We note firstly that the square can be partitioned into four 'sub' squares of sides one inch each. Given five points, at least two must occupy the same 1-inch square. Then we see from the diagram that the greatest distance between two points in a 1-inch square is the hypotenuse with length obtained from the Pythagoream theorem, i.e.
Ö (1) 2 + (1) 2 = Ö2 inches
Which is also the distance between diametrically opposite corners.
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