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 Thirty two barbarians (Posted on 2004-01-28)
There is a small town situated by a barbarian colony. The population in this town is very small, but they live well. Upon seeing the villagers in this town so happy, a group of thirty two barbarians sneak up and position themselves around the city. All the barbarians fired at exactly the same time, and every bullet went over 3 villager's heads before it killed another person, including anyone who may have been shot already. If no villager was at the same place at the time (and all villagers were in the town) when the simultaneous shooting occurred, what is the fewest amount of villagers in the town? (Note: "Around" means actually around. A line going around the city would work, but one going out of the city would not.)

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 upper bound | Comment 12 of 34 |
This time...

Assuming that ONLY villagers (not barbarians) can be killed.
Assuming that we can shoot the same villagers more than once.
Assuming that not all villagers must be killed.

The following diagram shows how to place 24 villagers in such a way that 36 (4 more than required) barbarians can line up on the outside and shoot them all (satisfying the constraints of the problem).

There are 18 lines of consequence:

AFSX
BDTV
CEUW
KLMN
GHIJ
OPQR
KOVX
GPTW
HLSU
DFMQ
BEIR
ACJN
ABGK
CDHO
EFLP
IMST
JQUV
NRWX

If we draw a circle around this "grid", each line will intersect this circle in two points (for a total of 36 intersections). We merely put a barbarian at any of these intersections, and each barbarian will shoot into the circle, along the line.
`. . . . . . . . . . . . . . .. . . . . . . A . . . . . . .. . . . . . B . C . . . . . .. . . . . . . . . . . . . . .. . . . . . D . E . . . . . .. . . . . . . F . . . . . . .. . G . H . . . . . I . J . .. K . . . L . . . M . . . N .. . O . P . . . . . Q . R . .. . . . . . . S . . . . . . .. . . . . . T . U . . . . . .. . . . . . . . . . . . . . .. . . . . . V . W . . . . . .. . . . . . . X . . . . . . .. . . . . . . . . . . . . . .`

There are 36 positions on which to place 32-36 barbarians.
This creates a new upper bound of 24 villagers required to fulfill the requirements.
 Posted by SilverKnight on 2004-01-28 19:34:41

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