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 midpoint of a line joining a pair of chosen points, is also a lattice point?
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Copyright © 19962018 Alexander Bogomolny
Five points are chosen at the nodes of a square lattice (grid). Why is it certain that at least
one midpoint of a line joining a pair of chosen points, is also a lattice point?
The midpoint of the line joining two grid points (x_{1}, y_{1}) and (x_{2}, y_{2}) is located at ((x_{1} + x_{2})/2, (y_{1} + y_{2})/2). The latter will be a grid point iff its coordinates are integers. The xcoordinate will be integer iff x_{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 ycoordinate. And out of the selected three points, at least two have ycoordinate with the same parity.
There is an Five Lattice Points of this problem.
References
 A. Engel, ProblemSolving Strategies, Springer Verlag, 1998, pp. 6162
 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 © 19962018 Alexander Bogomolny