# The sheep and the lions: is the conventional answer right? [duplicate]

I am challenging the solution to this classic puzzle:

There is an island filled with grass and trees and plants. The only inhabitants are 100 lions and 1 sheep. The lions are special:

1. They are infinitely logical, smart, and completely aware of their surroundings.
2. They can survive by just eating grass (and there is an infinite amount of grass on the island).
3. They prefer of course to eat sheep.
4. Their only food options are grass or sheep.

Now, here's the kicker:

1. If a lion eats a sheep he TURNS into a sheep (and could then be eaten by other lions).
2. A lion would rather eat grass all his life than be eaten by another lion (after he turned into a sheep).

Assumptions:

1. Assume that one lion is closest to the sheep and will get to it before all others. Assume that there is never an issue with who gets to the sheep first. The issue is whether the first lion will get eaten by other lions afterwards or not.
2. The sheep cannot get away from the lion if the lion decides to eat it.
3. Do not assume anything that hasn't been stated above.

So now the question: Will that one sheep get eaten or not and why?

Read the link if you are unfamiliar with it.

The bottom line of the solution is that an even number of lions won't eat the sheep, and an odd number will. I fully understand the recursive reasoning behind this.

But suppose there were an even number of lions, and one of the lions decided to eat the sheep anyway. This would violate one of the conditions of the riddle -- that they all act rationally.

But, if this were to actually happen, all the other lions would have to abandon their belief that all the others are rational. Then, they could no longer trust their previous logic. Clearly, none of the remaining 99 lions would eat that sheep -- since that would leave an even number of lions, and a sheep just got eaten by an even number of lions!

Then, no lion would eat the lion who behaved irrationally and ate the sheep. Was he, then, truly acting irrationally?

• Why is eating the sheep when there are 100 lions an irrational behavior? Commented Mar 2, 2017 at 7:28
• Why is this marked as a duplicate? This is a new question derived from the other question. It is the question why a given line of reasoning does not (or does it?) apply. Commented Mar 2, 2017 at 14:39
• Well, I cannot answer any more. So I answer here. My idea is that the reasoning doesn't work because if 100 lions effectively behave illogically in some sense that doesn't mean 98 also will. Commented Mar 2, 2017 at 14:42

## 2 Answers

One of the problem's assumptions is that the lions are all rational and aware of their surroundings.

They therefore know that the others are rational, and it would be irrational to eat a sheep if there are an even number of lions.

This means it is safe to eat a sheep if there are an odd number of lions.

You say that "all the other lions would have to abandon their belief that all the others are rational", but that doesn't make sense - the problem explicitly states that they are all rational and all fully aware of their surroundings, which includes the rationality of the other lions.

• I believe I am missing something. Why does the number of lions left change whether they would eat the sheep? Wouldn't it only matter whether you were the last lion left? Commented Mar 2, 2017 at 4:54
• @LaniKate: Details are here.
– Deusovi
Commented Mar 2, 2017 at 4:54
• Thank you, I actually just went and read the original puzzle page and now it makes sense to me. I love the concept. Commented Mar 2, 2017 at 4:56

If we factor in the premise that "all the other lions would have to abandon their belief that all the others are rational", then

that first lion cannot have been acting rationally by eating the sheep in any scenario.

Once the lions have to assume that their peers are capable of acting irrationally, all bets are off about whether they will be eaten as sheep regardless of the existing count. As such, the only "rational" behavior is to recognise the non-zero probability that a sheep will be eaten by some lion who suddenly decides to go wild, and NEVER become a sheep until all the other lions have succumbed to their irrationality, leaving you the last lion standing.