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# Lions and Sheep

On a magical island, there are 100 lions and 1 sheep, all of which can live by eating the plentiful grass on the island. Any lion that eats the sheep will magically turn into a sheep afterward, such that there will always be a sheep on the island.

Every lion would like to eat a sheep, but would much rather prefer to not be eaten (they wouldn’t mind turning into a sheep if they wouldn’t be eaten).

If all the lions act rationally and know all the other lions act rationally, how many lions will remain on the island in the end?

#### Solution

All 100 lions will remain!

Use backwards induction:

• If there was only 1 lion, it would eat the sheep and then happily live out its days as a sheep.
• If there were 2 lions, both of them know if they eat the sheep, the remaining lion will eat them. So neither of them will eat the sheep.
• If there were 3 lions, they all know that when there are 2 lions, neither of them will eat the sheep. So one of the lions would immediately eat the sheep.
• You can see that any odd number of lions means one of the lions will eat the sheep, and any even number of lions means none of the lions will eat the sheep (because there will be an odd number of lions left so they will also get eaten). Since 100 is an even number of lions, none of them will eat the sheep, and 100 will remain.