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# Math/Geometry brain teaser

Ryan McGuire
Ranch Hand
Posts: 1138
8
You're given a random number of points randomly distributed along a line. Where is the point that minimizes the sum of the distances to the original points?

Don't provide the mathematical formula that yields the answer, but rather describe the location qualitatively.

How did you arrive at the answer?

Ryan

Warren Dew
blacksmith
Ranch Hand
Posts: 1332
2
Hm ... seems to me half the time the answer is nonunique.

Jerry Young
Ranch Hand
Posts: 77
Any point on the line that has the same number of points to its right as the nubmer of points to its left.

You pair the points from the most left and most right, the second left and second right, ..., and you measure the sum of the distances of your chosen point to each pair, then you see this.

Ryan McGuire
Ranch Hand
Posts: 1138
8
Originally posted by Jerry Young:
Any point on the line that has the same number of points to its right as the nubmer of points to its left.

You pair the points from the most left and most right, the second left and second right, ..., and you measure the sum of the distances of your chosen point to each pair, then you see this.

Bing bing bing, we have a winner.

Originally posted by Warren Dew:
Hm ... seems to me half the time the answer is nonunique.

True. If there are initially an even number of "random" points, then anywhere on the line segment between the two middle points will satisfy the "minimum total distance" condition.

Ryan

Geoffrey Falk
Ranch Hand
Posts: 171
1
What is the probability distribution of the points? There is no such thing as a uniform distribution of n points on an infinite line.

Geoffrey

Ryan McGuire
Ranch Hand
Posts: 1138
8
Originally posted by Geoffrey Falk:
What is the probability distribution of the points? There is no such thing as a uniform distribution of n points on an infinite line.

Geoffrey

Point taken. Let's edit the problem so that first line is, "You're given a random number of points randomly distributed along a line segment."

Ryan