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A colleague and I are trying to figure out if the following statement is true and if so, which mathematical theorem and formulations are the one that apply here:

Given a hotel with N rooms of the same type. If for every day of a proposed reservation there is at least 1 room that is available. Then no matter what the other reservations are over that time period, I can arrange them so that the new reservation is made continuously on the same room.

When looking only at 2 rooms, this seems true: enter image description here

And "playing around with rectangles", it seems fine with more rooms too: enter image description here

But we're curious to see if there's a formal definition of this problem and a proof or cases that disprove this because so far all we have is "an intuition that this seems true".

Khepin
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2 Answers2

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For your set of reservations add in enough single night reservations to fill the hotel every night. Now the number of guests checking out each morning must equal the number of guests arriving that evening so they can check into the rooms that have been vacated. This assumes that no guest is assigned to a room before they arrive but it shows that you can accommodate all the guests without anybody changing rooms.

Ross Millikan
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Maybe I'm wrong, but I think you are describing a static planning problem instead of trying to dynamically accommodate new guests. To prove your conclusion is true, you can construct the following process:

  1. For the first day, find an empty room, call it $r_1$. This is where the new reservation is on throughout.
  2. On the second day, find the room where it is empty, call it $r_2$, swap the reservation from the second day in that room with r1, so that the new guest in $r_1$ can keep living in that room on the second day.
  3. Repeat that process on the third day and so on.
Kryvtsov
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