Infinite Room Hotel Problem – Webtekno

This concept of infinity led mathematicians to explain this subject in a more understandable way since the 1920s. The most influential of these efforts was that of German Mathematician David Hilbert. It became the “Infinite Hotel Paradox”.

Now imagine a hotel. Let this hotel have infinite rooms. “Such a thing is not possible!” We want you to think a little more before saying this. Moreover, each of these endless rooms is occupied. We are aware that it seems more complicated now, but we are sure that it will come to your mind once we examine it in more detail.

What happens when a person requests an empty room from a hotel that has every room occupied?

To a new guest When we consider what can be done, the answer will be as most of you predicted. But the logic of this hotel is a little different.

In hotel rooms, every guest to the room above his own number. It is possible to move. That is, if the guest in room 1 moves to room 2, room 1 will remain empty and make room for the new guest.

Well let’s make the question a little more complicated, in If a bus with 40 passengers comes to this hotel, how to arrange a seat?

Practically from the guest in room 1 You are asked to move to room number 2.then from room 2 to room 3 and so on.

Since there are an infinite number of rooms in the hotel, it is possible to find a new room for each guest. In a cycle that goes on like this Room number 1 becomes vacant for the new guest.

Actually, the logic is still the same.

However, this time, the guests were asked to stay in their own rooms. It should be moved above number 40. So, if the guest in room 1 moves to room 41, the guest in room 2 must move to room 42.

In this way, the first 40 rooms will be vacant. Let’s make the situation even more complicated, shall we?

We dreamed of the bus, now on a bus with “infinite” number of passengers What will the situation be like if encountered?

This time, since the number of new arrivals is uncertain and there is a situation of intertwined infinity, the following solution is found: to place each guest in a room that is twice the room number.

Well Guest number 1 moves to room number 2, Guest number 4 moves to room number 8 and so on. In this way, room is made for another infinity within infinity.

So in the hotel While even numbered rooms are occupied, odd numbered rooms remain empty and incoming guests can settle in those rooms. In this case, the hotel’s earnings remain constant.

If we push the logic once again, if an infinite number of buses with an infinite number of passengers arrive in front of the hotel and they want to stay at the hotel, can a place be arranged?

The answer as we initially thought No will be. Because it seems logically that we cannot add infinity to something that is infinite.

However, the people at the hotel responded to this situation by saying “He finds a solution with the logic “There are an infinite number of prime numbers in mathematics.” So how do they apply this rule in this kind of problem?

Actually, they need to make some changes inside before settling new guests into the hotel. This change requires all customers within the hotel to use their room numbers, as the smallest of the prime numbers is 2. to the power of 2 They are changing places.

So customer in room 1 is 2¹., Customer in room 9 2⁹. moving into the room. With this process, empty rooms are created in the hotel.

They place the passengers on the first bus as 3, on the second as 5, respectively, according to the prime number order.

In other words, the first passenger on the first bus is placed in rooms 3¹, the second passenger is placed in rooms 3², … Same operations Applicable to other prime numbered buses as well It goes on forever.

In this way, all customers are settled in the hotel and no room numbers conflict with each other. Because it is known that prime numbers can only be divided by 1 and themselves. Even rooms are left empty in the hotel, like rooms 6 or 12 are left empty because they are not exponents of prime numbers.

So how can we be sure that room numbers equal to powers of prime numbers will always be empty?

Because every positive integer can be written in a single form, as the product of prime numbers. In this case, if a room number in the hotel is a power of a prime number, it cannot be equal to the power of another prime number. No power of odd prime numbers is divisible by two and these rooms remain empty.

Hilbert does not actually tell us about infinity through this problem, How difficult it is to understand infinity and shows the right way to adopt this concept with various examples.

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