# Unbelievable Math Problem

During my first year of teaching one of my Algebra I students printed out this chain email and brought it to me to solve:

I don’t remember the exact conversation, but I do remember that it was something along the lines of

“My mom and sister and I spent the weekend trying to do this and we bet you can’t figure it out!”

Just in case you can’t read the problem in the photo, I’ll re-type it for you (but I have to admit I added the missing capitalization and punctuation . . . I just couldn’t leave alone!):

Wow, this is spooky.

UNBELIEVABLE MATH PROBLEM

Here is a math trick so unbelievable that it will stump you.  (Or at least it stumped me and I have a degree in math!)  Personally I would like to know who came up with this and where they had the time to figure this out.  I still don’t understand it!

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.

5. Multiply by 250.

8. Subtract 250.

9. Divide the number by 2.

Do you recognize the number?

*Disclaimer about this email . . . I was very motivated to explain to the student how this worked, because I’m not a fan of the whole “math is incredibly complicated and hard to figure out,” movement that seems to sweep across quite a few mathematics classrooms.  (We’ve talked about that in my “Calculating Tips is Calculus” post).

Anyway, let’s get to the bottom of this so-called “math trick.”  If you haven’t already go ahead and follow the steps in the email.  If you’re doing this on a calculator, as the email suggests be sure to press “enter” or “=” at the end of each step.

So, what happened?  Do you recognize the number?  It’s your phone number, right?

Hmmm, I wonder if this works with everyone’s phone number?  Try your mom’s phone number, or your gramma’s phone number, or your best friend’s phone number . . . does the trick work with their numbers too?

Well that’s tricky, isn’t it? (or is it?!?)

Let’s start at the beginning of the math trick email.  You start by entering the first 3 digits of your phone number.  Now, we know this trick already works for your phone number, and the other phone numbers you tested, but let’s see if we can figure out why/if it works for all phone numbers.  Instead of a specific number, I’m going to say the first 3 digits of my phone number are:

#\$%

This is important, do you know why I’m doing this (this being symbols instead of numbers)?

No?  Let me tell you; If I pick the numbers 123 as the first three digits of my phone number, I’m not explaining why this math trick works for all phone numbers.  I’m only showing that this math trick works for phone numbers that have 123 as their first 3 digits.  If I wanted to assign the first 3 digits actual numbers, I’d have to test this math trick for all possible combinations of the first three numbers!  That a total of 729 different first three numbers . . . that’s a lot of “math trick” tests (and it doesn’t even include the different combinations of first 3 digits, plus last 4 digits . . . that’d be a total of 4,782,969 different combinations to check this math trick for).  So while it seems funny to use symbols, since the symbols can represent any first 3 digits it ends up saving me a lot of time in the long run.

So, step 1: Grab a calculator (I won’t be able to use a calculator if I’m going to replace the digits with symbols, so I’ll grab a piece of paper instead).

Step 2: Instead of keying in the first 3 digits of my phone number, I’m going to use the symbols.

Step 3: Multiply by 80: 80(#\$%)

Step 4: Add 1: 80(#\$%) + 1

Step 5: Multiply by 250: 250[80(#\$%)+1)]–OK at this point the expression is getting a little ugly, so let’s go ahead and use the distributive property to simplify this mess!  When doing that, I get:

20,000(#\$%)+250

Step 6: Add the last 4 digits of your phone number: I’m going to use ^&*@ as the last 4 digits.  This means now I have:

20,000(#\$%)+250+^&*@–I know, I know its getting a little weird just stick with me!

Step 7: Add the last 4 digits of your phone number again: OK at this point I’m adding another round of ^&*@, giving me 2^, 2&, 2*, and 2@ so, I’m writing this expression as:

20,000(#\$%)+250+2(^&*@)

Step 8: Subtract 250: Thank goodness, that 250 was getting to be annoying!  Now you have

20,000(#\$%) + 2(^&*@).

Step 9: Divide by 2:  Well, what do you know?!?  The coefficient of #\$% and ^&*@ are both divisible by 2!  That means I’m left with

10,000(#\$%) + ^&*@

That’s a phone number!  Look, you take the first three digits of a phone number and multiply by 10,000.  Remember, multiplying anything by a power of 10 just moves the decimal point, so 10,000(#\$%) = #\$%0000 and what do you know; when I take #\$%0000+^&*@, I get #\$%^&*@–which is my phone number!

But, I have a challenge for you: Change the math trick so that it works if the person includes their area code.  When you come up with one, let me know!  I’d love to see it!