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