Cake Cutting for Three or more People Name Math is a piece of Cake This activity includes to procedures that result in fair division of an object when there are more than two parties. As you learn about these methods, you may want to use your clay to model the scenarios and discuss implications with your classmates. Lone Divider. This procedure is for three players. Player A cuts the cake into three pieces that he or she believes are of equal size. Player B makes it known to the group of one piece of the cake that he or she approves of and one piece that he or she disapproves of (feels neutral about the third piece). Player C repeats the approval disapproval rating based on his or her own preferences. If players B and C approve of different pieces, then each get their approved picce and the remaining piece goes to the player who cut the cake. If players B and C approve of the same piece, then the player who cut receives one piece that players disapprove of. The remaining two pieces are then put back together and the divide-and-choose method is conducted between players B and C to allocate the rest of the cake to these players. Last Diminisher. This procedure is for four or more players. Player A cuts a piece of cake that he or she believes represents one-fourth (or in for players) of the cake. The piece is handed to player B. Player B evaluates the piece. If he or she believes it is one- fourth or less the piece is passed to player C. If he or she believes it represents more than one-fourth of the whole cake, player B trims the excess from the slice, placing the excess back on the larger piece of cake, and passing the newly cut slice on to player C. Player C proceeds exactly as player B. passing the slice of cake on to player D. The last player must then decide if the piece is acceptable. If it is, this player takes the piece of cake and leaves the game. If this player does not think this piece is at least one-fourth of the entire cake he or she passes it back to the last player to cut from it. Hence, if player B trimmed and player C did not, the piece will go to player B, who will exit the game with this piece of cake. If neither player trims from it, the piece will go back to player A and player A will exit the game. A second picce of cake is now cut from the original cake that represents one-fourth of the whole and the procedure begins again exactly as above. When two players are left in the game, the divide-and- choose method is used to divide the remaining portion of 1. What seems to be the method of choice when only two players remain the game? Why do you suppose that is true? 2. Who is the only player who does not get to choose the specific picce of cake he or she wants during the lone divider method? Why is this fair? 3. Could the lone divider method be extended to four players? If so, what would it look like? If not, explain why. 4. Suppose that during the lone divider method, the cake is cut into three pieces, X, Y, and Z. Now suppose that player B believes that pieces X and Y are equally large in size (say four- ninths) and slice Z is smaller (say one-ninth). Player B chooses X as his or her approved piece. Player C also approves of piece X believing it is four-ninths of the whole cake. Player C, however, believes that piece Y is about two-ninths and piece Z is about three-ninths (or 1/3) of the cake. How can this scenario result in player B believing that he or she did not end up with at least one-third of the original cake? 5. If you were player B or Chow would you decide whether or not you would trim from the piece being passed around during the last diminisher method? If you decide to trim, what would be your strategy? 6. Again consider the last diminisher procedure. Would the last person ever decide to trim from the piece being passed to them? Explain your answer. 7. What happens to the trimmings that may exist during the last diminisher procedure? 8. Could the last diminisher method work for only three people? Explain why or why not. Cake Cutting for Three or more People Name Math is a piece of Cake This activity includes to procedures that result in fair division of an object when there are more than two parties. As you learn about these methods, you may want to use your clay to model the scenarios and discuss implications with your classmates. Lone Divider. This procedure is for three players. Player A cuts the cake into three pieces that he or she believes are of equal size. Player B makes it known to the group of one piece of the cake that he or she approves of and one piece that he or she disapproves of (feels neutral about the third piece). Player C repeats the approval disapproval rating based on his or her own preferences. If players B and C approve of different pieces, then each get their approved picce and the remaining piece goes to the player who cut the cake. If players B and C approve of the same piece, then the player who cut receives one piece that players disapprove of. The remaining two pieces are then put back together and the divide-and-choose method is conducted between players B and C to allocate the rest of the cake to these players. Last Diminisher. This procedure is for four or more players. Player A cuts a piece of cake that he or she believes represents one-fourth (or in for players) of the cake. The piece is handed to player B. Player B evaluates the piece. If he or she believes it is one- fourth or less the piece is passed to player C. If he or she believes it represents more than one-fourth of the whole cake, player B trims the excess from the slice, placing the excess back on the larger piece of cake, and passing the newly cut slice on to player C. Player C proceeds exactly as player B. passing the slice of cake on to player D. The last player must then decide if the piece is acceptable. If it is, this player takes the piece of cake and leaves the game. If this player does not think this piece is at least one-fourth of the entire cake he or she passes it back to the last player to cut from it. Hence, if player B trimmed and player C did not, the piece will go to player B, who will exit the game with this piece of cake. If neither player trims from it, the piece will go back to player A and player A will exit the game. A second picce of cake is now cut from the original cake that represents one-fourth of the whole and the procedure begins again exactly as above. When two players are left in the game, the divide-and- choose method is used to divide the remaining portion of 1. What seems to be the method of choice when only two players remain the game? Why do you suppose that is true? 2. Who is the only player who does not get to choose the specific picce of cake he or she wants during the lone divider method? Why is this fair? 3. Could the lone divider method be extended to four players? If so, what would it look like? If not, explain why. 4. Suppose that during the lone divider method, the cake is cut into three pieces, X, Y, and Z. Now suppose that player B believes that pieces X and Y are equally large in size (say four- ninths) and slice Z is smaller (say one-ninth). Player B chooses X as his or her approved piece. Player C also approves of piece X believing it is four-ninths of the whole cake. Player C, however, believes that piece Y is about two-ninths and piece Z is about three-ninths (or 1/3) of the cake. How can this scenario result in player B believing that he or she did not end up with at least one-third of the original cake? 5. If you were player B or Chow would you decide whether or not you would trim from the piece being passed around during the last diminisher method? If you decide to trim, what would be your strategy? 6. Again consider the last diminisher procedure. Would the last person ever decide to trim from the piece being passed to them? Explain your answer. 7. What happens to the trimmings that may exist during the last diminisher procedure? 8. Could the last diminisher method work for only three people? Explain why or why not