quiz #1

find the first 15 pairs of twin primes

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199)

lets start with the numbers 0, 0, 1 and generate future numbers in our sequence by adding up the previous three numbers. Write out the first 15 terms in this sequence, starting with the first 1. Use a calculator to evaluate the value of the quotients of consecutive terms (dividing the smaller term into the larger one). Do the quotients seem to be approaching a fixed number?"

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927 927/504 = 1.8392 504/274 = 1.8394 274/149 = 1.8398

You have 10 pairs of socks, 5 black pairs and 5 blue pairs, but they are not paired up. Instead, they are all mixed up in a drawer. It's early in the morning and you don't want to turn on the lights in your dark room. How many socks must you pull out to guarantee that you have a pair of one color? How many must you pull out to have two good pairs (each pair is the same color)? How many must you pull out to be certain that you have a pair of black socks?

1) pull out 3 socks - Either two will be black or two will be blue 2) pull out 5 socks 3) pull out 12 socks - you might be unlucky and pull out all the blue socks first

list the first 15 Fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610

suppose you are about to begin a game of Fibonacci nim. you start with 100 sticks. what is your first move?

100 = 89 + 8 + 3 so remove three sticks

suppose you are playing a round of Fibonacci nim with a friend. the game begins with 15 sticks. you remove 2; your friend takes one; you take two; your friend takes one; what should your next move be? can you make it w/out breaking rules? did you make a mistake?

15 - 2 - 1 - 2 - 1 = 9 9 = 8 + 1 --> taking one stick isn't breaking the rules; mistake was made during the second move. We should have taken one stick instead of two

what is the smallest natural number that has three distinct prime factors in its factorization?

2 x 3 x 5 = 30

it is july. what month will it be in 219 & 120,963 months? what month was it 89 months ago?

219 months from now = October (219 = 18 x 12 + 3) 120,963 months from now = October (120,963 = 10,080 x 12 + 3) 89 months ago = February (-89 = -7 x 12 - 5)

which of the following objects most closely resembles a golden rectangle? a 3x5 inch index card, an 8.5x11 or 11x14 or 11x17 inch paper?

3x5 inch index card because 5/3 = 1.6

suppose you are about to begin a game of Fibonacci nim. you start with 50 sticks. what is your first move?

50 = 34 + 13 + 3 so remove three sticks

what is the smallest number that looks prime but really isnt?

51 (3 x 17)

express each of the following natural numbers as a sum of distinct, nonconsecutive Fibonacci numbers: 52, 143, 13, 88

52 = 34 + 13 + 5 143 = 89 + 34 + 13 + 5 + 2 13 = 13 (already a Fibonacci) 88 = 55 + 21 + 8 + 3 + 1

fibonacci nim. the game starts with 90 sticks. you start by removing one stick; your friend takes two; you take three; your friend takes six; you take two; your friend takes one; you take two; your friend takes four; you take one; your friend takes two. how many sticks should you take?

90 - 1 - 2 - 3 - 6 - 2 - 1 - 2 - 4 - 1 - 2 = 66 66 = 55 + 8 + 3 so remove 3 sticks

Is it possible for a Fibonacci number greater than 2 to be exactly twice as big as the Fibonacci number immediately preceding it? Why or Why not? What would your answer be if "greater than 2" was removed?

Because the Fibonacci numbers keep getting bigger, the last fraction is always less than or equal to one. Fk + 1/Fk is equal to 2 only when Fk + 1 = 2

if n is an odd number greater than or equal to 3, can n + 1 ever be prime? what if n equals 1?

No, n + 1 will be an even number greater than 2, and so will have 2 as a factor If n = 1, then n + 1 = 2 which is prime

does a nonprime divided by a nonprime ever result in a prime? does a nonprime multiplied by a nonprime ever result in a prime? always? sometimes? never?

Sometimes. For example, 8/4 = 2 is prime, but 16/4 = 4 is not

order the following from smallest to largest:

States in the United States, honest congressmen (debatable), cars, telephones on the planet, people, grains of sand

You have 16 new CDs to put on your empty five-shelf CD rack. Can you place the CDs so that each shelf contains three or fewer CDs? Can you arrange them so that each shelf contains exactly three?

The answer to both questions is ''no.'' If each shelf had 3 (or fewer) CDs, then the total number of CD's would be (at most) 15.

can every odd number greater than 3 be written as the sum of two prime numbers? if so, prove it. if not, find the smallest counter example and show that the number given is definitely not the sum of two primes

The smallest counterexample is 11. 11 - 2 = 9, but 9 isn't prime.

100 people in your neighborhood always drive to work between 7:30 and 8 and arrive 30 minutes later. Why must two people always arrive at work at the same time, within a minute?

There are 100 people arriving between 8 and 8:30, leaving 31 separate one-minute slots to arrive in. There are more people than slots, so at least two people will arrive to work at the same time

explain what makes a rectangle a golden rectangle

a golden rectangle has side lengths of the golden ration, or 1.618

find the first 10 powers of 2 find the first powers of 5

first ten powers of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 first five powers of 5: 5, 25, 125, 625, 3125

explain the pigeonhole principle

if there are more pigeons than pigeonholes, and every pigeon must be in some pigeonhole, then there must be at least one pigeonhole with more than one pigeon

you own a watch that is currently flashing 3:00. what time will it be in 12, 14, 25, and 240 hours? what time is it when an elephant sits on it?

in 12 hrs = 3:00 in 14 hrs = 5:00 (14 = 12 + 2) in 25 hrs = 4:00 (25 = 12 x 2 + 1) in 240 hrs = 3:00 (240 = 20 x 12)

today is saturday. what day of the week will it be in 3724 days? what day of the week will it be in 365 days?

in 3724 days = Saturday (3724 = 532 x 7) in 365 days = Sunday (365 = 52 x 7 + 1)

suppose you are playing a round of Fibonacci nim with a friend. the game begins with 15 sticks. You start by removing two sticks; your friend then takes four. How many sticks should you take next to win?

there are nine sticks left. Because 9 = 8 + 1, so we remove one stick

you have an empty CD rack consisting of five shelves and you just bought five totally kickin' CDs. Can each CD go on a different shelf? What if you had six new CDs?

with 5 CDs, each one can have its own shelf. With 6 CDs, some shelf must have two

suppose you have a golden rectangle cut out of a piece of paper. now suppose you fold it in half along its base & then in half along its width. you have just created a new smaller rectangle. is that rectangle a golden rectangle? justify your answer.

yes, it will create a new golden rectangle. you are dividing each side by 2 1.618/2 = 0.809 new rectangle multiplied by 2 = 1.618

suppose we have a room filled with 370 people. will there be at least two people with the same birthday?

yes. 370 people and 366 days. There are more people than days, so there must be at least two people that share the same birthday