The Greedy Pirates! (Solution)

The idea is to reduce the number of participants (greedy pirates) and see what happens in that case and start reasoning based on that. One more thing to note here is that a pirate would accept a proposal only if he knows that in case he would not accept the proposal, he would get less of the treasure. Let’s say we identify the pirates by the number of years of experience they have.

Now consider the case when pirates 5, 4 and 3 killed and only 2 and 1are left. Now 2 has to get a vote from 1 to stay alive. However, 2 can never give 1 more number of coins that what he would get by killing 2 and hence 2 can never satisfy 1. Therefore, 2 would never want to be left alone with 1 and 3 being intelligent knows that!

Now consider the case when pirates 5 and 4 are killed and only 3, 2 and 1 are left. 3 has to get one more vote from either 2 or 1 to stay alive. And 3 knows that 2 never wants to stay alone with 1 and hence would always vote for 3. So 3 being so lucky could keep all the 1000 gold coins and give nothing to 2 and 1 and still stay alive. So 2 and 1 would not want 3 to be a proposer of the split even if they get one gold coin each and 4 being intelligent knows that!

Now consider the case when only the pirate 5 is killed and 4, 3, 2 and 1 are left. 4 has to get 2 more votes to get a majority which it can get from 1 and 2 by giving them 1 gold coin each and keeping the rest 998 with himself and giving 3 nothing. So 3 would not want 4 to propose even if he gets 1 gold coin for himself, and 1 and 2 would not want 4 to propose if they get more than 1 gold coin which 5 being intelligent knows!

Now consider the real problem wherein 5 has to propose so that he gets the maximum and doesn’t get killed. He needs 2 more votes minimum inorder to stay alive. He could get one vote from 3 by giving him one gold coin. And for one more vote he could give either 1 or 2 two gold coins as discussed above! So he keeps 997 gold coins for himself, gives 1 gold coin to 3 and give 2 gold coins to either 1 or 2!