Six friends were playing a game of indoor soccer together, but got into a fight during their game. Upset with each other, they decided to position themselves in the (square) 300x300 foot gymnasium so as to maximize the distance between the closest pair. Where should they each stand?
If we were maximizing the separation between 5 points, the solution would be to put one point in each corner and one in the middle, whereas if we were maximizing the separation between 7 points, an inscribed hexagon with a seventh point in the center would be the solution. So for 6 points, the solution might be a hibrid:
.o.......o
..........
.....o....
o.........
.........o
....o.....
The five lowerleft points form a series of equilateral triangles, with four of the five on the sides of the square, while the sixth point is in the upper right corner.
If the distance beween points is x, and distance from the central point to the bottom or left edge is y, then
y = xcos(15) and
y + x/sqrt2 = 300
thus
xcos(15) = 300  sqrt2
x(cos(15) + 1/sqrt2) = 300
x = 179.315
Edited on August 17, 2004, 2:13 pm
Edited on August 17, 2004, 2:15 pm
Edited on August 17, 2004, 2:16 pm

Posted by Bryan
on 20040817 14:11:46 