Solution to: Pirate Treasure

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 would be the only one left, he would get all the golden coins. If only pirates 1 and 2 would be 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). So:

                  |  Pirate 1  |  Pirate 2
  ----------------+------------+------------
  Number of coins |    1000    |     0
  ----------------+------------+------------
  Pirate agrees   |    No      |    Yes

When also pirate 3 would still be 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 did not. Therefore, pirate 3 should propose to keep everything for himself.

                  |  Pirate 1  |  Pirate 2  |  Pirate 3
  ----------------+------------+------------+------------
  Number of coins |     0      |     0      |    1000
  ----------------+------------+------------+------------
  Pirate agrees   |     No     |    Yes     |    Yes

When we have four pirates, pirate 4 should make two other pirates agree with his proposal. Therefore, 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 would not accept, they would get less.

                  |  Pirate 1  |  Pirate 2  |  Pirate 3  |  Pirate 4
  ----------------+------------+------------+------------+------------
  Number of coins |     1      |     1      |     0      |    998
  ----------------+------------+------------+------------+------------
  Pirate agrees   |    Yes     |    Yes     |     No     |    Yes

However, 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.

                  |  Pirate 1  |  Pirate 2  |  Pirate 3  |  Pirate 4  |  Pirate 5
  ----------------+------------+------------+------------+------------+------------
  Number of coins |     2      |     0      |     1      |     0      |    997
  ----------------+------------+------------+------------+------------+------------
  Pirate agrees   |    Yes     |     No     |    Yes     |     No     |    Yes

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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