The book's first puzzle is solved by Wade playing the arcade game Joust against a bot. Then when he wins, he's dropped into the movie WarGames, replacing Matthew Broderick's character, and he has to act out every scene to progress.
Seriously, that's it.
I will say the second book, The Dark Forest, is much better, but I'd read it several years ago and I haven't come back to it recently.
Whenever I recommend it to people, I just give them a quick synopsis of the first book and that's seemed to be sufficient for them.