Pirates want Gold
Five pirates have obtained 100 gold coins and have to divide up the loot. The pirates are all extremely intelligent, treacherous and selfish (especially the captain).
The captain always proposes a distribution of the loot. All pirates vote on the proposal, and if half the crew or more go "Aye", the loot is divided as proposed, as no pirate would be willing to take on the captain without superior force on their side.
If the captain fails to obtain support of at least half his crew (which includes himself), he faces a mutiny, and all pirates will turn against him and make him walk the plank. The pirates start over again with the next senior pirate as captain.
What is the maximum number of coins the captain can keep without risking his life?
The answer is 98.
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Let the pirates (in decreasing order of superiority) be Rob, Sansa, Arya , Bran, Rickon. Now , working backwards: 2 Pirates ( Bran and Rickon): Bran splits the coins 100 : 0 (giving himself all the gold). His vote (50%) is enough to seal the deal.
3 Pirates (Arya, Bran, Rickon): Arya splits the coins 99 : 0 : 1. Rickon will accept this deal (getting just 1 coin), because he knows that if he rejects the deal there will be only two pirates left and he gets nothing.
4 Pirates (Sansa, Arya, Bran, Rickon): Sansa splits the coins 99 : 0 : 1 : 0. By the same reasoning as before, Bran will support this deal. Sansa would not waste a spare coin on Arya, because Arya knows that if she rejects the proposal, she will pocket 99 coins once Sansa is thrown overboard. Sansa would also not give a coin to Rickon, because Rickon knows that if he rejects the proposal, he will receive a coin from Arya in the next round anyway.
5 Pirates (Rob, Sansa, Arya, Bran, Rickon): Rob splits the coins 98 : 0 : 1 : 0 : 1. By offering a gold coin to Arya and Rickon (who would otherwise get nothing) he is assured of a deal.