Three same colour side triangle.

There are six points in space, no three on the same line. Each pair of points is connected by a line segment. Each segment is colored black or white. Show that among the triangles formed by these segments there is always at least one with all of its sides the same color.


Answer:


Let A be any one of the six points. Note that there are five segments with A as one endpoint, each segment connects A to one of the other five points. Since every segment is colored black or white, at least three of these five segments must have the same color (Pigeon Hole Principle). Without loss of generality we can assume that there are at least three black segments with A as an endpoint. Label the three other endpoints of these black segments B, C, and D. If all three segments BC, BD, and CD are white then ...BCD is colored white. Otherwise at least one of the segments BC, BD or CD is black and then the triangle formed by A and that segment has all black sides.

The following animation is quite stimulating!!! Just a few frames of pictures giving rise to the illusion of rotating.