Magic Numbers

There can be subtle patterns and effects caused by using our Base-10 number system. Knowing mathematics can make you seem to have psychic powers.

Materials Needed

Paper, pen, phone book, envelope.

Setting Up

Before doing this demonstration, you need to write the ninth number on page 108 of the phone book onto a piece of paper. Then put it in the envelope and seal it.

Demonstration

The demonstration is done as follows.

1. Tell the audience some nonsense about you being able to predict the future or read their minds. Give someone the envelope. Ask them not to open it yet.
2. Ask for a three digit number, where all the digits are different. Write it down.
3. Write the number backwards.
4. Subtract the smaller from the larger. You will now have either a two or three digit number. It will have a 9 in the tens column.
5. Write this new number backwards. If it is a two digit number, you need to write a zero at the end. (e.g. 94 goes to 490).
6. Add these two numbers. You will get 1089.
7. Have someone read out the ninth phone number on page 108, write it where the audience can see.
8. Have the first person open the envelope to reveal the same number.

For example, if you are given 257:

Tips

The most common problem with this demonstration is the volunteer making a mistake with the phone book. Make certain it is clear that you want the ninth number, or the ninth person. You may even want to stick in a piece of paper into page 108, labelled “This one” with an arrow pointing ot the ninth one.

Further Information

This demonstration works because the number manipulation always leads to 1089.

To see why, we need to use some algebra. Below is a full proof of how and why this demonstration works. It can be technical, so you may want to skip straight to the further activities. There is both a general proof and a worked example, using the number 752 as a starting point.

First, let us perform the subtraction step. First we need to do some carrying. If we start with 752-257, 7>2 so carry from the tens column to get 12 in the ones column. Now carry from the hundreds to get 14 in the tens column. Next perform the subtraction to get 12-7=5 in the ones, 14-5=9 in the tens, and 6-2=4 in the hundreds.

In the more general case, start with x in the 100's, y in the 10's and z in the 1's. Let's assume that none of them is the same and that x>z.

Since x>z carry from the tens column to the ones to get z+10. Now we have y-1 in then tens column, so carry from the hundreds to get y-1+10=y+9, leaving x-1 in the hundreds column. Now we can subtract. Take x from z+10 and get z-x+10. In the tens column, take y from y+9 to get 9. Finally in the 100's column subtract z from x-1 to get x-z-1.

Becomes...

 Answer:

The next step is to reverse this new number then add the two numbers. This gives 5+4=9 in the ones column, 9+9=18 in the tens, so carry the one, then 1+4+5=10 in the hundreds; giving a sum of 1089.

In the general case, In both the 1's and the 100’s column we get z-x+10+x-z-1. The x’s and the z’s cancel, leaving 9. In the 10’s column we get 18, the one carries to the hundreds, giving ten, which then carries to the thousands. Thus we end up with 1089.

Or:

Further Activities

Try this puzzle a few times so people can see that it always leads to 1089. If students know other number tricks, use algebra to figure out why they work.

A trickier one: Can you work out why it is that if a number is divisible by nine, its digits add to a number divisible by nine?
(Hint: Any number, take away the sum of its digits, gives a number divisible by nine. Why?)