Analyzing Mind Reader

Try this trick a few times, and you'll realize that it doesn't matter what numbers you or your friend choose. The trick works no matter what.

Here's why. In a two-digit number, the first number (the number on the left) counts the number of sets of 10—the 10’s place—and the second number (the number on the right) counts the number of 1’s—the 1’s place. The number 38, for example, has three 10’s and eight 1’s.

All the math you told your friend to do was just a roundabout way of putting your friend's chosen number in the 10’s place and your number in the 1’s place.

To see what's really going on, let's go through the steps, one at a time.

Because it doesn’t matter what number your friend chooses, we’ll just represent it with a symbol, a . This triangle could be any number between 1 and 9.

First, you told your friend to double his or her number. That’s two times his number, or two times . You can write it mathematically like this:


2 x

Next you told your friend to add 2:

(2 x ) + 2

Then you told your friend to multiply this by 5:

5 ((2 x ) + 2)


This looks pretty complicated now, but we can simplify it by distributing the 5. That means multiplying the 5 times each number inside the (parentheses)



. . . is the same as this:

. . . is the same as this:

5 * ((2 * ) + 2)

5 * (2 * ) + 5*2

(10 * ) + 10


In the last step, you had your friend subtract the difference between 10 and your number. Suppose your number was 1. You ask your friend to subtract 9, the difference between 10 and 1.

(10 x ) + 10 - (10 - 1)
(10 x ) + 10 - (9)
(10 x ) + 1

What you end up with is your friend’s number times 10, plus your number—the same numbers that are on the cards when you flip them over!
Algebra - The Great Unknown
When numbers in math problems are represented by symbols, it’s called algebra. What’s great about algebra is it lets you find answers to math problems even when you don’t know all the numbers involved. (In this problem, your friend’s number and your number could be anything from 1 to 9.) Usually, unknown numbers in algebra problems are represented with letters, not triangles. But it works just the same.

X is the letter most commonly used to represent an unknown quantity. Have you ever heard of a TV show called "The X-Files"? It’s a show about weird, unexplainable things like alien creatures and people with strange powers. Just like in algebra, the X in "The X-Files" stands for the unknown.


to Mind Reader


For tips on how to use this trick with a class or group, go to the Teacher/Leader version.
Math Explorer Home


© Exploratorium