The Funny Thing About Math

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1 AMATYC - San Diego Session: S072 Friday - Nov. 10, :15 to 11:05 The Funny Thing About Math Volume II Terry Krieger Rochester Community and Technical College

2 Today s Presentation Some math from everyday life A card trick Revisiting a result from Funny Math Volume I Trying to do 5th-grade math Some family math Parting thoughts

3 Everyday Math

4 Shopko Back-to-School Display

No You re Staits Cloudy Organ Vegas Hollywood Idonno Goblin Valley Saguaro Hannah Montana Whyknot New York Rockies These two states are actually one state.

5 No You re Staits Cloudy Organ Vegas Hollywood Idonno Goblin Valley Saguaro Hannah Montana Whyknot New York Rockies These two states are actually one state. Nebula Wheat Okay Tex-Mex Part of Canada Corn Rogers Misery Arksomething Chicago Forgot Whodat? Elvis????? One of these is New Hampshire and the other is Vermont, I forget. Michelin Kendell Go Tribe!?? George New Mexico Pencil Virgel N. Carol S. Carol Florence???? Mane No one knows the names of these states. Siberia U.S. Virgin Islands? One of these is Alabama and the other is Mississippi, I forget.

6 Sneaky Math

7 Sneaky Math Math has become the vegetables in our academic diet.

8 Here s a Good Deal!

9 A Card Trick

10 Fitch s Card Trick (A Very Mathematically Sophisticated Card Trick) The pigeon hole principle Simple arithmetic Modular arithmetic Combinatorics

11 The Pigeon Hole Principle Given five cards, at least two must have the same suit.

12 The Main Idea No two cards in one suit are more than 6 apart.

13 Which Card Goes Face Down? Suppose we have the 2 and Jack of Diamonds. Locate the two cards on the wheel. Count clockwise from the target card to the unknown card in 6 or fewer steps. Place the unknown card face down on the table. Unknown Target Remember the count from the target card to the unknown card.

14 Communicating the Target Card Using the Four Remaining Cards Find the sum S of the face values of the cards, with Ace = 1 and Jack = Queen = King = 10. Then (S mod 4) gives the position (from left to right) of the target card (and suit). Target Card Jack of Diamonds

15 The Agreement Possible arrangements of 3 cards = = 6 Low Medium High (LMH) = 1 Low High Medium (LHM) = 2 Medium Low High (MLH) = 3 Medium High Low (MHL) = 4 High Low Medium (HLM) = 5 High Medium Low (HML) = 6

16 Let s Try Some (MHL = 4) (LHM = 2)

17 A Difficult Case (MLH = 3)

18 Revisiting an Interesting Result

19 The Number of Sixes Find the number of integers from 1 to 1,000,000 for which at least one of its digits is a six. Examples: 6, 26,411, 3661, 606,062 HINT: Start with one-digit numbers, move on to two-digit numbers, three-digit numbers, and so on. Look for a pattern.

20 The Number of Sixes After a huge amount of effort 1-digit: 1 = 1 2-digit: = 18 3-digit: (19) = digit: [ (19)] = digit: 10, [ [ (19)]] = 37,512 6-digit: 100, [10, [ [ (19)]]] = 427, ,559

21 Why don t you just realize that it s much easier to count the integers that don t have sixes and then subtract that from 1,000,000? One Million in base ten: 10 6 = 1,000,000 With no sixes, use base nine: 9 6 = 531,441 The difference 4 468,559

22 So we can calculate the percentage of integers from 1 to 10 6 that contain at least one six by computing X %. And, in general, we can calculate the percentage of numbers from 1 to 10 n that contain at least one six by computing 10 n 9 n 10 n X 100.

23 What does this mean? 10 n 9 n 10 n X 100 n What is calculating: Percent 1 Percent of integers from 1 to 10 with a six 10 2 Percent of integers from 1 to 100 with a six 19 3 Percent of integers from 1 to 1000 with a six Percent of integers from 1 to 10,000 with a six Percent of integers from 1 to 100,000 with a six Percent of integers from 1 to 1,000,000 with a six Percent of integers from 1 to with a six Percent of integers from 1 to with a six Percent of integers from 1 to with a six 99.9

24 Let n Get Large For positive integer, n: lim q n ª` 10 n 9 n 10 n X 100 r = 100% We conclude that ALL positive integers contain at least one digit that is a six!

25 When will 40% of Positive Integers Contain at Least One Six? We need to solve: 1 10 n 9 n 10 n 9 q r 10 n = 0.40 = = 0.9 n ln 0.6 = ln 0.9 n ln 0.6 n = ln 0.9 n Note: ,528

26 Will 40% of of the first 70,528 Positive Integers Contain at Least One Six? Let s Investigate!

27 Trying 5th-grade Math

28 I Can t Help My 5th-Grader with His Math Homework

29 Answer: They travel the same number of miles each day. Explain: Because, um, that s what it says.

30 I Have Some Questions What month is this month? How many days does this month have? How many weekends does this month have? How many weekdays does this month have. What day of this month are we on today?

31 Suppose that this month is October, 2016, and that today is the 4th. William Jason William travels only on Saturdays and Sundays and has flown 1020 miles this month. => 510 mi/day Jason travels every weekday and has flown 1200 miles this month. => 600 mi/day Who flies more miles per day? Clearly it is Jason.

32 Suppose that this month is October, 2016, and that today is the 6th. William Jason William travels only on Saturdays and Sundays and has flown 1020 miles this month. => 510 mi/day Jason travels every weekday and has flown 1200 miles this month. => 300 mi/day Who flies more miles per day? Clearly it is William.

33 They must mean the entire month. William Jason William travels only on Saturdays and Sundays and has flown 1020 miles this month. => 102 mi/day Jason travels every weekday and has flown 1200 miles this month. => 57.1 mi/day OK, it s William, but these are some really short flights!

34 An Interesting Fact If you take all the veins out of your body and lay them end-to-end you will die.

35 Some Family Math Children Can Be Tricky

36 Does Daniel Cheat at Animal Bingo?

37 The Game My son randomly chooses cardboard disks from a fabric bag. Each disk has the picture of an animal on it. The first player to cover the animals on his three cards is the winner.

38 The Issue Daniel s Cards Dad loses by 9 My Cards Dad loses by 8 Dad gets clobbered every game!

39 What are the Chances? This game is equivalent to randomly drawing cards from a deck that has 18 red cards and 18 blue cards. What is the probability that, after drawing 27 cards, all 18 red cards will appear and only 9 of the blue cards will appear? C(18, 18) C(18, 9) C(36, 27) Red Cards 18 Blue Cards

40 Maybe I m Just Unlucky! What is the probability that dad loses two consecutive but independent games of Animal Bingo losing the first game by 9 and the second game by 8? Dad loses by 9 Dad loses by 8 C(18, 18) C(18, 9) C(36, 27) C(18, 18) C(18, 10) C(36, 28)

41 BTW The other clue that leads me to believe that my son was cheating was his regular use of the phrase: Hey dad, look the other way for a second.

42 Is Joey an Unsafe Railroad Engineer?

43 The Game Diagram acquired from the Internet Use ALL of our track pieces to create a connected train track with no dead ends. (I mean, how hard can it be, right?)

44 The Issue Dead End Dead End Dead End All of my track configurations seem to leave one or more dead ends.

45 Five Dead Ends Dead End Dead End Dead End Dead End Dead End

46 Can it be done? As a mathematician, it seems wise to find a way to figure out if this can be done without trying every permutation of 42 pieces of track. P(42, 42) = 42! Actual Value 1,405,006,117,752,879,898,543,142,606,244,511,569,936,384,000,000,000 (1.4 Sexdecillion)

47 Small Tangent How do we know, without calculating, that the number 42! ends in nine zeros? 42! = Actual Value 1,405,006,117,752,879,898,543,142,606,244,511,569,936,384,000,000,000

48 A Look at the Track Pieces These pieces don t bother me but these do!

49 There are Seven of These Guys

50 What Euler has to Say... A closed network must have an even number of odd nodes.

51 A Closed Network 1 Odd Node Odd Node 2 Even Node Even Node Even Node An even number of odd nodes.

52 Revisiting the Pieces

53 There are Seven Odd Nodes An odd number of odd nodes.

54 It Can t Be Done!

55 Parting Thoughts

56 Proofs: All Odds >2 are Prime Algebra Student: Physicist: Engineer: 3 is prime, 5 is prime, 7 is prime, the rest by induction. 3 is prime, 5 is prime, 7 is prime, (9 is experimental error), 11 is prime, 3 is prime, 5 is prime, 7 is prime, 9 ~ we can build a workaround, 11 is prime, Social Scientist: 3 is prime, 5 is prime, 7 is prime, 9 is prime, A computer scientist was able to generate the following proof: 3 is prime, 3 is prime, 3 is prime, Politician: 1 is prime, 2 is prime, 3 is prime 4 is prime,

57 A Mathematical Limerick 3 3 q 1 z 2 dz r q cos 3p 9 r = ln e 3 The integral of z-squared dz From 1 to the cube root of three Times the cosine Of three pi over nine Equals log of the cube root of e. Thank You for Attending!

Activity 1: Play comparison games involving fractions, decimals and/or integers.

$Activity 1: Play comparison games involving fractions, decimals and/or integers.$ Students will be able to: Lesson Fractions, Decimals, Percents and Integers. Play comparison games involving fractions, decimals and/or integers,. Complete percent increase and decrease problems, and.