You are quite right Selmo (although it's found in "Flies and Spiders"), in The Hobbit there's a whole list of "quiet games of the aiming and throwing sort"that Hobbits play: quoits, dart-throwing, shooting at the wind, bowls, and ninepins. Now I have no idea what half of those are, but still, they're nice Hobbit-games.
The Moria orcs might've played "Pin the wings on the Balrog".
(well,
I think it's funny...)
I'm in the process of making up a hand game for the little girls of Minas Tirith to play, though I doubt that counts as a sport.