A defining features of learning mathematics is being able to read, write, and discover proofs. A proof is a method of communicating a mathematical truth to another person who speaks the "language" of mathematics. It's a way of showing whether a statement made is absolutely true or false.

I recently came across a fallacious proof attempting to show that the integer one equals two – shown are the steps to accomplish this. Common sense dictates that this is impossible, but the proof is somewhat convincing to the untrained eye. However, most people with basic algebraic knowledge should be able to pick out the mistake, so take a minute or two to see if you can identify the incorrect step. If you're stuck, don't give up so quickly! It's easier th ...

At some point in your life, you've probably played this game of logic known as the missing square puzzle, or Curry's paradox - named after its inventor Paul Curry. This optical illusion is commonly used in mathematics classes to help students reason about geometrical figures, or rather to teach them to not reason using figures, but only using the textual description thereof and the axioms of geometry. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled triangle, but one has a 1×1 hole in it.

As depicted in the animation, the illusion works because the diagonal of the two internal triangles is not a continuous slope from corner to corner. The overall connected shap ...