(b) Provide a good allocation of the sexes to the tables. 18. The game or puzzle of Clue is played with a deck of 21 cards. At the beginning of a game, three of the cards are randomly selected and placed face down in the center of the table. The remaining cards are distributed face down as evenly as possible to the players. Each player may look at his or her own cards. The object of the game is to correctly guess the three cards in the center. At each player's turn, the player is allowed to either guess the identity of the three cards in the center or ask any other player a question of the form: "Do you have any of the following three cards?" (The asking player then publicly lists the three cards.) If the asked player has one of the three identified cards, then the asked player must show one of the cards to the asking player (and only to the asking player). Otherwise, the asked player simply responds "No". If a player correctly guesses the three cards in the center, then that player wins. If a player incorrectly guesses the three cards in the center, the player is out of the game. Deductions about the identity of various cards can be made if we define X(i 1 if player i has card j, else 0 Arbitrarily define the three cards in the center as player 1. Thus, we can initially start with the constraints 21 2X1,13 For each card,j = 1, 2, , 21 : (a) Suppose player 3 is asked: "Do you have either card 4, 8, or 17?" and player 3 responds (b) Suppose in response to your question in (a), player 3 shows you card 17. What constraint (c) What LP would you solve in order to determine whether card 4 must be one of the cards "No." What constraint can be added? can be added? in the center? Note, in the "implementation" of the game marketed in North America, the 21 cards are actually divided into three types: (i) six suspect cards with names like "Miss Scarlet"ii) six weapons cards with names like "Revolver," and (c) nine room cards with names like "Kitchen." This has essentially no effect on our analysi s above (b) Provide a good allocation of the sexes to the tables. 18. The game or puzzle of Clue is played with a deck of 21 cards. At the beginning of a game, three of the cards are randomly selected and placed face down in the center of the table. The remaining cards are distributed face down as evenly as possible to the players. Each player may look at his or her own cards. The object of the game is to correctly guess the three cards in the center. At each player's turn, the player is allowed to either guess the identity of the three cards in the center or ask any other player a question of the form: "Do you have any of the following three cards?" (The asking player then publicly lists the three cards.) If the asked player has one of the three identified cards, then the asked player must show one of the cards to the asking player (and only to the asking player). Otherwise, the asked player simply responds "No". If a player correctly guesses the three cards in the center, then that player wins. If a player incorrectly guesses the three cards in the center, the player is out of the game. Deductions about the identity of various cards can be made if we define X(i 1 if player i has card j, else 0 Arbitrarily define the three cards in the center as player 1. Thus, we can initially start with the constraints 21 2X1,13 For each card,j = 1, 2, , 21 : (a) Suppose player 3 is asked: "Do you have either card 4, 8, or 17?" and player 3 responds (b) Suppose in response to your question in (a), player 3 shows you card 17. What constraint (c) What LP would you solve in order to determine whether card 4 must be one of the cards "No." What constraint can be added? can be added? in the center? Note, in the "implementation" of the game marketed in North America, the 21 cards are actually divided into three types: (i) six suspect cards with names like "Miss Scarlet"ii) six weapons cards with names like "Revolver," and (c) nine room cards with names like "Kitchen." This has essentially no effect on our analysi s above