Two men and two boys wish to cross a river. Their small canoe will only carry one man or two boys.

 

What is the least number of canoe trips needed to get anyone across??

 

 

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?

 

Solution

The only lockers that remain open are perfect squares (1, 4, 9, 16, etc) because they are the only numbers divisible by an odd number of whole numbers; every factor other than the number's square root is paired up with another. Thus, these lockers will be "changed" an odd number of times, which means they will be left open. All the other numbers are divisible by an even number of factors and will consequently end up closed.

So the number of open lockers is the number of perfect squares less than or equal to one thousand. These numbers are one squared, two squared, three squared, four squared, and so on, up to thirty one squared. (Thirty two squared is greater than one thousand, and therefore out of range.) So the answer is thirty one.

 

 

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?

 

Solution

Use the first two cuts to cut an 'X' in the top of the cake. Now you have four pieces. Make the third cut horizontal, which will divide the four pieces into eight. Think of a two by two by two Rubik's cube. There's four pieces on the top tier and four more just underneath it.

 

 

 

Two trains travel toward each other on the same track, beginning 100 miles apart. One train travels at 40 miles per hour; the other travels at 60 miles an hour. A bird starts flight at the same location as the faster train, flying at a speed of 90 miles per hour. When it reaches the slower train, it turns around, flying the other direction at the same speed. When it reaches the faster train again, it turns around -- and so on. When the trains collide, how far will the bird have flown?

 

Solution

Since the trains are 100 miles apart, and the trains are travelling toward each other at 40 and 60 mph, the trains will collide in one hour. The bird will have been flying for an hour at 90 miles per hour at that point, so the bird will have travelled 90 miles.

 

 

 

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

The problem is with the division that takes place between the fourth and fifth equations. Since a = b, a - b is zero, and you can't divide by zero.

 

 

 

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

Let c be the number of chickens, and r be the number of rabbits.

          r + c = 72         4r + 2c = 200 

To solve the equations, we multiply the first by two, then subtract the second.

        2r + 2c = 144              2r = 56               r = 28               c = 44 

So there are 44 chickens and 28 rabbits in the cage.

 

 

 

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?

 

Solution

It should only take nine minutes to cook the egg. If you want to try to figure out how it is done in this short amount of time before seeing the answer, stop reading now. To start, flip both hourglasses over and put the egg in the water. When the four minute hourglass runs out, flip it back over immediately. When the seven minute hourglass runs out, flip that back over immediately too. One minute later, the four minute hourglass will run out again. At this point, flip the seven minute hourglass back over. The seven minute hourglass had only been running for a minute, so when it is flipped over again it will only run for a minute more before running out. When it does, exactly nine minutes will have passed, and the egg is done.

 

 

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 meter at all points along the rope. How much more rope does he need? Assume the earth is perfectly spherical.

 

Solution

The circumference of a circle is 2πr, where r is the radius of the circle. If you want a rope that is one meter above the ground, this radius is larger by one meter. Let R be this new radius. So R = r + 1.

Let x be the amount of extra rope required by the eccentric's rival. So:

     x = (2π(r + 1)) - (2πr)      x = (2πr) + (2π) - (2πr)      x = 2π 

So x is about 6.2832 meters. Note that this answer does not depend on the radius of the circle. If the eccentric and his rival were attempting to tie up a baseball rather than the earth, the amount of additional required rope would be the same amount.