Greg's Mathematical Brain Food

Solutions at bottom

Problems

Number one

Can you place six X's on a Tic Tac Toe board without making three-in-a-row in any direction?

Number two

Nine dots are arranged in a three by three square. Connect each of the nine dots using only four straight lines and without lifting your pen from the paper.

Number three

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?

Number four

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?

Number five

Arrange the numbers 1 through 9 on a tic tac toe board such that the numbers in each row, column, and diagonal add up to 15.

Number six

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 

Number seven

A box contains two coins. 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?

Number eight

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

Number nine

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?

Number ten

You own a pet store. If you put in one canary per cage, you have one canary too many. If you put in two canaries per cage, you have one cage too many. How many canaries and cages do you have?

Solutions

Number one

Number two

Number three

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.

Number four

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.

Number five

There is only one solution, discounting mirror image solutions and rotations:

Number six

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.

Number seven

For starters, the answer is not 1/2. When the coin is drawn, there are four possibilities, each of which is equally likely:

The problem tells us that the last possibility did not occur. Therefore, there are three remaining possibilities, each of which is equally likely. Of the three, two of the possibilities will show heads on the other side; only one will show tails on the other side. So the probability that the other side of the coin is heads is two thirds.

Number eight

99 99/99

Number nine

Let S be the number of sheep and C be the number of chickens. So:

           2S = C
      5S + 3C = 99 

We can rephrase the first equation thusly:

      6S - 3C = 0 

And then we can add this to the second equation, which yields:

          11S = 99 

By solving for S, we find that S equals 9. By substituting back in one of the original equations, we find that C equals 18. So there are nine sheep and eighteen chickens.

Number ten

Let b be the number of birds, and c be the number of cages. The problem gives us two equations, which we can merge together to solve for c:

           b = c + 1
           c = b/2 + 1
       c - 1 = b/2
      2c - 2 = b
      2c - 2 = c + 1
      2c - c = 1 + 2
           c = 3 

Now we substitute 3 for c in one of the original equations to discover that b equals 4. So there are four canaries and three cages.

Back to Greg's Brain Food