# Serial Problem

At Very Long Hotel in Florida, there are n rooms located along a very long corridor and numbered consecutively from 1 to n. One night after a party, n people, who have been likewise numbered from 1 to n, arrived at this hotel and proceeded as follows: Guest 1 opened all the doors. Then Guest 2 closed every second door beginning with door 2. Afterwards, Guest 3 changed the position of every third door starting with the third door (that is, the guest opened the doors that were closed and closed the doors that were open). In a similar way, Guest 4 changed the position of doors 4, 8, 12,.... This process continued until each person had walked the length of the corridor. Of course, the last person, Guest n, merely strolled to the end of the corridor, where the guest changed the position of door n. The question is, "Which doors were left open and which ones were left closed at the end of the process?"

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#### Solution Preview

We can interpret the problem into this form:

A serial of numbers: 1, 2, 3, 4, 5 ... n (all numbers are positive, which stands for closing door.)

Each time a guest multiplies the serials by -1 that is a multiplier of its own number.

Therefore, if the number finally turns to be negative, the door is open and if positive then closing.

Now we are facing the following situation:

First time, all the numbers will times -1 ...

#### Solution Summary

A problem is solved using serials.