The McKay conjecture

Math is full of enormously complicated abstract objects that are impossible to study in their entirety. 

But often, mathematicians have discovered, it’s enough to look at a small fragment of such an object to understand its broader properties. 

In the third century BCE, for instance, the ancient Greek mathematician Eratosthenes estimated the circumference of the Earth —roughly 25,000 miles —by measuring shadows cast by the sun in just two cities about 500 miles apart. 

Similarly, when mathematicians want to understand an impossibly convoluted function, they might only need to look at how it behaves for a small subset of possible inputs. 

That can be enough to tell them what the function does for all possible inputs.