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## Apartment Block Fire

Math brain teasers require computations to solve.

 Puzzle ID: #28484 Fun: (2.74) Difficulty: (2.67) Category: Math Submitted By: Dave

Dave, the janitor of an apartment block, notices that the boiler pressure is rising, and will shortly explode.
He phones two of the residents, tells them the news, asks them to do the same and phone just two more people, and then get out of the building.
Assume that each of the residents is in, and answers the phone immediately.
If each phone call takes 30 seconds, and it takes each person 90 seconds to get out of the building, how long will it take to empty all 375 apartments?

7 minutes.

In the first 30 seconds, Dave rings one other person, call him A.

In the second 30 seconds, Dave rings B, and leaves the building. A rings C.

3rd: A rings D, and leaves. B and C ring E and F.

4th: B and C ring G and H. D, E and F ring I, J, K.

5th: D, E, F ring L, M, N. G, H, I, J, K ring O, P, Q, R, S.

The number of residents informed in each time period is as follows:
1, 2, 3, 5, 8, ...
These are known as Fibonacci Numbers, where each number is the sum of the preceding two.
The total number of people informed is:
1, 3, 6, 11, 19, ...
Each number is the sum of the previous two, plus 2.
Continuing this series we get:
32, 53, 87, 142, 231, 375.

Thus 375 people were informed after 11 periods of phone calls.
11 calls at 30 seconds each = 5.5 minutes, plus 90 seconds for the last people to exit the building gives 7 minutes.

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