Math Is Fun Forum

Here's the original problem.

Here's the proposed solution.

The problem claims that "If 50% or more of the pirates vote for [the proposal], then the coins will be shared that way." But what if one pirate remains? The problem continues: "Otherwise, the pirate proposing the scheme will be thrown overboard..." If we take "otherwise" to include scenarios in which no vote takes place (effectively a "nil" vote rather than a sub-50% vote), then we must assume that the last pirate, who is locked into this coin distribution system and has no choice but to propose a scheme to himself, on which he cannot vote, is forced to walk the plank.

Let's call the pirates A, B, C, D, and E in descending order of age.

E must walk the plank if the proposal comes to him. E will avoid this scenario if at all possible.

D knows this, and will propose a 100 : 0 scheme because E values life above gold and will accept it.

C knows that D will vote against whatever he proposes, because if the vote comes to D, D gets all the gold. E only requires 1 coin to guarantee his vote, because otherwise he gets nothing. C therefore proposes 99 : 0 : 1.

B knows that C will automatically oppose his scheme for the same reason that D opposes C's. B requires two of the three remaining pirates to vote for his scheme, however, so merely appeasing D is not enough. B must make both D's and E's cuts slightly more favorable than what C would offer, so B proposes 97 : 0 : 1 : 2.

A (here's our final solution) proposes 97 : 0 : 1 : 2 : 0. B will vote against the proposal regardless, so it's not necessary to waste precious gold on him. C would get nothing if the proposal failed, so offering him one coin is all A needs to secure his vote. Appeasing E isn't necessary, because E is (albeit slightly) more expensive to sway than D, who only requires 2 coins. Votes from C and D make the necessary 50%, because only four pirates are allowed to vote.

Thoughts?