Pirate Problem Generalization

Logic Level 3

A group of pirates have obtained some number of gold coins. They are now going to decide how many coins to give to each pirate, by voting. Here is how the vote works:

The first pirate proposes a way to split the coins. All pirates including that first one are allowed to vote, by saying yes or no. If the number of people saying no outnumbers the number of people saying yes, the first pirate gets killed. Then, the same procedure applies and everyone votes for the second pirate's plan, and so on. If in any vote the number of people saying yes is the same as, or greater than, the number of people saying no, then that plan will be adopted.

The main goal of each pirate is to survive. Their secondary goal is to maximize the number of coins they each gain. Every pirate is very logical and knows that all other pirates are also logical.

Let \(f(x) \) be the least number of pirates with \(x\) gold coins (where \(x\) is an integer) such that, no matter what the proposed plan is, the proposer will die.

If \(f(x) = ax + b,\) where \(a\) and \(b\) are constants, find \(a+b\).

Inspired by this problem and this problem.


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