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Dealing
Someone (anyone) shuffles the deck, and then each player takes one card and places it face-up on the table. The player with the highest card is the dealer. He shuffles the cards and hands them to the player to his right, who cuts them and returns them to the dealer. The cards are dealt one at a time, starting with the player to the dealer's left and moving in a clockwise rotation until each player has 13 cards.
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Bidding for tricks
In this phase, players bid for the number of tricks they think they can take. (It's like being at an auction.) Because each player has 13 cards, 13 tricks must be fought over and won in each hand. The bidding starts with the dealer and moves to his left in a clockwise rotation. Each player gets a chance to bid, and a player can either bid or pass when it's his turn. The least you can bid is for seven tricks, and the maximum you can bid is for all 13. The bidding goes around and around the table, with each player either bidding or passing until three players in a row say "Pass" after some bid has been made.
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Playing the hand
The player who buys the contract, determined by the bidding, is called the declarer. The declarer is the one who will play the hand. The player seated to the left of the declarer puts down the first card face up in the middle of the table; this is the opening lead. The play moves clockwise. The next player, the dummy, places her cards face-up on the table in four vertical rows, one row for each suit, and completely bows out of the action. In other words, only three people are playing.
Once the lead is on the table, the declarer plays any card from dummy in the suit that was led; third hand does the same, and fourth hand, the declarer, also does the same. Whoever has played the highest card in the suit wins the trick and leads any card in any suit desired to the next trick. The same process goes on for all 13 tricks. The rule is you have to follow suit if you have a card in the suit that has been led. If you don't have a card in that suit, you can throw away (discard) any card you wish from another suit, usually some worthless card. After 13 tricks have been played, each team counts up the number of tricks it has won.
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Scoring
After the smoke clears and the tricks are counted, you know soon enough whether the declarer's team made its contract by taking at least the number of tricks they bid. You then register the score. The deal moves in a clockwise manner; the player to the left of the person who has dealt the previous hand deals the next one.