8. Birgit has ten candles. Each candle has a burning duration of three hours. At a certain moment, she lights one of those candles. From that moment on, she wants to burn the ten candles under the following conditions:
1) she only lights the next candle after a whole number of hours;
2) there must always be at least one candle burning;
3) no candle is blown out;
4) if a certain candle or combination of candles is burning during a period between two whole hours, then in the next period between two whole hours, not exactly the same candle or combination of candles may burn.

How many hours will it take under these conditions for all ten candles to be completely burned out?
A) 12
B) 15
C) 16
D) 17
E) 18

## B-questions
A8. D) 17 First note that at the beginning of each hour, something must happen; either a candle must be lit or a candle must go out (or both). Since there are 10 candles, this amounts to 20 events. This process can therefore take a maximum of 19 hours. However, it is the case that in the first 5 hours (thus in the first six events), there must be a moment when two events occur simultaneously. We prove this by contradiction.

Assume that exactly one event takes place each hour. At $t=0$, Birgit lights a candle. At $t=1$ and $t=2$, she has no other choice but to light a second and third candle. At $t=3$, the first candle goes out, so Birgit does nothing further. At $t=4$ and $t=5$, the second and third candles go out, so Birgit still does not light any new candles. Between $t=5$ and $t=6$, no candles are burning, but this is in contradiction with the data given in the problem.

The same argument can be used to show that there must also be a moment with two events occurring simultaneously in the last 5 hours. This means that there are two moments with two events occurring simultaneously. The process can therefore take a maximum of 17 hours.

The diagram below shows that it is indeed possible in 17 hours. Each horizontal line represents the three burning hours of a candle. Note that there is only one event each hour, except at $t=4$ and at $t=13$, when a candle both goes out and is lit.

![](https://cdn.mathpix.com/cropped/2024_04_17_a4374d618d1fd957f1eeg-3.jpg?height=254&width=1556&top_left_y=1986&top_left_x=244)