Although Bhargava uses his office primarily for meetings, the mathematical toys decorating its surfaces are more than just a colorful backdrop. When he was a graduate student at Princeton, they helped him solve a 200-year-old problem in number theory.

If two numbers that are each the sum of two perfect squares are multiplied together, the resulting number will also be the sum of two perfect squares (Try it!). As a child, Bhargava read in one of his grandfather’s Sanskrit manuscripts about a generalization of this fact, developed in the year 628 by the great Indian mathematician Brahmagupta: If two numbers that are each the sum of a perfect square and a given whole number times a perfect square are multiplied together, the product will again be the sum of a perfect square and that whole number times another perfect square. “When I saw this math in my grandfather’s manuscript, I got very excited,” Bhargava said.

There are many other such relationships, in which numbers that can be expressed in a particular form can be multiplied together to produce a number with another particular form (sometimes the same form and sometimes a different one). As a graduate student, Bhargava discovered that in 1801, the German mathematical giant Carl Friedrich Gauss came up with a complete description of these kinds of relationships if the numbers can be expressed in what are known as binary quadratic forms: expressions with two variables and only quadratic terms, such as x2 + y2 (the sum of two squares), x2 + 7y2, or 3x2 + 4xy + 9y2. Multiply two such expressions together, and Gauss’ “composition law” tells you which quadratic form you will end up with. The only trouble is that Gauss’ law is a mathematical behemoth, which took him about 20 pages to describe.

Bhargava wondered whether there was a simple way to describe what was going on and whether there were analogous laws for expressions involving higher exponents. He has always been drawn, he said, to questions like this one — “problems that are easy to state, and when you hear them, you think they’re somehow so fundamental that we have to know the answer.”

The answer came to him late one night as he was pondering the problem in his room, which was strewn with Rubik’s Cubes and related puzzles, including the Rubik’s mini-cube, which has only four squares on each face. Bhargava — who used to be able to solve the Rubik’s Cube in about a minute — realized that if he were to place numbers on each corner of the mini-cube and then cut the cube in half, the eight corner numbers could be combined in a natural way to produce a binary quadratic form.

There are three ways to cut a cube in half — making a front-back, left-right or top-bottom division — so the cube generated three quadratic forms. These three forms, Bhargava discovered, add up to zero — not with respect to normal addition, but with respect to Gauss’ method for composing quadratic forms. Bhargava’s cube-slicing method gave a new and elegant reformulation of Gauss’ 20-page law.

Additionally, Bhargava realized that if he arranged numbers on a Rubik’s Domino — a 2x3x3 puzzle — he could produce a composition law for cubic forms, ones whose exponents are three. Over the next few years, Bhargava discovered 12 more composition laws, which formed the core of his Ph.D. thesis. These laws are not just idle curiosities: They connect to a fundamental object in modern number theory called an ideal class group, which measures how many ways a number can be factored into primes in more complicated number systems than the whole numbers.

“His Ph.D. thesis was phenomenal,” Gross said. “It was the first major contribution to Gauss’ theory of composition of binary forms for 200 years.” ]]>