Catching the snitch requires a combination of trivia knowledge and the ability to solve a small logic puzzle.
Imagine a 13x13 grid, within which the seekers will fly. At the beginning of the game, the seekers get the introduction to the puzzle, and the question that has to be answered. The seekers start in opposite corners, and they move one square each turn. They can move up, down, left, or right, into a new square of the grid.
Some squares will be empty. Most, however, will not be. The majority of the squares on the grid are Trivia Squares. When a seeker enters a Trivia Square, it triggers a trivia question. The question looks the same as it would during Quaffle play - the countdown, etc. Same rules apply: one letter rule, etc. Both seekers answer when a trivia question is triggered, not just the one who entered the square. When a seeker "wins" a trivia question, he gains a "trivia point". Once a seeker reaches 5 trivia points, the location of the next clue to the logic puzzle is revealed to him.
The clues, then, are on a second type of square - the Clue Square. While it is possible to accidentally or randomly land on a Clue Square and get a clue to the solution of the logic puzzle, it's unlikely - the best way to do it is to correctly answer enough trivia questions so that the seeker can get the location and deliberately "fly" there. The clues all add up to one single answer - there aren't separate answers for the various clues. There are always 5 clues for a puzzle, so there are 5 Clue Squares.
After every 10 minutes of play, the 2 seekers get a 2 minutes break to work on the puzzle. They may still work on it during regular Snitch play, but it's much more difficult!
Once a seeker has solved the puzzle, they submit their answer to the referee. If the seeker has solved it correctly, they obtain the location of the Snitch Square. Part of the challenge of being a seeker is multitasking - the seeker must to continue with the trivia part of the game and solve the puzzle at the same time. Seekers cannot answer the puzzle until they find the final clue. If a seeker attempts to answer the puzzle and solves it incorrectly, they may not submit their answer again until the other seeker has attempted an answer OR until 17 minutes have passed.
UNLIKE the Clue Squares, it is impossible to accidentally land on the Snitch Square ... it's not enabled until the Seeker solves the puzzle; before then it looks like an Empty Square.
In addition to the Empty Squares, Clue Squares, the Snitch Square, and Trivia Squares, there are two more types of squares... First, there's the Bludger Square. Landing on it loses the Seeker three turns. A bludgered seeker can't move, but can still answer trivia questions. And finally, there's the Wronski Feint Square - there are only two of these on the board. Landing on this one causes the opposing seeker to lose 5 turns.
Landing on the Snitch Square (after having solved the puzzle) ends the Quidditch match immediately and gives 50 more points to the team of the Seeker who caught the Snitch.