Chapter 2: Number Contemplation 2.1 Counting: The Pigeonhole Principle Monday, January 19, 2009
Numbers Natural or counting numbers Integers Rational Numbers Real Numbers
Estimation How many blades of grass on a football field? How many cars would be needed to line them up, bumper to bumper from New York City to San Francisco? How many golf balls could fit in this room?
Estimation How many blades of grass on a football field? (Kentucky Bluegrass is 1/8” wide) How many cars would be needed to line them up, bumper to bumper from New York City to San Francisco? (Total Est. Distance: miles) How many golf balls could fit in this room? (Diameter = 1.86”)
Example Order the following quantities from smallest to largest: 1. Number of telephones in the US 2. Number of US Congressmen 3. Number of people in the US 4. Number of grains of sand 5. Number of states in the US 6. Number of cars in the US
Quantification Consider your everyday activities Can you view them quantitatively rather than qualitatively? Skipping class $39,434 for tuition =$19,717 per semester 4 courses per semester = $ per course 42 days of class per semester = $ per day of class
Pigeonhole Principle If we have an antique desk with slots for envelopes (known as pigeonholes), and we have more envelopes than slots, the certainly some slot must contain at least two envelopes. Examples: Same SAT scores, same zip codes, leaves on trees, temporal twins, etc.
Birthday Problem Suppose we had a room filled with 400 people. Will there be at least 2 people who celebrate their birthday on the same day?
Birthday Problem In a group of at least 23 randomly chosen people, there is more than 50% probability that some pair of them will both have been born on the same day. For 57 or more people, the probability is more than 99%, and it reaches 100% when the number of people reaches 367 (there are a maximum of 366 possible birthdays).
Birthday Problem
A Historical Note Ramanujan & Hardy The Intrigue of Numbers “Every natural number is interesting”
Some Problems In a standard deck of 52 cards, what is the smallest number of cards you must draw in order to guarantee that you have at least one pair? What is the smallest number of cards you must draw in order to guarantee that you have five cards of one suit?
Some Problems Someone offers to give you a million dollars in $1 bills. To receive the money, you must lie down. The bills will be placed on your stomach. If you keep the money on your stomach for 10 minutes, the money is yours. Do you accept the offer?
Answer Each note no matter what the denomination weighs one gram. There are 454 grams to a pound. One pound would have 454 notes. Take 1,000,000 dollar notes divided by 454 and come out with approximately pound for one million dollars in one dollar bills
More Dollar Facts How high will the stack be? The Dollar Bill is cm wide by cm long. One bill is cm thick Approximately 232 Will stack up in an inch if new. 1,000,000 ∕232 = in = ft
Animal Crackers Each box of animal crackers contains exactly 24 crackers. There are exactly 18 different shapes made. Are there always two crackers of the same shape in each box? Explain why or why not.
Problem of the Day What is the next term in the following sequence?