# Pirate democracy at its finest

With our pirate crew becoming too big, the captain grew very concerned about splitting all the treasure - we continued to split it equally, but, of course, each crew member got less and less with the time.

When the captain got sick of it, he gathered all the crew and made an announcement:

Pirates! There are already 999 of you in my crew! That's way too much!

I gave each of us a different number from 1 to 1000 according to how much you do for the crew. I myself got the 1000, and Michael, the one sleeping in that corner, got the 1.

From now on we will each day vote on executing the lamest member of our crew, in the order: 1, 2, 3, etc... The one who is judged doesn't vote! If the strict majority (> 0.5) of others decides to execute the lamest member, we do so, and then continue. If not - we stop the process altogether.

That is a completely democratic way to clear the crew of the weakest members. The ones alive will totally benefit from it, for their share in the treasures we pillage will highly increase!

Given that every pirate was very clever and predictive, how many pirates died in the process?

I have a hunch that the answer is

489, so 511 pirates remain.

Explanation:

When there is 1 pirate left, obviously nothing happens.
When there are 2 pirates left, the one with the higher number will vote to execute the other so he gets a larger share.
When there are 3 pirates left, the one with the second-highest number will not vote to execute, since that would leave him in the previous situation where he will be executed. Since the votes are tied, the process stops here.
When there are 4 pirates left, the three pirates will vote to execute the last one in order to get a larger share. There's no risk that they will be executed, since the process will stop at 3.
When there are 5 pirates left, the one with the fourth-highest number cannot stop the other three to reach the situation with 4 pirates. The same holds for 6 pirates.
When there are 7 pirates left, the three pirates 4-6 can vote not to execute in order to stop being executed themselves in the next steps. So 7 is again a 'stable' number.

Continuing this way, we see that

when there are $$2^n-1$$ pirates left, execution will stop.
Since 511 is the largest such number smaller than 1000, 511 pirates will remain alive and 489 will die.

• This is the same exact problem (actually off-by one due to the voting rules) of Pirates and gold coins with 0 coins. This is a great answer by SQB which explains the logic in detail: and gets the same result as Glorfindel's. Commented May 23, 2019 at 20:39
• It's faster to break out yer musket than compute that...
– smci
Commented May 24, 2019 at 23:55