Frederick Giarrusso | Brain Teaser: Proof That 2 = 1

This is a classic brain teaser I learned in middle school (or “junior high school” where I grew up).

Let’s prove that:

$2 = 1$

Here’s the proof:

Let’s start by establishing some variables:

$a = 1$

$b = 1$

Okay, so if that’s true, then we know:

$a = b$

In algebra, we learned we could multiply both sides of the equation, so:

$a \times a = b \times a$

$a^2 = b a$

And of course, we can also subtract a number from both sides of the equation:

$a^2 - b^2 = ba - b^2$

Now, we can factor that equation to get:

$(a + b)(a - b) = b(a - b)$

Of course, $(a - b)$ cancels out, so:

$(a + b) = b$

Putting back in the numbers, we see that:

$(1 + 1) = 1$

$2 = 1$

And there is our proof!

To misquote Groucho Marx:

“An eighth grader could understand that! Someone fetch me an eighth grader!”

2 Comments

Rob Brown on July 13, 2012 at 5:26 pm

When I was a math teacher, I gave my students the following little ditty:

In your attempt to prove your theorem
And show you are the hero
Be careful how you do your math
And don’t divide by zero.

🙂

Brain Teaser: Proof That 2 = 1