# Painting a plane!

Paint the points on a plane with three colors, so that the points on each line are a maximum of 2 colors, and all three colors are used. (Math Festival 1990)

You can do this by painting as follows:

Paint the origin blue, the rest of the x-axis red, and everything else green.

Any horizontal line will be either all green, or red and blue. Any nonhorizontal line will be all green except where it intersects the x-axis, so it will only have two colors: green and something else.

(You could actually paint the x-axis in any combination of red and blue you wanted, as long as it used both of them.)

• You beat me to it by a matter of seconds. I'll just add that any line at all can be used in lieu of the x axis. – msh210 Jan 6 at 13:18
• You could also paint both x and y axes red (except the origin), or any number of lines going thru the center. – Florian F Jan 6 at 13:29
• @Deusovi why a horizontal line could be blue?? – MathR Jan 6 at 13:39
• @Reza It cannot be entirely blue. It can be both red and blue, if it's the line at y=0. – Deusovi Jan 6 at 13:41
• @Deusovi Thanks – MathR Jan 6 at 13:42