Imagine you are hanging out with one of your friends--let's say a guy named Elliott. Elliott claims that he is the fastest person in your group of friends. This is a good example of a conjecture--Elliot believes that it's true, and you also believe it might be true. Still, you challenge Elliot to a race to the end of the street anyway.

One of two things could happen: you could beat Elliot in that race, or he could beat you. If you beat Elliot, you have a counterexample to his claim. If he beats you, though, he hasn't proven his claim. Elliot will have provided a single example of his claim, showing that he's faster than one in the group--you. However, he would need to beat every single one of your friends in a race to prove that he is the fastest person in your group. If none of the others are around to challenge Elliot, because, for instance, they're running in a track meet, we can never know for sure if Elliott is the fastest.

In mathematics, it is often quite easy to come up with examples that support a conjecture you've made. You might even have 1,000 examples of the claim The sum of two even numbers is even (e.g. 12+246 = 258). However, if all you can do is find examples to support that claim, the conjecture cannot be proved. To prove a conjecture, you have to remove all doubt, and logic is needed to accomplish that.

For some conjectures, you can use logical rules to create a generic way to represent a situation and then prove that it is true always, in all situations. If you follow the rules of logic and no one can find an error in your work, then your conjecture has a proof. A proof starts with a conjecture but includes all of the logical steps that were used to prove that it is a true statement.