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."