Five lattice points and a hidden midpoint

Question

Scatter five points anywhere in the plane, with the only rule that each point has integer coordinates (both x and y are whole numbers). The points can be anywhere you like, near or far, in any pattern. Prove that some two of your five points have a midpoint that also has integer coordinates. And show that five is the smallest number for which this is guaranteed.

Accepted Answer

Sort points by the parity of their coordinates. For any point (x, y), look only at whether x is even or odd, and whether y is even or odd. There are just four possible "parity types": (	ext{even},	ext{even}),\quad (	ext{even},	ext{odd}),\quad (	ext{odd},	ext{even}),\quad (	ext{odd},	ext{odd}). Apply the pigeonhole principle. You have five points but only four parity types, so at least two points share the same type, call them (x1, y1) and (x2, y2). Same type means x1 and x2 have the same parity, and y1 and y2 have the same parity. Check the midpoint. Two numbers of the same parity alw…

Five lattice points and a hidden midpoint

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