## 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?

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Copyright © 1996-2018 Alexander Bogomolny
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 _{1}, y_{1})_{2}, y_{2})_{1} + x_{2})/2, (y_{1} + y_{2})/2)._{1} and x_{2} 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

- A. Engel,
*Problem-Solving Strategies*, Springer Verlag, 1998, pp. 61-62 - A. Soifer,
*Mathematics as Problem Solving*, Springer, 2009 (2nd, expanded edition) - A. Soifer,
*Geometric Etudes in Combinatorial Mathematics*, Springer, 2010 (2nd, expanded edition)

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Copyright © 1996-2018 Alexander Bogomolny