials. pigeons = MMC students pigeonholes = possible first/last initials #(pigeons) 2,100 ≈ #(pigeonholes) = 676 Warning: Doesn’t mean every student has an “initial twin”! T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 11 / 21

  12. Examples Someone’s been using my initials. Someone’s been using my initials. How many first/middle/last name initials are there? 26 possible letters. Some people have no middle names, so include “blank” for middle initial. 26 × 27 × 26 = 18 , 252 possible triples of initials. CLAIM: At least 2 students at Cornell University have the same first/middle/last initials. pigeons = CU students pigeonholes = possible first/middle/last initials #(pigeons) 20,600 ≈ #(pigeonholes) = 18,252 T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 12 / 21

  13. Examples Hairs in NYC Hairs in New York City CLAIM: At any time in New York City, there are 2 people with the same number of hairs. pigeons = people in New York City = possible # of hairs pigeonholes #(pigeons) 8,363,000 ≈ #(pigeonholes) 7,000,000 < T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 13 / 21

  14. Examples Triangular dartboard A triangular dartboard Dartboard = equilateral triangle with side length of 2 feet CLAIM: If you throw 5 darts (no misses), at least 2 will be within a foot of each other. T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 14 / 21

  15. Examples Triangular dartboard A triangular dartboard Divide triangle into 4 sub-triangles. Darts in same sub-triangle are within 1 foot of each other. = darts (5) pigeons pigeonholes = sub-triangles (4) T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 15 / 21

  16. Examples A party problem A party problem Set-Up: Party with 10 people. Each guest counts how many guests she/he has met before. Cool Fact: At least 2 people will have met the same number of guests before! T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 16 / 21

  17. Examples A party problem A party problem Cool Fact: At least 2 people will have met the same number of guests before! Why? = party guests pigeons pigeonholes = possible number of guests met before How many guests has each person met before? (0 – 9) 0 = met no one before. 9 = met everyone before. 0 and 9 can’t happen at the same party! number of guests met before: only nine possiblities! (0 – 8 or 1 – 9) T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 17 / 21

  18. Examples A party problem A party problem Cool Fact: At least 2 people will have met the same number of guests before! pigeons = party guests (10) = possible number of guests met before (9) pigeonholes T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 18 / 21

  19. Examples Birthdays Birthday twins! Question: How many people do you need to guarantee 2 of them share a birthday? T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 19 / 21

  20. Examples Birthdays What are the odds? So: 366 + 1 = 367 people � 100% chance of shared birthday It’s amazing! 23 people 50% 57 people 99% 100 people 99.9999% 200 people 99.999999999999999999999999999% This is called The Birthday Problem . Not really Pigeonhole Principle, but still about counting things. T. Goldberg ( Cornell ) The pigeonhole principle April 14, 2010 20 / 21

  21. THE END Thank you for listening. For many more Pigeonhole puzzles and examples, please see the Internet.

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