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## Chips Ahoy

Alice, Bonita, and Carmen have just finished playing three rounds of a game. In each round there was only one loser. Alice lost the first round, Bonita lost the second round, and Carmen lost the third round. After each round, the loser was required to double the number of chips of each of the other two players by giving away some of their own chips.

Question 1: After three rounds, each of the women had 24 chips. How many chips did each of the women have at the start?

Question 2: Suppose that each woman had 20 chips at the finish rather than 24. Now how many chips did each of the women have at the start?

Question 3: Give another possible situation for the number of chips each woman has at the end of three rounds of this game. Are there other situations...explain?

Question 4: Give an impossible situation for the number of chips each woman has at the end of three rounds of this game. Are there other impossible situations...explain?

Question 5: Investigate this problem for the following different options:

• 2 players and 2 rounds, 4 players and 4 rounds, ...n players and n rounds?
• 3 players and 6 rounds....n players and m rounds (and what are the conditions on n and m)?

Hint: Take a guess for the starting number of chips....and simulate the three rounds being played...

Can you develop a "backwards" argument, from the end of the three rounds back to the start of the game?

Solution Commentary: Why ruin your fun....check your results with others. Also, if you have a solution, you can easily test it by "role-playing" the three games.