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Mathematical Reasoning Puzzles

These puzzles require both logical and mathematical reasoning. Can you solve them?

#11

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 `

Solution

#12

The following is what seems to be a mathematical proof that ten equals 9.999999.... What's wrong with it?

`                a =  9.999999...              10a = 99.999999...          10a - a = 90               9a = 90                a = 10 `

Solution

#13

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?

Solution

#14

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?

Solution

#15

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?

Solution

#16

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?

Solution

#17

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

Solution

#18

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?

Solution

#19

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?

Solution

#20

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?

Solution