老虎与羊
Tiger and Sheep
题目详情
岛上有 100 只老虎和 1 只羊。老虎吃草能活，但更想吃羊。
规则：一次只能有 1 只老虎吃羊，吃掉羊的老虎立刻变成羊。所有老虎都理性且想活下去。
问：这只羊最终会被吃掉吗？
One hundred tigers and one sheep are put on a magical island that only has grass. Tigers can eat grass, but they prefer eating sheep. Assume: (1) Only one tiger can eat one sheep at a time, and that tiger then turns into a sheep; (2) All tigers are smart and rational, and they want to survive. Will the single sheep eventually be eaten?
解析
不会。
关键规律：老虎数为偶数时羊能活，为奇数时羊会被吃。
- 1 只老虎：会吃羊。
- 2 只老虎：任何一只吃羊都会变羊并被另一只吃掉，因此都不吃。
- 3 只老虎：有动机先吃，使剩下变成“2 老虎局”，对自己安全，因此会吃。
归纳得：100 为偶数，所以羊不会被吃。
Original Explanation
No, the sheep will not be eaten. The key is to notice that if the number of tigers is even, no tiger wants to reduce the total tiger count from even to odd (because an odd number of tigers makes the sheep’s position hopeless). By backward reasoning:
- 1 tiger + 1 sheep: the tiger eats the sheep (safe for the tiger).
- 2 tigers + 1 sheep: if one tiger eats the sheep, it becomes a sheep, leaving 1 tiger + 1 sheep, and then it gets eaten. So the tigers do not eat.
- 3 tigers + 1 sheep: one tiger can eat the sheep, turning into a sheep, leaving 2 tigers + 1 sheep, which is safe for those 2 tigers. Hence, they do eat.
- 4 tigers + 1 sheep: similar reasoning shows they do not eat.
By induction, with 100 tigers (an even number), they will not eat the sheep.