Cracking The Algebra Nut 

By Mike O’Hern, Center Director of Mathnasium of West Knoxville

My students think I work for peanuts.  Literally.  Lately I’ve begun the habit of having peanuts for a snack in the afternoon, and I guess I’m the type to become somewhat predictable because at around 3:50 or so some of them will start saying, “it’s almost time for your four o’clock peanuts!”

Anyway, as I was eating my “four o’clock peanuts” the other day, I saw a fun little math trick.  It went like this.  Think of a number between 1 and 10.  Now double it.  Now add 10 to that.  Now cut that in half.  Now subtract your original number.  Your number has been magically transformed into 5!

Now I’m not “Super Math Guy” or anything, but I have to say that one really isn’t that tough to figure out.  If we were bold enough to start talking about algebra, we might approach this using a variable like, say, N, standing for “number.”  But we aren’t that bold because even uttering the word “algebra” will scare off readers.  So instead we’ll use P, standing for peanut.

Start with a number: P

Double it: 2P

Add ten: 2P + 10

Divide by two: (2P + 10)/2 = P + 5

Subtract original number: P + 5 – P = 5

Voila!  No matter what the original number is, the answer is five.  If you get it, you get a peanut!

But if you want to really impress someone I’ve stumbled upon one that isn’t nearly as obvious as to how it’s done.  Think of two numbers between 1 and 10.  Add them together.  Multiply that number by 10.  Add the larger original number, then subtract the smaller original number.  Tell me the number you’ve ended on and I’ll tell you the two original numbers.

If we were bold enough to start talking about algebra, we might approach this using a variable like, say, N, standing for “number.”

Now you need to do a couple of quick mental math steps.  I’ll show you by using an example.  126 is their final number.  First add the last digit to the previous digits, 6 + 12 = 18.  Then cut that in half to get 9.  This is the larger of their two original numbers.  Now subtract the last digit of their answer from 9 to get their other original number, 9 – 6 = 3.  They started with 9 and 3!  Try again with 106.  (10 + 6)/2 = 8.  8 – 6 = 2.  Starting numbers were 8 and 2.

Now to figure out how all of this works, we start with a peanut snack.  No, sorry, mind wandering.  I’ll put the steps here, then walk you through it.

Start with two numbers: P and N

Add them together: P + N

Multiply by ten: 10(P + N)

Add the larger: 10(P + N) + P

Subtract the smaller: 10(P + N) + P – N

Now that their calculations are done, notice a couple of things in that last line.  First, the first part being multiplied by 10 guarantees a zero at the end of the number, right?  So when we do our mental calculation, what we’re calculating is really (P + N + P – N)/2.  Surely you’ve noticed that the N goes away (add and N, then take it away), leaving 2P/2, which is P – voila!  The larger number!

And because we guaranteed a zero at the end for the first part, we also guaranteed that the last digit of their calculated number would be P – N.  So when we subtract the last digit of the calculated number from the larger number, we’re calculating P – (P – N) which leaves us with N – the smaller number!  (Go back to the Knoxville Parent archive in your basement and look over the January 2015 issue if you don’t get that last step.)

Mike O’Hern, Center Director of Mathnasium of West Knoxville, earned his Bachelor’s Degree in Metallurgical Engineering at the University of Tennessee, Knoxville in 1988. He pursued graduate studies in Materials Science & Engineering while on the Research Staff at Oak Ridge National Laboratory. Mike has had a life-long love of mathematics and teaching, and feels that math is not about learning to be ready for the next math class – it’s about learning to think.