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There is absolutely, unequivocally nothing illegitimate about keeping track of seen cards in blackjack.

Ditto.

The casinos in Nevada will send large suited gentlemen over to your table to ask you to play a different game if they think you're counting, because counting gives you a slight (fractions of a percentage) advantage over the house in certain variations of the game. The reason they are allowed to do this has nothing to do with it being bad or illegal or nefarious or anything. It's merely because they can: state law in Nevada says that any business can choose to refuse service to any customer for any reason.

The casinos in Atlantic City are governed by a different state law that says "games of skill" are not allowed in casinos. So they can't kick you out for posessing a skill in the game.

There are ways to essentially negate the effects of card counting, and most casinos employ these wherever possible. At the indian casino that I frequent near here, they deal from a continuous-shuffle machine, where the cards behind the shoe are constantly being randomized and you don't have any starting reference point for the count. The casinos that have hand-dealt decks will train their dealers to count, and when the count goes player-favorable, they will preferentially shuffle the deck at that time.
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Tony Fabris