## Five Points in Square Lattice

Five points are chosen at the nodes of a square lattice (grid). Why is it certain that at least one mid-point of a line joining a pair of chosen points, is also a lattice point?

Solution Five points are chosen at the nodes of a square lattice (grid). Why is it certain that at least one mid-point of a line joining a pair of chosen points, is also a lattice point?

The midpoint of the line joining two grid points (x1, y1) and (x2, y2) is located at ((x1 + x2)/2, (y1 + y2)/2). The latter will be a grid point iff its coordinates are integers. The x-coordinate will be integer iff x1 and x2 have the same parity, i.e., iff they are either both even or both odd. Out of 5 points, at least three satisify this condition. But the same is true of the y-coordinate. And out of the selected three points, at least two have y-coordinate with the same parity.

There is an Five Lattice Points of this problem.

### References

1. A. Engel, Problem-Solving Strategies, Springer Verlag, 1998, pp. 61-62
2. A. Soifer, Mathematics as Problem Solving, Springer, 2009 (2nd, expanded edition)
3. A. Soifer, Geometric Etudes in Combinatorial Mathematics, Springer, 2010 (2nd, expanded edition) • 200 points have been cosen on a circle, all with integer number of degrees. Prove that the points there are at least one pair of antipodes, i.e., the points 180° apart.
• If each point of the plane is colored red or blue then there are two points of the same color at distance 1 from each other.
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