Your answer to the "1 gold coin" problem is incomplete.
Your answer to the "1 gold coin" problem is incomplete. It doesn't explain why the 6th pirate has to join. Your brief solution seems to imply that the captain can always save his (her?) skin, regardless of the number of pirates. But in fact, in the 5-pirate case the captain (pirate 5) cannot find two allies and must die.
That is in turn the secret to the 6-pirate case: pirate 6 can count on pirate 5 as an ally, since if pirate 6 is killed off then we are back to the 5-pirate case and pirate 5 would die too. So pirate 6 only has to buy off one of the other pirates; he can give the coin to pirate 1 (as you suggest) or pirate 4.
-Jason
p.s. Pirate 6 cannot necessarily buy off pirate 2 or 3 in this way. If he tried, then pirate 4 would tell that pirate, "We'll both be happier if we mutiny, since pirates 5 and 6 will then be dead and you'll be just as rich. Why will you be just as rich? Because I promise I'll give you the coin anyway when I'm captain and have to decide which of pirate 2 and pirate 3 to pay off as an ally."
Would pirate 2 or 3 actually accept pirate 4's deal? I'm not sure, but if pirate 6 isn't sure either, then pirate 6 would be prudent not to risk it.
The reason I'm not sure has to do with the enforceability of pirate promises.
If pirate 4 has the ability to make an Unbreakable Vow, as in the Harry Potter books, then it's in pirate 4's interest to promise the coin to pirate 2 or 3 in that way, so that pirate 2 or 3 will be willing to accept the deal and more people will be killed in the ensuing mutiny. But if pirate 4 cannot do this, then pirate 2 or 3 may worry that pirate 4 would take sadistic pleasure in double-crossing him over the gold, and will reluctantly reject the chance to see two colleagues die in order to preserve the sure thing of a gold coin offered by pirate 6.
Pirate 4 may not work. P4 knows that if he were to refuse, P5 is next in line and there is no way P5 can find an ally, so he's done. So when its P4's turn he's going to give it to P3 and that will be good enough to live. In both cases, P4 gets nothing, but P4 will prefer the latter since two pirates are killed and we know the pirates like that. Hence in my humble opinion, Pirate 6 giving P1 the coin is his only way out. Do correct me if I'm wrong.
For the complete argument, you first need to notice the following:
1. If there were only two pirates left (P1 and P2), P2 could keep the coin to himself and P1 could do nothing about it.
2. If there were three pirates (P1, P2, and P3), P3 would have to give the coin to P1. Any other choice would make both P1 and P2 want to kill him.
3. If there were four pirates (P1 through P4), P4 would have to give the coin to P2 or P3. He can't please P1 no matter what he does because if he got eliminated, P1 could count on getting the coin from P3, plus he'd have the satisfaction of seeing P4 die first. On the other hand, either P2's or P3's good will could be bought for a gold coin, as both would get nothing if P4 were voted down. This is assuming that P3 plays his cards right, but we are told he is very intelligent, so he will.
4. If there were 5 pirates, P5 would face certain death, as Jason already pointed out. The best he could do is buy off one of the pirates that would get no money in the 4-pirate situation. The other three pirates would still want to kill him either because they get no money, or just because they enjoy murder. Hence P5 has a pretty compelling reason to want P6 to live.
5. Now for the 6-pirate scenario. According to the problem, P4 should prefer to get the gold coin and not kill anyone to getting no money and killing P6 and P5. So Sam is wrong about P4 wanting to kill P6 even if P6 gives P4 the coin. P6 giving the coin to P4 is a good move. P4 will be happy and won't want to kill P6. P5 already has good reason not to kill P6. So P6 has two allies this way, which is what he needs to survive.
P6 would be unwise to give the coin to P3. P3 would think this way. I have a coin. If I vote to kill P6, then P5 won't be able to find enough allies, so he'll die too. Then P4 will give me the coin. In the end, I'll have the coin anyway, and I'll enjoy killing P5 and P6.
P6 could do well by giving the coin to P2 as well. P2 will figure out that if P6 dies, P5 does too. Then P4--being very intelligent--will give the coin to P3 and will win the vote. So P2 would get nothing that way. If P6 proposes to give the coin to P2, P2 should enthusiastically vote for his proposal. P5 will vote for anything P6 proposes. So P6 has the required two allies this way.
The same argument shows that P6 giving the coin to P1 would be making a good move.
So in fact, he could give the coin to P1 or P2 or P4.
> Why can not the captain just say whomever kills the other pirates will be paid the coin?
Because it wouldn't be consistent with the rules as they are stated in the problem. Why can't the captain just say he will have to think about what do until the next day, then sneak up at night and kill the other pirates one by one while they sleep? For the same reason. Answers like these do not address mathematical/logical challenge the puzzle poses.
Why do we follow rules when we play any kind of game/sport? Because it is more interesting that way.
Actually, the captain could give the coin to any of the pirates, but P5
Looking back at my allegedly complete solution, I've just realized that P6 could in fact give the coin to P1, P2, P3, or P4. If it came down to 4 pirates at one point, then P4 would have to give the coin to either P2 or P3 as I explained above. Either would be an acceptable choice. So if either P2 or P3 is offered the coin by P6, it will be wise of them to accept it, as neither can be certain that (s)he'll be offered the coin again if P6, and consequently P5 are killed.
So the only bad moves for P6 are to try to keep the coin or offer it to P5, whose good will is already guaranteed anyway.
That is in turn the secret to the 6-pirate case: pirate 6 can count on pirate 5 as an ally, since if pirate 6 is killed off then we are back to the 5-pirate case and pirate 5 would die too. So pirate 6 only has to buy off one of the other pirates; he can give the coin to pirate 1 (as you suggest) or pirate 4.
-Jason
p.s. Pirate 6 cannot necessarily buy off pirate 2 or 3 in this way. If he tried, then pirate 4 would tell that pirate, "We'll both be happier if we mutiny, since pirates 5 and 6 will then be dead and you'll be just as rich. Why will you be just as rich? Because I promise I'll give you the coin anyway when I'm captain and have to decide which of pirate 2 and pirate 3 to pay off as an ally."
Would pirate 2 or 3 actually accept pirate 4's deal? I'm not sure, but if pirate 6 isn't sure either, then pirate 6 would be prudent not to risk it.
The reason I'm not sure has to do with the enforceability of pirate promises.
If pirate 4 has the ability to make an Unbreakable Vow, as in the Harry Potter books, then it's in pirate 4's interest to promise the coin to pirate 2 or 3 in that way, so that pirate 2 or 3 will be willing to accept the deal and more people will be killed in the ensuing mutiny. But if pirate 4 cannot do this, then pirate 2 or 3 may worry that pirate 4 would take sadistic pleasure in double-crossing him over the gold, and will reluctantly reject the chance to see two colleagues die in order to preserve the sure thing of a gold coin offered by pirate 6.
1. If there were only two pirates left (P1 and P2), P2 could keep the coin to himself and P1 could do nothing about it.
2. If there were three pirates (P1, P2, and P3), P3 would have to give the coin to P1. Any other choice would make both P1 and P2 want to kill him.
3. If there were four pirates (P1 through P4), P4 would have to give the coin to P2 or P3. He can't please P1 no matter what he does because if he got eliminated, P1 could count on getting the coin from P3, plus he'd have the satisfaction of seeing P4 die first. On the other hand, either P2's or P3's good will could be bought for a gold coin, as both would get nothing if P4 were voted down. This is assuming that P3 plays his cards right, but we are told he is very intelligent, so he will.
4. If there were 5 pirates, P5 would face certain death, as Jason already pointed out. The best he could do is buy off one of the pirates that would get no money in the 4-pirate situation. The other three pirates would still want to kill him either because they get no money, or just because they enjoy murder. Hence P5 has a pretty compelling reason to want P6 to live.
5. Now for the 6-pirate scenario. According to the problem, P4 should prefer to get the gold coin and not kill anyone to getting no money and killing P6 and P5. So Sam is wrong about P4 wanting to kill P6 even if P6 gives P4 the coin. P6 giving the coin to P4 is a good move. P4 will be happy and won't want to kill P6. P5 already has good reason not to kill P6. So P6 has two allies this way, which is what he needs to survive.
P6 would be unwise to give the coin to P3. P3 would think this way. I have a coin. If I vote to kill P6, then P5 won't be able to find enough allies, so he'll die too. Then P4 will give me the coin. In the end, I'll have the coin anyway, and I'll enjoy killing P5 and P6.
P6 could do well by giving the coin to P2 as well. P2 will figure out that if P6 dies, P5 does too. Then P4--being very intelligent--will give the coin to P3 and will win the vote. So P2 would get nothing that way. If P6 proposes to give the coin to P2, P2 should enthusiastically vote for his proposal. P5 will vote for anything P6 proposes. So P6 has the required two allies this way.
The same argument shows that P6 giving the coin to P1 would be making a good move.
So in fact, he could give the coin to P1 or P2 or P4.
Because it wouldn't be consistent with the rules as they are stated in the problem. Why can't the captain just say he will have to think about what do until the next day, then sneak up at night and kill the other pirates one by one while they sleep? For the same reason. Answers like these do not address mathematical/logical challenge the puzzle poses.
Why do we follow rules when we play any kind of game/sport? Because it is more interesting that way.
So the only bad moves for P6 are to try to keep the coin or offer it to P5, whose good will is already guaranteed anyway.
By the way good puzzle.