I really think that Tolkien just put chess in there because it makes middle earth seem like it actaully did exist at one point. Same with the armor and weapons as Bombur said. It just seems more realistic. Just because tolkien puts chess in ME, doesn't mean tolkien invented it.
Quote:
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He charged the ranks of the goblins of Mount Gram in the Battle of the Green Fields, and knocked thier king Golfimbul's head clean off with a wooden club. It sailed a hundred yards through the air and went down a rabbit-hole, and in this was the battle was won and the game of Golf invented at the same moment.
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This quote is taken from The Hobbit in the first chapter. It's talking about Bullroarer inventing the game of Golf. Of course this is not true. There was no ME and therefore, there was never a goblin who golf was named after. But the point is that adding things (in this case games) from our world into ME only makes it seem more realistic.