Give an example where a mathematician, unable to solve a certain math problem, ended up solving a related but different one.

Sure! Here is an example where a mathematician, unable to solve a certain math problem, ended up solving a related but different one:

In the early 20th century, the mathematician David Hilbert posed a famous problem known as Hilbert's tenth problem. This problem asked whether there exists an algorithm to determine whether a given polynomial equation with integer coefficients has integer solutions. For many years, mathematicians struggled to find a solution to this problem.

However, in the 1970s, a mathematician named Yuri Matiyasevich made a breakthrough. He was not able to directly solve Hilbert's tenth problem, but he was able to prove a related result known as Matiyasevich's theorem. This theorem states that there is no algorithm to determine whether a given Diophantine equation has integer solutions.

A Diophantine equation is a polynomial equation with integer coefficients that asks for integer solutions. Matiyasevich's theorem essentially showed that Hilbert's tenth problem was undecidable, meaning that there is no general algorithm that can solve it.

So, while Matiyasevich did not directly solve Hilbert's tenth problem, his work on a related problem led to a significant result in the field of mathematics. This example demonstrates how mathematicians sometimes make progress by working on related problems, even if they are unable to solve the original problem they set out to tackle.