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player A : 4,4
player B : A,10
board : K,K,Q,Q,2

Which player wins? I guess Player B but player A claims he has a pair of 4 which is a bigger kicker. who is correct?

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Player B wins. All that matters is the best 5 card hand you can make with the combined 7 cards. So player A does NOT have a pair of 4 as a kicker... player A has K,K,Q,Q,4. Player B has K,K,Q,Q,A. Both identical 2 pairs, but player B has A which beats 4.

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The best "hand" of five cards wins. That is two pair: KK QQ A kicker, and player B wins.

Player A has a pair of fours that "doesn't play" because there are already two pairs on the board (and no room for a third pair in a five card hand). But player B's ace kicker does play (and beat's Player A's 4-kicker).

Change the board to KKQ32, and Player A wins, because he has two pair, kings over fours, with a Q kicker, while player B has one pair, KK with an ace kicker. The second queen "counterfeited" Player A's second pair by giving player B a second (higher) one.

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