Pirate loot problem
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The Pirate loot problem is a lateral thinking puzzle that involves logic and strategic thinking to solve. The problem can be varied to include any number of pirates, and any number of gold pieces. This article will present the problem with five pirates and one hundred pieces of gold.
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[edit] The problem
Imagine you are the oldest in a band of five pirates, where no two pirates in the group are of the same age. It is your responsibility as the oldest pirate to decide how to divide the group's booty, which comprises one hundred pieces of gold. Once you announce how you will allocate the gold, all the pirates in the group (including you) vote either "yes" or "no" on your decision. If at least half of the votes are "yes", the pirates divide the loot as you directed, after which everyone carries on about his business. But, if more than half of the votes are "no", you are then killed, and the task of deciding the allocation of gold lies with the next oldest pirate in the group.
Knowing that all the other pirates are just as smart, and just as logical as you, and furthermore, knowing that they all want the largest amount of gold they can get for themselves, how do you divide the gold so that you get the maximum amount of gold pieces possible, and guarantee that you will receive enough "yes" votes to stay alive?
[edit] The solution
The first step in solving the problem is to imagine the situation with only two pirates. Knowing how the older pirate will divide the loot, and understanding both pirates' viewpoints in that situation, will then form the basis of the solution to the problem when there are three pirates, and then when there are four pirates, and then finally when there are five pirates.
[edit] Two pirates
In the event that there are two pirates (the second youngest being pirate B, and the youngest being pirate A), pirate B can fearlessly announce that all the gold will go to him. Pirate A will undoubtedly vote "no" on this decision, but pirate B's "yes" vote is at least half of the votes, and suffices enough for the decision to go through. Observing the problem from each pirate's point of view, we can conclude that a situation where there are only two pirates is very advantageous for pirate B.
[edit] Three pirates
This is precisely the reason why, in a situation where there are three pirates (pirate C being the oldest), pirate B wants pirate C dead. So, pirate C will not bother allocating any gold to pirate B, as no amount of gold besides the full amount can convince pirate B to vote "yes" (and pirate C will certainly not opt to give the full amount to pirate B, as he is trying to receive as much as he can for himself). Pirate A's "yes" vote is therefore crucial, so pirate C will allocate one gold piece to pirate A. Pirate A will agree to this, and vote "yes", because that is one more gold piece than what he would receive in the previous situation, with just him and pirate B.
[edit] Four pirates
Pirate C receives ninety-nine pieces of gold in the situation above, when he is the oldest pirate of three, with one piece having to go to pirate A to assure his "yes" vote. Therefore, pirate C wants pirate D (the oldest pirate in a situation with four pirates) to be killed, and so pirate D should waste no gold trying to buy pirate C's "yes" vote. Pirate B would not benefit from pirate D's death, as he would get no gold when he is the second oldest of three pirates. So, pirate D can then buy pirate B's "yes" vote with one piece of gold. Pirate D now has his two "yes" votes (his own, and pirate B's), and does not need to worry about pirate A voting "no".
[edit] Five pirates
Pirate E, however, in a situation where he is the oldest of five pirates, needs three "yes" votes to keep his band from killing him. We can conclude by now that the second youngest pirate is always going to vote "no", because he knows that if he ever became the oldest he could decide on the allocation and maximize his profits. Pirate E could supposedly assure pirate D's "yes" vote by giving him all one-hundred pieces, but he would not be maximizing his own profits, so no gold is given to pirate D. Pirate C does not benefit from pirate E's death, as it would make him the second oldest in the previous situation, in which pirate D would give him no gold, therefore pirate C's vote can be bought with one piece of gold. Pirate B benefits from a situation where he is the second youngest of four pirates in that he receives one gold piece from pirate D. Therefore pirate B's vote could be bought with two pieces of gold. But, if we consider that pirate A receives no gold pieces when he is the youngest of four pirates, we know that his vote can be bought with just one piece of gold. We therefore give one piece to him, instead of two pieces to pirate B.
Therefore, the allocation of gold which is most profitable for the oldest of five pirates, and will receive three "yes" votes, is as follows: ninety-eight pieces to the oldest pirate, one piece each to pirates C and A, and nothing to pirates D and B.
