Usually, you can verify a solution to a problem. Whether it’s using multiplication for division or plugging the answer in for a variable, math teachers tell you to check your work using your answer in every school math class.

But let’s say you can verify a solution easily, is it just as easy to solve for that solution?

This is the P versus NP problem, a Millenium Prize Problem where the solver will receive a million dollars if valid proof is provided.

What is P versus NP?

In computer science, the efficiency of algorithms is very important. Most algorithms are believed to be “fast” if solvable in a standard called polynomial time. Polynomial time is when a problem is solvable in steps scaled by a factor of a polynomial given the complexity of input. So let’s say the complexity of input is some number n, a polynomial time algorithm will be able to solve a problem in nk steps.

Essentially, P vs NP is asking the question: Are problems that can have solutions verified in polynomial time, also have their answers solved in polynomial time?

NP-Completeness

An Euler Diagram showing the cases for NP-Completeness for P ≠ NP and P = NP. Credit: Behnam Esfahbod, Wikimedia Commons (CC