# Professor Knowfair's square dissection assignment

While at the public library with professor Knowfair, he also told me his plans for this week's assignment. He said:

Dissecting a triangle is all fine and dandy, but squares are the real deal. For next week's assignment, I'm going to ask the $n$-th student to dissect a square into $n$ equal-area triangles.

You can't help but wonder: this time, is the professor fair? How many students will be able to complete their tasks?

## 1 Answer

I believe the answer is

The professor is unfair - only students where n is even can complete their tasks.

because

For all even n, divide the square on the diagonal, then use the solution to the previous triangle puzzle on each half symmetrically. For odd n, 1 and 3 are clearly impossible. I suspect that pattern continues, although I did not verify it any further.

Edit: Monsky's Theorem apparently states this. How convenient!

• Even if the pattern didn't hold for every odd, the Professor would still be unfair to at least 2 students, so he'd be unfair in general. Commented Mar 22, 2016 at 1:00
• Well, I thought it'd take longer. Commented Mar 22, 2016 at 2:00