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# Math/Geometry brain teaser

Bartender
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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

blacksmith
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Hm ... seems to me half the time the answer is nonunique.

Ranch Hand
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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
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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

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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
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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

 Don't get me started about those stupid light bulbs.
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