Starter Asked on 9th September 2016 in Puzzles.
I think it was be 50 (considering that its a rule that the captain would get the highest number of gold coin)
Let the pirates in order of seniority be A, B, C, D, E
The first captain wld be A. First he would propose that he takes 98 and give 1 each to C and E (as mentioned by john123)
In this case, B would tell E to reject the proposal and he would give him more than 1(so, at least 2). E knows that he would get nothing if he gets A,B and C killed, as D would take 100 for himself and vote for the proposal himself. That would gain him half of the vote and the proposal would be accepted
So, if it comes down to C, he would get a max of 1 coin, as he would choose 1 over 0.
B offers him more than 1. E knows that B is sure to keep his promise as if B offers him 0 coins, there is no way E would accept it and if B offers him 1 coin, he can reject B and move to C’s proposal, where he surely will get 1 coin, and there still a very, very small chance that he gets more than 1.
This can be sure by an example: if a greedy man is given 2 options of boxes, one sure to contain $10 and the other sure to have a minimum of 10, the man would choose the second one as he would not face a loss in comparison to the first one, and would also have a chance to gain a bonus.
So, this shows that to survive, B has to give at least 2 coin to E and would surely do it. C will think all this and come to a conclusion that if B distributes, he would get 0, so it is better to except 1. There is no way E would reject B so B is bound to get accepted and thus he can’t gain more than 1. This secures 1 vote for A
Now thinking all this, A would think that to get E to accept his proposal, A had to give E the max number he can get. In worse case, as captain had to keep the highest number, if B were to secure 1 vote, B would only divide it among him and E, which would be 51 for B and 49 for E. Also, E knows that after A gets killed, B won’t give more than 2 coins as proved above( in C’s proposal).
So, if A offers E 49, E would sure to get 49 and will not reject the proposal for B.
Another example for this, if the greedy man is offered 2 boxes, 1 sure to contain 10 rupees and the other might have less than 10 but a max of 10, he would choose the first box, as he the second box may indulge in a loss in comparison to the first box.
Thus, A would propose 50 coins for himself, 1 for C and 49 for E
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