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本帖最后由 永恒的蓝色梦想 于 2020-8-2 08:25 编辑
Hilbert's New Hotel
An infinite number of people (numbered 1, 2, 3, etc.) are lined up to get a room at Hilbert's newest infinite hotel. The hotel contains an infinite number of floors (numbered 1, 2, 3, etc.), and each floor contains an infinite number of rooms (numbered 1, 2, 3, etc.).
Initially the hotel is empty. Hilbert declares a rule on how the nth person is assigned a room: person n gets the first vacant room in the lowest numbered floor satisfying either of the following:
- the floor is empty
- the floor is not empty, and if the latest person taking a room in that floor is person m, then m + n is a perfect square
Person 1 gets room 1 in floor 1 since floor 1 is empty.
Person 2 does not get room 2 in floor 1 since 1 + 2 = 3 is not a perfect square.
Person 2 instead gets room 1 in floor 2 since floor 2 is empty.
Person 3 gets room 2 in floor 1 since 1 + 3 = 4 is a perfect square.
Eventually, every person in the line gets a room in the hotel.
Define P(f, r) to be n if person n occupies room r in floor f, and 0 if no person occupies the room. Here are a few examples:
P(1, 1) = 1
P(1, 2) = 3
P(2, 1) = 2
P(10, 20) = 440
P(25, 75) = 4863
P(99, 100) = 19454
Find the sum of all P(f, r) for all positive f and r such that f × r = 71328803586048 and give the last 8 digits as your answer.
题目:
希尔伯特最新的无穷旅馆里有无穷多个人 (编号为 1, 2, 3 等) 排队等待入住。酒店有无穷多个楼层 (编号为 1, 2, 3 等) , 每个楼层包含无穷多个房间 (编号为 1, 2, 3 等) 。
一开始旅馆是空的。希尔伯特规定了如何为客人分配房间:第 n 个人将会入住满足以下任一条件的最低楼层中的编号最小的空房间:
- 这层楼是空的
- 这层楼不是空的。假设这层楼上一个入住的人是 m,那么 m + n 是完全平方数
因为第 1 层是空的,所以第 1 个人住在 1 层的第 1 间 。
因为 1 + 2 = 3 不是完全平方数,所以第 2 个人没有住在 1 层的第 2 间 。
因为第 2 层是空的,所以第 2 个人住在 2 层的第 1 间 。
因为 1 + 3 = 4 是完全平方数,所以第 3 个人住在 1 层的第 2 间 。
最终,排队的每个人都能在酒店入住。
如果第 n 个人入住了 f 层的第 r 间,那么定义 P(f, r) 为 n 。如果没有人入住这个房间,那么定义 P(f, r) 为 0 。以下是几个例子:
P(1,1) = 1
P(1,2) = 3
P(2,1) = 2
P(10,20) = 440
P(25,75) = 4863
P(99,100) = 19454
找出所有满足 f × r = 71328803586048 的正数 f 和 r,求出所有的 P(f, r)之和,给出它的最后 8 位数字作为答案。 |
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