Red/Blue Neon Triangle

Six points are scattered around in space so that no three of them are collinear and no four of them are coplanar. The points are named 1, 2, 3, 4, 5, and 6. Each point is connected to each other point by a tubular neon light that is either red or blue. Show that there is some neon triangle that has all of its sides the same color.

Show/Hide Solution

From point A, the most diverse colors would be three one way and the other. Suppose AB, AC, and AD are red.

*This picture is as if all the points are dropped down onto a plane.

If BC or CD or BD are red, we’re done. So, assume they’re all blue, and we’re done again.

Any other configuration of red and blue will lead to a triangle in a similar way.

Leave a Reply

Your email address will not be published. Required fields are marked *