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An island is filled with apple trees and other plants. The only inhabitants are 160 foxes and 1 chicken.

The foxes are smart, logical, and completely aware of their surroundings. They have two food options: apples and chicken.

Foxes are happy to eat apples to survive but would rather eat chicken.

If a fox eats a chicken, it turns into a chicken, and could then be eaten by other foxes.

A fox prefers to eat apples forever than be eaten by another fox (if it turned into a chicken).

Assume that there is never an issue with who gets to a chicken first, because the fox that is closest would eat it, if it decided to do so. Also, a chicken cannot escape if a fox decides to eat it.

Question: Will that one chicken on the island be eaten or not and why?

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3 Answers 3

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Let's start with a different scenario:

Assume there is only one fox, and one chicken. In that case, the fox will obviously eat the chicken, and then it will simply live out the rest of its life as a chicken. There are worse outcomes.

Now,

if there are two foxes, still with only one chicken, neither fox will want to eat the chicken - if they do, they will turn into a chicken, and then it will just be them and the other fox, and they'll get eaten.

Therefore,

with three foxes, and of course, one chicken, the foxes will be incentivized to eat the chicken, as once one of them does, there will be two foxes and one chicken, and as we just saw, the chicken will be safe.

However,

in the case of four foxes (and just the one chicken, obviously), none of the foxes will want to eat the chicken, because then there will be three foxes left with the fox-turned-chicken, and as we just saw, that will quickly mean only two foxes and one fox-turned-chicken.

I'm beginning to see a pattern here:

If there are an even number of foxes, the foxes will not eat the chicken because that will put them in the scenario in which there are an odd number of foxes, in which the foxes will eat the chicken, because that will make an even number of foxes, in which case the foxes will not eat the chicken because, well, you get the point.

Therefore, in our original scenario with 160 foxes,

our prodigious poultry protagonist will prosper in perpetuity. Phew!

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  • $\begingroup$ Great answer!!! $\endgroup$
    – Charlie
    Dec 1, 2023 at 14:56
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    $\begingroup$ Another +1 for the alliteration at the end, nice one, haha $\endgroup$
    – justhalf
    Dec 1, 2023 at 15:46
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The answer is:

the chicken is not eaten.

  1. One fox can always safely eat the chicken.

  2. With two foxes, neither wants to eat the chicken, because the other can then safely eat the transformed fox.

  3. With three foxes, the closest fox can safely eat the chicken, because it reduces the problem to 2. where neither fox wants to eat the chicken.

...

Extending this to larger numbers of foxes, we can see that for any even number of foxes they don't want to eat the chicken. Therefore, with 160 foxes the chicken remains safe.

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It appears that the chicken would

remain uneaten.

This conclusion is drawn from the fact that

the foxes, being entirely logical and cognizant of their surroundings, would individually decide against eating the chicken, since that would likely lead to them being devoured by another fox.

While I initially considered the idea of

foxes coordinating to maximize the overall consumption of chickens, their preference for apples, combined with the aversion to being eaten themselves, suggests they would opt for apples indefinitely.

Thus bringing me to my conclusion that the chicken will

not be eaten.

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  • $\begingroup$ This doesn't explain e.g. the 3 fox/1 chicken case $\endgroup$
    – bobble
    Dec 1, 2023 at 14:44
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    $\begingroup$ "the foxes, being entirely logical and cognizant of their surroundings, would individually decide against eating the chicken." Why? I am logical and cognizant of my surroundings but that does not adequately explain the reason that I don't walk blindly into traffic. :) $\endgroup$
    – LeppyR64
    Dec 1, 2023 at 15:06
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    $\begingroup$ Because you prefer being alive, which, having perfect logic, the foxes would too. $\endgroup$
    – Someone
    Dec 1, 2023 at 15:33
  • $\begingroup$ Of course, but what the other comments are getting at is that the foxes' preference for staying alive isn't enough to explain why the chicken doesn't get eaten. After all, if the chicken doesn't get eaten now perhaps that means that a fox that eats this chicken won't get eaten. $\endgroup$ Dec 1, 2023 at 16:06

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