Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.

The n-queens problem is about finding how many different ways queens can be placed on a chessboard so that none attack each other. A mathematician has now all but solved it.

Legend says the Chinese military once used a mathematical ruse to conceal its troop numbers. The technique relates to many deep areas of modern math research.

The concept of dimension seems simple enough, but mathematicians struggled for centuries to precisely define and understand it.