I think that most people are familiar with Sudoku puzzles. There are some interesting questions that can be asked such as what is the minimum number of the 81 boxes in the grid that need to be filled in initially (the ‘clues’) in order to be able to obtain a unique solution?
It is known that for at least some puzzles one can obtain a unique solution with 17 clues, and no one had found any puzzles with 16 clues that had unique solutions. But is 17 the minimum number required? If you could show that none of the 16-clue puzzles had unique solutions, then you would have proven that the minimum number was 17.
One way to prove it is by brute force, using a computer to attempt to solve every possible 16-clue combination. But it turns out that the number of such possibilities is about 3.3×1016, much less than the 6.7×1021 total number of possible Sudoku arrangements but still staggeringly large, and this would require a huge amount of computer time, about 300,000 years.
So mathematician Gary McGuire used the various symmetries that exist in these puzzles to reduce the number of different puzzles to be solved by a computer. Even then the final number of puzzles took 7 million computer hours and took about a year to carry out. You can read his paper here and an article about it here.
Here is an entertaining video from mathematician James Grime explaining the problem and how it was solved.