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# Logical Pirates

Suppose you and a friend want to split something up; it could be a candy bar, a cake, or a pile of gold coins. How would you fairly split it up? Of course, the obvious solution is to split evenly--half a candy bar, half a cake, or half a pile of gold coins. But what if you and your friend value things differently? For example, maybe you really like cake filling and your friend really likes frosting.

This quiz deals with situations involving "fair" division by voting, which isn't necessarily a good system at all!

Two pirates find 100 gold coins, and need to divide them up. The first pirate will split the coins into two piles, and the second pirate will choose which pile to take. How many coins does the first pirate get?

NOTE: On this (and every problem of this quiz) assume all pirates have perfect knowledge of the result of any action and are greedy (that is, they each want the maximum amount of coins they can get given the situation).

Three pirates find 100 gold coins, and need to split them up. The first pirate will propose a division, and then all pirates (including the first) vote on whether to accept the division or throw the first pirate overboard. If at least half of them vote to accept, the division will happen. Otherwise, the first pirate is thrown overboard and the second pirate proposes a division, and the process continues.

Each pirate's priorities, in order, are to stay alive, get as much gold as possible, and see other pirates thrown overboard. In other words, if they would get the same amount of gold from either vote, they will vote to throw the pirate overboard.

How much gold can the first pirate get?

Five pirates find 100 gold coins, and need to split them up. The first pirate will propose a division, and then all pirates (including the first) vote on whether to accept the division or throw the first pirate overboard. If at least half of them vote to accept, the division will happen. Otherwise, the first pirate is thrown overboard and the second pirate proposes a division, and the process continues.

Each pirate's priorities, in order, are to stay alive, get as much gold as possible, and see other pirates thrown overboard. In other words, if they would get the same amount of gold from either vote, they will vote to throw the pirate overboard.

How much gold can the first pirate get?

The pattern in the previous problems continues to hold for bigger and bigger numbers. For instance, if there were 100 pirates splitting 100 gold coins, the first pirate could get 51 coins by proposing the distribution 51, 0, 1, ..., 0, 1, 0, earning all the votes of the pirates getting 1 coin (as they would otherwise get nothing).

However, for very large numbers of pirates, the pattern breaks, because there simply aren't enough gold coins to go around...

Suppose 201 pirates find 100 gold coins and need to split them up. The first pirate will propose a division, and then all pirates (including the first) vote on whether to accept the division or throw the first pirate overboard. If at least half of them vote to accept, the division will happen. Otherwise, the first pirate is thrown overboard and the second pirate proposes a division, and the process continues.

Each pirate's priorities, in order, are to stay alive, get as much gold as possible, and see other pirates thrown overboard. In other words, if they would get the same amount of gold from either vote, they will vote to throw the pirate overboard.

How much gold can the first pirate get?

Adding just a few pirates more to the last problem makes things tricky, be careful!

Suppose 204 pirates find 100 gold coins and need to split them up. The first pirate will propose a division, and then all pirates (including the first) vote on whether to accept the division or throw the first pirate overboard. If at least half of them vote to accept, the division will happen. Otherwise, the first pirate is thrown overboard and the second pirate proposes a division, and the process continues.

Each pirate's priorities, in order, are to stay alive, get as much gold as possible, and see other pirates thrown overboard. In other words, if they would get the same amount of gold from either vote, they will vote to throw the pirate overboard.

How much gold can the first pirate get?

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