A sheep must choose between two islands full of lions. To survive, it must reason exactly as they do.
Two Islands, One Sheep
Riddle statement
A sheep must choose between two islands: one is inhabited by 19 lions and the other by 100.
All the lions are perfectly rational and know that the others are rational too. If one eats the sheep, it immediately turns into a new sheep; only one lion can attack at a time.
Each lion prefers, in this order:
- to eat the sheep and survive;
- not to attack;
- to eat the sheep and later be eaten.
Therefore, if attacking guarantees survival, one of them will do it; if every attacker would eventually be eaten, none will attack.
Which island should the sheep choose in order to survive?
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Solution
Answer: the sheep should choose the island with 100 lions.
With one lion, the sheep dies: the lion eats it and, once transformed into a sheep, no lion remains to eat it.
With two lions, neither attacks. Any attacker would become a sheep facing the other lion and would then be eaten.
With three lions, however, one of them attacks: after transforming, it would face two lions, and we have just seen that a sheep survives against two lions.
The pattern continues. Every additional lion reverses the outcome, because an attack turns the situation into the same problem with one fewer lion.
Therefore:
- with an odd number of lions, one of them attacks;
- with an even number, none will risk it.
Since 19 is odd, the sheep would die on that island. Since 100 is even, no lion will attack on the other.
The sheep should take refuge on the island with 100 lions.