The shape of a military base is an equilateral triangle of side kilometers. Security constraints make cellular phone communication possible only within kilometers. Each of soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.
Solution
Divide any side of the equilateral triangle into four equal segments as follows. Then construct through the interior points to the sides parallel lines to the equilateral triangle.
In this way we get a net of the military base consisting in equilateral triangles of side kilometers each. Since there are soldiers, according to the Pigeonhole Principle, at each moment at least two of them are in a triangle of the previous net, hence they can communicate.
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