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The Pirates' Share (illustration)
FILE No.129Unsolved

The Pirates' Share

Difficulty ★★★about 5 minLogic PuzzleWorking backwardVotes in favor
Five pirates split 100 gold coins. The most senior proposes, and a majority vote passes it. What is the most pirate 1 can secure?

Case Record

There are five pirates, called Pirate 1 through Pirate 5 in order of seniority. The rules for splitting 100 gold coins are as follows.

  • Starting with Pirate 1, each proposes a way to split the coins.
  • If half or more of the pirates, including the proposer, vote in favor (3 or more of 5), it passes.
  • If it is rejected, the proposer is eliminated with no share, and the next most senior pirate makes a proposal.

Everyone is smart and wants to maximize their own share. If the share is the same, they prefer to eliminate the proposer.

Your question

What is the most coins Pirate 1 can secure for himself?

Case Files

Diagram

Five pirates and 100 gold coinsNo. 1No. 2No. 3No. 4No. 5Oldest →→ youngestRulesProposals start with No. 1Passes if at least half agree, proposer included (3 of 5)If it fails, the proposer is out with nothing; the next oldest proposesAll are clever and greedy; with equal shares they prefer to eliminate theproposer
Diagram

Hints

If you get stuck, open the hints one at a time, in order.

Open hint 1

Start by thinking about the last two pirates.

Open hint 2

Each pirate compares "how many coins would I get under the next proposal?"

Open hint 3

To buy a vote, give one coin more than the next proposal would.

Solve It

Your question

What is the most coins Pirate 1 can secure for himself?

Choose the answer closest to the truth.

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The Truth

98 coins

The answer is "98 coins."

Work backward. With two pirates (4 and 5), Pirate 4 votes for his own proposal and it passes, so Pirate 4 gets 100 and Pirate 5 gets 0. With three pirates (3 to 5), Pirate 3 pays Pirate 5 one coin and Pirate 5 votes yes (99, 0, 1). With four (2 to 5), Pirate 2 pays Pirate 4 one coin and it passes (99, 0, 1, 0).

With five, Pirate 1 needs 3 votes. He pays one coin each to Pirates 3 and 5, who would get 0 under the next proposal, and keeps 98. (98, 0, 1, 0, 1)

With 100 coins he gets no votes except his own and is rejected. 51 or 20 coins are far less than what he can secure.

Work backward from the fewest piratesLeftNo. 1No. 2No. 3No. 4No. 52 (No. 4, 5)———10003 (No. 3–5)——99014 (No. 2–5)—990105 (all)980101No. 1 pays 1 coin each to Nos. 3 and 5 (who would get 0 otherwise),wins 3 votes, and keeps 98.
Diagram of the solution

Key Points

  • Work backward from the end
  • Compare with the others' "next share"
  • Buy only the number of votes you need

Why people get fooled

Because you think "split it fairly" or "give to the majority."

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