Karim and the Genie: Karim was taking a nap under a tree. He had a dream about a magical lamp and a genie... [Story about doubling coins and paying 8 coins each round] (i) How many coins did Karim initially have? (ii) For what cost per round should Karim agree to the deal, if he wants to increase the number of coins he has? (iii) Through its magical powers, the genie knows the number of coins that Karim has. How should the genie set the cost per round so that it gets all of Karim's coins?
Karim initially had 8 coins.
Karim should agree to a cost per round that is less than his starting number of coins (c < S).
The genie should set the cost per round at 8S/7 (where S is Karim's starting coins).
Explanation
(i) Working backwards or using algebra: 8S - = 8, giving S = 8. (ii) For Karim to profit: 8S - 7c > S, which simplifies to 7S > 7c, or S > c. (iii) For genie to get all coins: 8S - 7c = 0, so c = 8S/7. This ensures Karim ends with exactly 0 coins after 3 rounds.