Question 28057: suppose 5 points with integer coordinates are chosenat random from the xy-coordinate system. Show that the midpoint of a line segment that connects at least one pair of these points has integer coordinates.
Answer by bmauger(101)   (Show Source):
You can put this solution on YOUR website! Question 27929
I answered the question as explicitly as I could just yesterday at the link abouve. The crux of the answer rests at the fact that every integer must be either odd or even, and that when you pair two evens or two odds together, their midpoint will also be an integer. By analogy if it were in one dimention (a number line) you'd only need 3 points to assure that the midpoint of one of the segments would be on an integer because if the first point were even and the second one were odd a third point would have to be either even or odd, and consequently have an integer midpoint with one of them. 4 points assures that at the worst, all of the even odd pairings will be present with 0 matches. Adding the fifth point assures that it will match one of the 4 points and will have an integer midpoint with it.
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