Unbelievable Math Problem

A friend sent the following “unbelievable math problem” via email the other day. It was one of those emails that have been traveling around the Internet for many years.

I decided that I’d try to decode the mysteries of this trick and, in doing so, perhaps refresh some of my early high school algebra. Since my sons will soon enter high school and may want some math help from yours truly, it seemed like a fun and perhaps helpful exercise.

Here’s the email:
Here is a math trick so unbelievable that it will stump you. Personally I would like to know who came up with this and why that person is not running the country.
1. Grab a calculator. (you won't be able to do this one in your head)
2. Key in the first three digits of your phone number (NOT the Area code)
3. Multiply by 80
4. Add 1
5. Multiply by 250
6. Add the last 4 digits of your phone number
7. Add the last 4 digits of your phone number again.
8. Subtract 250
9. Divide number by 2
Do you recognize the answer?

The answer is, of course, your seven digit phone number. Here’s an example, using the number 765-4321 for illustration purposes.

The first three digits are 765.
765 x 80 = 61,200
Add 1 to get 61,201
61201 x 250 = 15,300,250
Add the last four digits, or 4321, to get 15,304,571
Add the last four digits, or 4321, again to get 15,308,892
Subtract 250 to yield 15,308,642
Divide by 2 and we’re back to 7,654,321, or 765-4321
Very cool!

So, how does it work?

Well, clearly there are two variables: the first three digits of the phone number and the last four digits of the phone number. I called these “X” and “Y” respectively. The entire phone number I called “Z”.

Quickly you can see that the formula for z is:
Z = (X x 10,000) + Y
Why 10,000? Well, because we have to move the first three digits, or X, over to the left by four spaces/digits. We then add the last four digits, or Y.
So, in our example:
Z = (765 x 10,000) + 4,321 … which equals … 7,650,000 + 4,321 … which equals … 7,654,321 or 765-4321
Thus, our formula works. Indeed, Z = (X x 10,000) + Y.

My next step was to translate the directions in the email into an expression. Here it is:

Z = ((((X x 80) + 1) x 250) + Y + Y - 250) / 2

Now all I had to do was adjust or simplify the above expression, as presented in the email, to Z = (X x 10,000) + Y.

I started by applying the division-by-2 to all elements within the braces.

Z = ((((X x 80) + 1) x 250) / 2) + Y/2 + Y/2 - 250/2

This can be quickly simplified to:

Z = ((((X x 80) + 1) x 250) / 2) + (Y+ Y)/2 - 125

And further simplified to:

Z = ((((X x 80) + 1) x 250) / 2) + Y - 125

Now focusing within the braces, we can apply the division-by-2 to the 250.

Z = (((X x 80) + 1) x 250/2) + Y - 125

Which further simplifies to:

Z = (((X x 80) + 1) x 125) + Y - 125

Now focusing within the braces, we can apply the multiply-by-125 to the two elements.

Z = (((X x 80) x 125) + (1 x 125)) + Y - 125

Which simplifies to:

Z = (((X x 80) x 125) + 125) + Y - 125

And further simplifies to:

Z = (X x 80 x 125) + 125 + Y - 125

And further simplifies to:

Z = (X x 10,000) + Y

Which of course, is what we were after. Now, if only I had a better recall of integrals, matrices and the other joys of high school algebra.