大酒店

The Grand Hotel

专题: Brainteaser / 脑筋急转弯
难度: L3
来源: OpenQuant

第 1 小问

题目详情

假设你拥有一家只有 1 层、但有无限多个房间的酒店，而且所有房间都已住满。

现在来了一位新客人，你要怎样安排他入住？

Suppose you own a hotel that has 1 floor and an infinite number of rooms that are all occupied.

A new customer wants to check in. How will you accommodate this customer?

解析

因为酒店有 $\infty$ 个房间，并且 $\infty + 1 = \infty$ ，你只需要让住在房间 $i$ 的客人搬到房间 $i+1$ ，这样 1 号房就空出来了。这个过程也可以重复使用来接纳任意有限数量的新客人。更一般地说，如果有 $k$ 位新客人入住，只需把每位原住客从房间 $i$ 挪到房间 $i+k$ 。

Original Explanation

Since there are $\infty$ rooms and $\infty + 1 = \infty$ , you just need to ask each person in room $i$ to move to room $i+1$ which will make the first room vacant. This procedure can be repeated to accommodate any finite number of new guests. In the general case, when $k$ new guests would like to move in, the hotel can simply move every guest from room $i$ to room $i+k$ .

第 2 小问

题目详情

假设你拥有一家只有 1 层、但有无限多个房间的酒店，而且所有房间都已住满。

现在有无限多个人想入住这家酒店，你要怎样安排所有人？

Suppose you own a hotel that has 1 floor and an infinite number of rooms that are all occupied.

An infinite number of people want to check into the room. How will you accommodate all of them?

解析

因为 $\infty + \infty = \infty$ ，只需让住在房间 $i$ 的客人搬到房间 $2i$ 。这样 1 号房的人搬到 2 号房，2 号房的人搬到 4 号房，3 号房的人搬到 6 号房，依此类推。这样所有奇数号房间都会被腾出来，供新的无限多位客人入住。

Original Explanation

Since $\infty + \infty = \infty$ , we just need to ask each person in room $i$ to move to room $2i$ . In this case, the person in room 1 will move to room 2, the person in room 2 will move to room 4, the person in room 3 will move to room 6, etc. This will free up all the odd-numbered rooms for the new guests.

第 3 小问

题目详情

假设你拥有一家只有 1 层、但有无限多个房间的酒店，而且所有房间都已住满。

现在有无限多辆巴士到达酒店，而且每辆巴士上又都有无限多名乘客。你要怎样安排所有人入住？

Suppose you own a hotel that has 1 floor and an infinite number of rooms that are all occupied.

An infinite number of buses arrive at the hotel, each having an infinite number of people. How will you accommodate all of them?

解析

因为 $N \times N$ 是可数集，所以可以建立一个从 $N$ 到 $N \times N$ 的一一映射。因此，即使有无限多辆巴士、每辆巴士上又有无限多名乘客，酒店仍然可以把所有人都安排进去。

Original Explanation

Since $N \times N$ is a countable set, we can get a 1 to 1 mapping from $N$ to $N \times N$ . Therefore, we can also accommodate an infinite number of buses each with an infinite number of people.