# Flawed Proof

The following is what seems to be a mathematical proof that two equals one. What's wrong with it?

a  =  b
aa  =  ab
aa  -  bb  =  ab  -  bb
(a  +  b)(a  -  b)  =  b(a  -  b)
a  +  b  =  b
a  +  a  =  a
2a  =  a
2  =  1

# Mathematicians Kids Ages

Two mathematicians, Albert and Isaac, chat. Isaac says he has three children who all have the same birthday (but who weren't necessarily born in the same year). Albert asks their ages. Isaac replies, "The product of the ages of my children is 72." Albert points out that this is not enough information to determine their ages. Isaac responds with another clue -- he tells Albert the sum of the ages of his children. But Albert again points out that there is not enough information. Finally Isaac says, "My youngest child is named Galileo." At last, Albert correctly determines the ages of Isaac's children. What are the ages?

# Clockwork

A clock is observed. The hour hand is exactly at the minute mark, and the minute hand is six minutes ahead of it. Later, the clock is observed again. This time, the hour hand is exactly on a different minute mark, and the minute hand is seven minutes ahead of it. How much time elapsed between the first and second observations?

# Chicken and Rabbit Farm

There are several chickens and rabbits in a cage (with no other types of animals). There are 72 heads and 200 feet inside the cage. How many chickens are there, and how many rabbits?

# Grass on the Field

Grass grows in a field at some rate r, where r is the units of grass grown per day. It is known that if 10 sheep are turned out in the field, the grass will be gone in 20 days. On the other hand, if 15 sheep are turned out in the field, the grass will be gone in 10 days. If 25 sheep are turned out in the field, when will the grass be gone?

# Squeegee Time

You've been asked to buy 100 squeegees, using 100 dollars to do so. You may buy no more or less than 100 squeegees, and the total price must be exactly 100 dollars. There is no sales tax. Red squeegees cost \$6.00. Yellow squeegees cost \$3.00. Blue squeegees cost \$0.10. How many of each must you buy?

On a man's tombstone, it is said that one sixth of his life was spent in childhood and one twelfth as a teenager. One seventh of his life passed between the time he became an adult and the time he married; five years later, his son was born. Alas, the son died four years before he did. He lived to be twice as old as his son did. How old did the man live to be?

# Mathematicians Vacation

Isaac and Albert wanted to take a vacation. They were debating how they could get to their hotel in the fastest manner. Isaac said, "We should go by train." But Albert said, "No, the train reaches the end of the line half way to the hotel -- we would have to walk the rest of the way. We should bike to the hotel instead." Isaac disagreed. So Albert biked the whole way to the hotel, while Isaac took the train for the first half of the journey and walked for the remainder. The speed of the train turned out to be four times that of the bike's speed. The bike's speed turned out to be two times faster than walking speed. Who got to the hotel first?

# Unattainable Nuggets

At McDonald's you can order Chicken McNuggets in boxes of 6, 9, and 20. What is the largest number of nuggets that it is not possible to obtain by purchasing some combination of boxes?

# Loose Marbles

A toy store ordered 7 small bags and 18 large bags of identical marbles. When the marbles arrived, it was discovered that the bags had broken during shipping, and all 233 of the marbles were rolling around loose in the box. How was the store's manager was able to determine how many marbles were supposed to go in each of the small bags and how many were supposed to go in each of the large bags?

# Coins In A Box

A box contains two quarters. One is a double-headed coin, and the other is an ordinary coin, heads on one side, and tails on the other. You draw one of the coins from a box and look at one of the sides. Assuming it is heads, what is the probability that the other side shows heads also?

# Kitchen Timer

You're a cook in a restaurant in a quaint country where clocks are outlawed. You have a four minute hourglass, a seven minute hourglass, and a pot of boiling water. A regular customer orders a nine-minute egg, and you know this person to be extremely picky and will not like it if you overcook or undercook the egg, even by a few seconds. What is the least amount of time it will take to prepare the egg, and how will you do it?

I'm going to buy one-cent, two-cent, three-cent, five-cent, and ten-cent stamps. I'm going to buy four of each of two sorts, and three of each of the rest, and I have exactly enough to buy them -- just this handful of dimes. How many of each type am I going to buy?

# Hot Dog Eating

If a boy and a half can eat a hot dog and a half in a minute and a half, how many hot dogs can six boys eat in six minutes?

# Divying the Booty

Five pirates raid the ship of a wealthy bureaucrat and steal his trunk of gold pieces. By the time they get the trunk aboard, dusk has fallen, so they agree to split the gold the next morning. But the pirates are all very greedy. During the night one of the pirates decides to take some of the gold pieces for him. He sneaks to the trunk and divides the gold pieces into five equal piles, with one gold piece left over. He puts the gold piece in his pile, hides it, puts the other four piles back in the trunk, and sneaks back to bed. One by one, the remaining pirates do the same. They sneak to the trunk, divide the coins into five piles, with always one coin left over. Each pirate puts the gold coin in his own pile, hides it, and puts the remaining four piles back in the trunk. What is the smallest number of coins there could have been in the trunk originally?

# Math Reasoning

Reason why 3041409320171337804361260816606476884437764156896051207829102700 cannot possibly be the value of 50 factorial, without actually performing the calculation.

# Math

What two numbers have a product of 48 and, when the larger number is divided by the smaller, a quotient of 3?

# Change

A man has \$1.15 in six coins. He can't make change for a dollar, a half dollar, a quarter, a dime, or a nickel. What coins did he have?

# Apple Money

Two boys sell apples. Each sells thirty apples a day. The first boy sells his apples at two for fifty cents (and therefore earns \$7.50 per day). The second boy sells his apples at three for fifty cents (and therefore earns \$5.00 per day). The total received by both boys each day is therefore \$12.50. One day, the first boy is sick, and the second boy takes over his apple selling duties. To accommodate the differing rates, the boy sells the sixty apples at five for a dollar. But selling sixty apples at five for a dollar yields only \$12.00 earnings at the end of the day. What happened to the other fifty cents?

# Dozen

The expression, "Six of one, half a dozen of another," is commonly used to indicate that two alternatives are essentially equivalent, because six and a half dozen are equal quantities. But are "six dozen dozen dozen" and "a half dozen dozen dozen" equal?

# Alligator Breeders

Two alligator breeders were talking one day, and in the course of their conversation they noticed a curious coincidence. If alligator breeder #1 were to sell alligator breeder #2 seven alligators, then breeder #1 would have exactly as many alligators as breeder #2. On the other hand, if alligator breeder #2 were to sell alligator breeder #1 seven alligators, then breeder #1 would have exactly twice as many alligators as breeder #2. How many alligators did each breeder have?

# One Hundred

How can you express the number 100 using six nines and no other digits?

# Sheep and Chickens

A drove of sheep and chickens have a total of 99 heads and feet. There are twice as many chickens as sheep. How many of each are there?

# Rope 'round the World

An eccentric individual makes it is life's work to tie a rope around the earth's equator. He buys a lot of rope and makes the attempt. A rival of his, not to be outdone, decides he wants to tie a rope around the earth's equator that is elevated from the ground by one yard at all points along the rope. How much more rope does he need? Assume the earth is perfectly spherical.

# This Big

A fish is fifteen inches long. Its head is as long as its tail. If the head were twice as long as it really is, the head and tail would together be as long as what's in between. How long is each part of the fish?

# Coins

A man has 100 coins (there are no quarters or nickels) that add up to \$5.00. How many coins did he have of each denomination?

# Eating Chocolates

One boy can eat 100 chocolates in half a minute, and another can eat half as many in twice the length of time. How many chocolates can both boys eat in 15 seconds?

# Rolling Dice

A solo dice game is played thusly: one each turn, a normal pair of dice is rolled. The score is calculated by taking the product, rather than the sum, of the two numbers shown on the dice. On a particular game, the score for the second roll is five more than the score for the first; the score for the third roll is six less than that of the second; the score for the fourth roll is eleven more than that of the third; and the score for the fifth roll is eight less than that of the fourth. What was the score for each of these five throws?

# School Lockers

A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony: There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?

# Birthday Cake

You must cut a birthday cake into exactly eight pieces, but you're only allowed to make three straight cuts, and you can't move pieces of the cake as you cut. How can you do it?

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