Quiz: Can you describe a square with 1 equation?

I was hanging out with my friends, and somehow the topics wandered into something that prompted a mischievous grin from one of the guys.

“Eh, I have a maths problem for you.” He could probably blind someone with that playful twinkle in his eye.

Can you describe a square with just 1 equation?

“What?” My mind was already racing to solve the problem.

“Just take it as a square in the 2D Cartesian plane. The centre of the square will be at the origin (0,0) to simplify the equation.” He seemed delighted to have stumped me.

I asked if there were any boundary conditions. No. Was the equation elegant? As in fairly simple when looked at, no mangling of terms. He gave it a second, and… yes, it was elegant.

Then he said the most important piece of information so far. “The answer might anger mathematicians. It is probably one of the fundamental differences between an engineer and a mathematician.”

Our conversations left this topic. But my brain was subconsciously still working on it. Then I blurted, “I think I solved it.” I described my way of thinking, and he said that’s along the correct line of thought. Then he just gave me the answer.

Yes, I agree. The answer will probably ruffle the feathers of the math purists.

As of this writing, I have yet to come up with the analytical formula of an equation to describe a square. I will post my findings, and my friend’s answer in a week’s time. You are encouraged to come up with your own solution. Post in the comments. Better yet, write about it in your blog. Tell your friends about it. Show them how awesomely clever you are.

Can you describe a square with 1 equation?

Bonus fun: My friend tried to explain a bit more. “There’s only one equal sign.” I think he was trying to insult me or something…

Update: Find out the answer.


  1. Mike Anderson says:

    How about:

    max(abs(x),abs(y)) = c

  2. Christopher Tay says:

    How about the L1 norm?

    |x| + |y| = C ?

  3. Chris McAloney says:

    Well, Mike Anderson got the one I was thinking of, but abs(x) + abs(y) = c also forms a square, just one that’s rotated by pi/4.

  4. Wow, you guys are good! I wasn’t even thinking of those equations.

    I’m going to meditate on those, and think hard on how to make my theory work too. Actually, mine seems to be quite complicated, but I thought it would be exact.

    Now I feel ashamed of how inelegant my solution is… Still, I’ll work on it, and will post the answer (together with my friend’s answer) in a few days.