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Five Points Living In a Unit Square


Suppose you have placed five random points all in, or on, a square of side length 1.

True or False: The distance between at least two of the points will be less than or equal to SQRT(2)/2.

Once you have solved the above problem, consider this extension...

Suppose you have placed nine random points all in, or on, a cube of side length 1.

True or False: The distance between at least two of the points will be less than or equal to SQRT(3)/2.

And, can you generalize your results to n-dimensional unit squares?


A small tidbit (thanks to Alexander Bogomolny): In any set of 51 points inside a unit square, there are always three points that can be covered by a circle of radius 1/7.

 

Source: E. Barbeau et al's Five Hundred Mathematical Challenges (1995)


Hint: Draw a square...think about where the points could land...

Also, think about the special distance of SQRT(2)/2...how is it related to a unit square?

 


Solution Commentary: The Barbeau et al text suggests: "Divide the square into four squares of side 1/2. By the Pigeonhole Principle, one of these four squares contains at least two of the points, whose distance apart must be no greater than the diagonal of the square of side 1/2, namely SQRT(2)/2."

Clever!