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Grid exercise (Posted on 2016-05-18) Difficulty: 2 of 5
Plot five points at random at the intersections of a coordinate grid. Between each pair of points a line segment can be drawn.

Prove that the midpoint of at least one of these segments occurs at an intersection of grid lines.

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

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Solution Solution | Comment 1 of 2
Each coordinate of a grid point is either odd or even, making four possible parities for a point: (odd, odd), (odd, even), (even, odd), and (even, even). 

(odd+odd)/2 and (even+even)/2 are both integers.  This implies given two grid points with the same parity then their midpoint must also be on a grid point.

Because there are five given points in all then there must exist at least one pair whose parities match and subsequently that midpoint must be a grid point.

  Posted by Brian Smith on 2016-05-18 11:58:48
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