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Challenge 254: A Colouring Challenge

a Consider the 3 x 3 set of points both of whose co-ordinates are positive integers from 1 to 3 [the set {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}].

Show that it is possible to colour six of these points red in such a way that none of the red points is the midpoint of any other pair of red points [for instance, if (1, 3) and (3, 1) are coloured red, then (2, 2) cannot be, as it is the midpoint of these two].

Show that it is not possible to colour more than six of these points red in such a way that none of the red points is the midpoint of any other pair of red points.

b Now consider the 4 x 4 set of points both of whose co-ordinates are positive integers from 1 to 4.

What is the maximum number of these points that you can colour red in such a way that none of the red points is the midpoint of any other pair of red points?

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