Tuesday, July 8, 2008
Pirates and Treasure
Five pirates stumble across 1000 gold coins. They decide to divide up the coins amongst themselves in the following way. The first pirate will suggest a distribution of the coins. Let's say he suggests: 100 for pirate 5, 100 for pirate 4, 100 for pirate 3, 100 for pirate 2, and 600 for himself. After making a suggestion, all the pirates will vote on whether they agree to the distribution. If a CLEAR majority agree with the distribution, this distribution stays and the game is over. If not, then the pirates kill the first pirate, and the second pirate suggests a distribution for the remaining four pirates. Once again all the pirates vote and only if there is a clear majority does the second pirate live and the distribution stay. Note that a clear majority means over 50%, so if there are four pirates total then three half to agree for the pirate making the suggestion to live. This continues until a distribution is agreed on by a majority of the pirates. In addition to this set-up, all of the pirates have the same list of priorities. First, each is interested in self-preservation, meaning he will vote yes for a plan if he believes the alternative would result in him eventually dying. Second, the pirates are greedy, meaning that they will vote against a plan if they believe they can get more coins later on. Finally, all the pirates are bloodthirsty. This means that if they believe they can live and get the same number of coins later on, they would rather see another pirate die. Given these rules, what is the most number of coins that the first pirate can suggest for himself to guarantee that the distribution is accepted and he lives?
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ANSWER:
The pirate can keep 997 for himself.
The idea behind the solution to this puzzle is that a pirate will accept a proposal only if he knows that in case he would not accept the proposal, he would get less of the treasure.
If pirate 1 is the only one left, he would get all the golden coins.
If only pirates 1 and 2 are left, pirate 2 would die for sure, since pirate 1 is bloodthirsty and will reject all proposals of pirate 2 (since he will get all coins anyway).
When pirate 3 is also alive, he needs the agreement of one of the other two. Pirate 2 will agree with every proposal since, as we have seen, he would die if he didn't. So pirate 3 should propose to keep everything for himself.
When four pirates are alive, pirate 4 should make two other pirates agree with his proposal. So he proposes to give one coin to pirate 1, one coin to pirate 2, and the rest to himself. Pirates 1 and 2 will accept the proposal, since they are greedy, and if they wouldn't accept, they would get less.
But as we know, there are five pirates. If pirate 5 gives both pirate 1 (or 2) and pirate 3 one coin more than in the previous case, they are willing to accept the proposal. Then a majority (three out of five) of the pirates will support the proposal, and pirate 5 can keep the rest of the treasure to himself.
Conclusion: Pirate 5 should propose to give two coins to pirate 1 (or 2), one coin to pirate 3, and the remaining 997 coins to himself.
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