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A person was trapped in a building, and you have to save him. You are a police officer and you will arrive at the building at 7 o'clock in the morning. The building looks like this:

enter image description here

Every 2 minutes, you, the person and the interior walls will become invisible. You don't know where the person is but you know that he will move to an adjacent room every 2 minutes. However, you can check any room in the building every 2 minutes because you have a cop car.

Unfortunately, at 7:29 am, the entire building will explode, killing both of you! You have to find a way to save the person, there is not much time left! Note: Your first checking is at 7:02 a.m.

Matheinstein
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    What's the role of the building interior turning invisible every 2 minutes? It strikes me as very odd, but I can't see how that affects the problem in any way - you can still only search one room at a time. – Nuclear Hoagie Nov 19 '20 at 18:13
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    If all the interior walls turn invisible, why can't you just park yourself in room 3, wait for the walls to turn invisible, and look all the way down each wing in turn to find out where the missing person is? – Michael Seifert Nov 19 '20 at 19:39
  • @NuclearHoagie well there are no doors, so – Matheinstein Nov 19 '20 at 23:43
  • @MichaelSeifert the missing person and you are also invisible. – Matheinstein Nov 19 '20 at 23:51
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    My point is that if you're in Room 3, and all the walls are invisible, you can see into Room 2 and see if the person is there. And you can see all the way through Room 2 into Room 1 and see if he's there. In fact, if all the interior walls are invisible, you can see the entire interior of the building from Room 3. I assume this isn't what you intended; I merely say this to point out that the "invisible walls" thing is confusing, and you might want to rephrase it for clarity. – Michael Seifert Nov 19 '20 at 23:51
  • With your most recent comment, the intent makes more sense. I would recommend editing that into the puzzle. – Michael Seifert Nov 19 '20 at 23:52
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    @Matheinstein Now I understand it even less. So this is a building with no doors where the interior walls and everybody inside turns invisible every two minutes. What's the point of any of that information? You can't see anything except the exterior wall when the invisibility turns on, so I don't see how it helps or hinders you in any way. It seems like the question would be unchanged if it were a regular building, with doors between rooms and opaque walls. – Nuclear Hoagie Nov 20 '20 at 13:30
  • @NuclearHoagie The interior walls can only let you check every 2 minutes – Matheinstein Nov 20 '20 at 13:38
  • @Matheinstein But... they don't. When the walls turn invisible, so does the person you're looking for, so anytime you're able to see, there's nothing to look at. Are you getting at a way to avoid seeing the person when they switch rooms? So if you're in Room 4 and the person is in Room 5 at 7:02, you still won't have found them at 7:04 if you move to Room 5 and they move to Room 4? I think a much simpler and more realistic solution is to have the lights turn off or on each minute. – Nuclear Hoagie Nov 20 '20 at 15:51
  • @NuclearHoagie I am trying to make it so that you can't see the person when they switch rooms. Also, the question mentions that you have a cop car, and a cop car glows, so if you have the lights turn on and off each minute, the person will notice the cop car. – Matheinstein Nov 21 '20 at 11:58
  • I don't understand the solution has already been found. – PDT Nov 22 '20 at 14:55
  • The wording of https://puzzling.stackexchange.com/questions/455/why-does-this-solution-guarantee-that-the-prince-knocks-on-the-right-door-to-fin could have been borrowed to make this puzzle easier to understand. Also, if you've seen that puzzle before this one is easy to figure out. – Rob Watts Nov 25 '20 at 17:03

2 Answers2

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If the person can stay in those 2 minutes, the problem will be unsolvable because he can only move if and only if we check his room.

Otherwise, here is a solution with exactly 15 14 (thanks @JaapScherphuis!) checkings, just in time!

Let $x$ be his possible rooms and the yellow cell is the room to be checked.

enter image description here

athin
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This is not a new solution but an easy way to understand how it works:

The first thing to do is a standard checkerboard coloring so the person to catch will alternate colors with every move. Let's assume the center square is black.

If we knew parity it would be easy: At black moves we sit in the center, that way the person cannot change arms without running into us. At white moves we check the arms one after the other. This is possible because there is only one white room per arm. We can start this procedure at a white square and it will take 7 moves.

Now, we do not know parity, but there are only 2 possibilities. we can simply check them one after the other and will be finished after at most 2x7 = 14 moves. enter image description here

Paul Panzer
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