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Thirty two barbarians (Posted on 2004-01-28) Difficulty: 3 of 5
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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Solution Full Solution (re: Solution) | Comment 9 of 34 |
(In reply to Solution by SilverKnight)

The following diagram shows how to place 16 villagers in such a way that 16 barbarians can line up on the outside and shoot them all (satisfying the constraints of the problem).

There are 8 lines of consequence:
ABFI
CDIK
NMHF
POKH
BGHL
DEFG
MLKJ
EIJO

You'll notice that *EACH* of the 16 letters is at the end of *ONE* (and only one) of these lines.

If we draw a circle around this "grid", each line will intersect this circle in two points (for a total of 16 intersections). We merely put a barbarian at these intersections, and each barbarian will shoot into the circle, along the line. Each of the 16 barbarians will kill a different villager.

. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . A . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . B . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. C . . . . . . . . . . . . . . . .
. . . D . . . E F . G . . . . . . .
. . . . . . . . . . H . . . . . . .
. . . . . . . I . . . . . . . . . .
. . . . . . . J . K L . . . M . . .
. . . . . . . . . . . . . . . . N .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . O . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . P . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
This shows how 16 barbarians kill 16 peasants.

It is now a simple matter to rotate the whole diagram (by say, 5 degrees) and overlay it on top of itself, to double the number of barbarians shooting and villagers killed.
  Posted by SilverKnight on 2004-01-28 17:12:11
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