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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?

Submit your solution

Please do send in your solution to this problem to  weeklymaths@kcl.ac.uk  You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide you with feedback on your solution.

 

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