New Scientist

Rubik’s Quadratic Forms

How a toy inspired an award-winning mathematician’s Eureka moment.

Rubik’s cubes can be life changing. 

Photo courtesy of Rubik’s

This article originally appeared in New Scientist.

Manjul Bhargava has just won a Fields medal, the most coveted prize in mathematics. Now 40, he was one of the youngest people to be made a full professor at Princeton University, at age 28. He explains the link between Rubik’s cube and his groundbreaking work in number theory.

Does the Fields medal mean more to you than any other award you have won?

Any award is a milestone, which encourages one to go further. I don’t know that I think of any award as meaning more to me personally than any other. The mathematics that led to the medal was far more exciting to me than the medal itself.


The award citation says that you were inspired to extend Carl Friedrich Gauss’ law of composition in an unusual way. Can you explain what that is, and what you did?


Gauss’ law says that you can compose two quadratic forms, which you can think of as a square of numbers, to get a third square. I was in California in the summer of 1998, and I had a 2-by-2-by-2 mini Rubik’s cube. I was just visualizing putting numbers on each of the corners, and I saw these binary quadratic forms coming out, three of them. I just sat down and wrote out the relations between them. It was a great day!

Have any of your other discoveries had unusual origins?


I do tend to think about things very visually, and the Rubik’s cube is a concrete example of that visual approach. But that one is probably the most unusual and unexpected origin of all.

You have proved several theorems. Do you have a favorite?

Mathematicians often say that choosing a favorite theorem is like choosing one’s favorite child. Although I don’t yet have any children, I understand the sentiment. I enjoyed working on all the theorems I have proved.

Are there any mathematicians, living or dead, that you have particularly looked up to?

My mother [Mira Bhargava, a mathematician at Hofstra University in Hempstead, New York] has been a source of inspiration to me from the very beginning. She was always there to answer my questions, to encourage and support me, and she taught me how much the human mind is capable of.


This year a woman, Maryam Mirzakhani at Stanford University, has finally won one of the Fields medals, after 52 consecutive male winners. How do you view this achievement?

This is long overdue! Hopefully in a few years we will not even need to discuss this, as more and more females receive the award. I am honored to be a recipient in the same year as Maryam. It has been a pleasure to know her—we overlapped for a year early in our careers at Harvard, and later at Princeton. Her work is absolutely fantastic. I hope the media will not speak of her only as a top-rate female mathematician, but also as a top-rate mathematician who is doing truly groundbreaking work.