Chattanooga Math Circle. Heunggi Park, Covenant College April 23, 2017

Transcription

1 Chattanooga Math Circle Heunggi Park, Covenant College April 23, 2017

2 Warming-up: A Brain Exercise

3 The Intrigue of Numbers 1

4 The Intrigue of Numbers Prove that every natural number is interesting. 1

5 The Intrigue of Numbers Prove that every natural number is interesting. Proof: 1

6 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? 1

7 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. 1

8 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? 1

9 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. 1

10 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? 1

11 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? Well, there are only two possibilities: Either 3 is interesting or not. 1

12 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? Well, there are only two possibilities: Either 3 is interesting or not. Let s assume 3 is not interesting. 1

13 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? Well, there are only two possibilities: Either 3 is interesting or not. Let s assume 3 is not interesting. Well, then notice this: 1

14 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? Well, there are only two possibilities: Either 3 is interesting or not. Let s assume 3 is not interesting. Well, then notice this: 3 has this spectacular property that it is the smallest natural number that is not interesting. Thus we see that 3 is quite interesting as well. 1

15 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? Well, there are only two possibilities: Either 3 is interesting or not. Let s assume 3 is not interesting. Well, then notice this: 3 has this spectacular property that it is the smallest natural number that is not interesting. Thus we see that 3 is quite interesting as well. What then? 1

16 The Intrigue of Numbers Prove that every natural number is interesting. Proof: Is 1 interesting? Of course, it is interesting since it is the first natural number and it is the only number with property. How about 2? Is 2 interesting? Well, 2 is the first even number. So, I think, 2 is interesting. Now let s consider the number 3. Is it interesting? Well, there are only two possibilities: Either 3 is interesting or not. Let s assume 3 is not interesting. Well, then notice this: 3 has this spectacular property that it is the smallest natural number that is not interesting. Thus we see that 3 is quite interesting as well. What then? Knowing that 1, 2, and 3 are interesting, we can make a similar argument for 4 or any other number. 1

17 2 2 Magic Square

18 2 2 Magic Square Fill in boxes with 1 s and 1 s so that the columns and rows and the diagonals all have different sums: 2

19 2 2 Magic Square Fill in boxes with 1 s and 1 s so that the columns and rows and the diagonals all have different sums: Here are 16 all possible such squares: (Why 16? Well, 16 = 2 4.)

20 2 2 Magic Square Fill in boxes with 1 s and 1 s so that the columns and rows and the diagonals all have different sums: Here are 16 all possible such squares: (Why 16? Well, 16 = 2 4.) Surprise? 2

21 2 2 Magic Square Fill in boxes with 1 s and 1 s so that the columns and rows and the diagonals all have different sums: Here are 16 all possible such squares: (Why 16? Well, 16 = 2 4.) Surprise? It can t be done! 2

22 3 3 Magic Square Fill in boxes with 1 s and 1 s so that the columns and rows and the diagonals all have different sums: 3

23 Pigeonhole Principle

24 Pigeonhole Principle: Statement 4

25 Pigeonhole Principle: Statement The following general principle was formulated by the famous German mathematician Dirichlet ( ): 4

26 Pigeonhole Principle: Statement The following general principle was formulated by the famous German mathematician Dirichlet ( ): The Statement: If (N + 1) (or more) pigeons occupy N pigeonholes, then some pigeonhole must have at least 2 pigeons. 4

27 Pigeonhole Principle: Statement The following general principle was formulated by the famous German mathematician Dirichlet ( ): The Statement: If (N + 1) (or more) pigeons occupy N pigeonholes, then some pigeonhole must have at least 2 pigeons. Proof (?) 4

28 Revisited: (Slightly Better Version) 2 2 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: 5

29 Revisited: (Slightly Better Version) 2 2 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: We already saw that it is impossible by exhausting all possible 2 2 such squares. 5

30 Revisited: (Slightly Better Version) 2 2 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: We already saw that it is impossible by exhausting all possible 2 2 such squares. Pigeons: 2 rows + 2 columns + 2 diagonals = 6 pigeons 5

31 Revisited: (Slightly Better Version) 2 2 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: We already saw that it is impossible by exhausting all possible 2 2 such squares. Pigeons: 2 rows + 2 columns + 2 diagonals = 6 pigeons Pigeonholes: Smallest sum ( 1) + ( 1) = 2 and largest sum = 2. So 2, 1,0,1,2 5 pigeonholes. 5

32 3 3 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: 6

33 3 3 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: Pigeons: 6

34 3 3 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: Pigeons: 3 rows + 3 columns + 2 diagonals = 8 pigeons 6

35 3 3 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: Pigeons: 3 rows + 3 columns + 2 diagonals = 8 pigeons Pigeonholes: 6

36 3 3 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: Pigeons: 3 rows + 3 columns + 2 diagonals = 8 pigeons Pigeonholes: Smallest sum ( 1) + ( 1) + ( 1) = 3 and largest sum = 3. So 3, 2, 1,0,1,2,3 7 pigeonholes. 6

37 3 3 Magic Square Fill in boxes with 1 s, 0 s and 1 s so that the columns and rows and the diagonals all have different sums: Pigeons: 3 rows + 3 columns + 2 diagonals = 8 pigeons Pigeonholes: Smallest sum ( 1) + ( 1) + ( 1) = 3 and largest sum = 3. So 3, 2, 1,0,1,2,3 7 pigeonholes. Generalization? 6

38 Example: Cheaper by the Dozen

39 Cheaper by the Dozen 7

40 Cheaper by the Dozen In the movie Cheaper by the Dozen," there are 12 children in the family. 7

41 Cheaper by the Dozen In the movie Cheaper by the Dozen," there are 12 children in the family. (a) Prove that at least two of the children were born on the same day of the week. 7

42 Cheaper by the Dozen In the movie Cheaper by the Dozen," there are 12 children in the family. (a) Prove that at least two of the children were born on the same day of the week. (b) Prove that at least two family members (including both parents) are born in the same month. 7

43 Cheaper by the Dozen In the movie Cheaper by the Dozen," there are 12 children in the family. (a) Prove that at least two of the children were born on the same day of the week. (b) Prove that at least two family members (including both parents) are born in the same month. (c) Assuming there are 4 children s bedrooms in the house, show that there are at least three children sleeping in at least one of them. 7

44 Cheaper by the Dozen In the movie Cheaper by the Dozen," there are 12 children in the family. (a) Prove that at least two of the children were born on the same day of the week. (b) Prove that at least two family members (including both parents) are born in the same month. (c) Assuming there are 4 children s bedrooms in the house, show that there are at least three children sleeping in at least one of them. 7

45 Example: Pigeonhole Middle School

46 Pigeonhole Middle School Pigeonhole Middle School has 400 students. Show that at least two students were born on the same day of the year. 8

47 Hairs in Tennessee

48 Hairs in Tennessee, Part 1 9

49 Hairs in Tennessee, Part 1 And even the very hairs of your head are all numbered Matthew 10:30 (NIV) 9

50 Hairs in Tennessee, Part 1 And even the very hairs of your head are all numbered Matthew 10:30 (NIV) Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? 9

51 Hairs in Tennessee, Part 2 10

52 Hairs in Tennessee, Part 2 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? 10

53 Hairs in Tennessee, Part 2 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? So How Hairy Are We? 10

54 Hairs in Tennessee, Part 2 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? So How Hairy Are We? The average human head is 6 to 7 inches wide and 8 to 9 inches long. The average circumference is 21 to 23 inches. Males generally have a slightly larger head than females. 10

55 Hairs in Tennessee, Part 2 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? So How Hairy Are We? The average human head is 6 to 7 inches wide and 8 to 9 inches long. The average circumference is 21 to 23 inches. Males generally have a slightly larger head than females. One of the two authors of the beautifully written book titled The Heart of Mathematics" actually counted the number of hairs on a 1/4-inch 1/4-inch square area on his scalp and counted about 100 hairs that s roughly 1600 hairs per square inches! 10

56 Hairs in Tennessee, Part 3 11

57 Hairs in Tennessee, Part 3 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? How Many People in Tennessee 11

58 Hairs in Tennessee, Part 3 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? How Many People in Tennessee According to 2010 Census Data found in censusviewer.com/state/tn, the total population of the state of Tennessee is as follows: 11

59 Hairs in Tennessee, Part 3 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? How Many People in Tennessee According to 2010 Census Data found in censusviewer.com/state/tn, the total population of the state of Tennessee is as follows: Total Population of Tennessee in Population by Race White alone African American alone Hispanic/Latino Origin Asian alone

60 Hairs in Tennessee, Part 4 12

61 Hairs in Tennessee, Part 4 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? 12

62 Hairs in Tennessee, Part 4 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? How Many Hairs on Head of an Average Person? 12

63 Hairs in Tennessee, Part 4 Question: Do there exist two people in Tennessee who have exactly the same number of hairs on their heads? How Many Hairs on Head of an Average Person? How Many People in Tennessee 12

64 Generalization

65 Generalized Pigeonhole Principle If you have N pigeons in K holes (N > K), and (N/K) is not an integer, then some hole must have strictly more than (N/K) pigeons. 13

66 Generalized Pigeonhole Principle If you have N pigeons in K holes (N > K), and (N/K) is not an integer, then some hole must have strictly more than (N/K) pigeons. So 16 pigeons occupying 5 holes means 13

67 Generalized Pigeonhole Principle If you have N pigeons in K holes (N > K), and (N/K) is not an integer, then some hole must have strictly more than (N/K) pigeons. So 16 pigeons occupying 5 holes means some hole has at least 4 pigeons. 13

68 More Examples

69 First/Last Name Initials 14

70 First/Last Name Initials How many first and last name initials are there? 14

71 First/Last Name Initials How many first and last name initials are there? Claim: 14

72 First/Last Name Initials How many first and last name initials are there? Claim: At least two students at Covenant College have the same first and last name initials. 14

73 A Triangular Dartboard 15

74 A Triangular Dartboard Consider a dartboard which is an equilateral triangle with side length of 2 feet. If you throw 5 darts with no misses, then at least two will land within a foot of each other. 15

75 A Party Problem 16

76 A Party Problem There are 8 guests at a party and they sit around an octagonal table with one guest at each edge. If each place at the table is marked with a different person s name and initially everybody is sitting in the wrong place, prove that the table can be rotated in such a way that at least 2 people are sitting in the correct places. 16

77 A Party Problem There are 8 guests at a party and they sit around an octagonal table with one guest at each edge. If each place at the table is marked with a different person s name and initially everybody is sitting in the wrong place, prove that the table can be rotated in such a way that at least 2 people are sitting in the correct places. 16

78 Bonus: Classical Rational Approximation Theorem

79 Classical Rational Approximation Theorem Show that for any irrational x R and positive integer n, there exists a rational number p x q with 1 q n such that p < 1 q nq. 17

80 References The Heart of Mathematics: An Invitation to Effective Thinking by Edward B. Burger and Michael Starbird Timothy E. Goldberg, The pigeonhole principle, math.cornell.edu/~goldberg/talks/pigeonholes.pdf Mark Flanagan, The pigeonhole principle, Su, Francis E., et al. Pigeonhole Principle." Math Fun Facts. The art of problem solving. Pigeonhole Principle." artofproblemsolving.com/wiki/index.php?title= Pigeonhole_Principle An unknown source 18