Challenge 254: A Colouring Challenge
You don't need to join all the dots in this challenge, just work out how to colour them in to solve the problem! As half term is coming up, the next weekly maths challenge will appear in two weeks time.
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?