This page is designed to be read as a link from Creativity Diamond – Outside the box and will probably not make much sense if you haven’t either read that or my book Opening the Creativity Diamond. Please check them out and come back when you have. You’re welcome to stay but may feel as though you’re missing something!
I have two solutions to the problem of joining nine dots with three lines; the first is for real people, the second for mathematicians (who may or may not be real – I am undecided).
Three lines for real people
So, what if the dots aren’t points but actually have a size? This means that (with careful angles – you may want to expand the picture) you can join up a row of three dots with a line that clips the top of the first dot and the bottom of the second. Extend this line out until it ends level with the centre of the middle row then draw a horizontal line back through that row.
Finally come in again with another angled line clipping the dots on the bottom row.
As an aside the lines (1) and (3) are parallel – something of more interest to mathematicians.
I like this solution and in my experience it is easy to explain to people. When I’ve used this in training you can see people’s expressions change when they get the answer.
Three lines for mathematicians
With apologies to non-mathematicians here is a more esoteric solution.
As far as I understand it (would any mathematicians reading please feel free to correct me) we could choose to have a non-Euclidean space. Still with it? If so we might have the situation that parallel lines can meet at infinity. This means that all (!) you have to do is draw a semi-infinite line through the top row, an infinite line back through the middle then a semi-infinite line back through the bottom.
And you thought it was tricky?!