Abstract
The following problem was posed in the 27th International Mathematics Olympiad (1986): One is given a finite set of points P n in the plane, each point having integer coordinates. Is it always possible to colour some of the points red and the remaining points white in such a way that, for any straight line L parallel to either one of the coordinate axes, the difference (in absolute value) between the number of white points and red points on L is not greater than 1? It is not hard to see that the answer to the above question is “yes”. In this note we generalize this result, and show that P n can be coloured with m ( m⩾2) colours in such a way that for any straight line parallel to either one of the coordinate axes, the difference (in absolute value) between the number of points coloured i and the number of points coloured j is at most 1, 1⩽ i< j⩽ m. A conjecture for the higher dimensional case is presented.
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