Five Points in Square Lattice
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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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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 interactive version of this problem.
References
- A. Engel, Problem-Solving Strategies, Springer Verlag, 1998, pp. 61-62
Copyright © 1996-2009 Alexander Bogomolny
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