Point Coloring

Probability Level 2

Is it possible to color every point on the plane by $\color{#D61F06}\text{red}$ , $\color{#20A900}\text{green}$ or $\color{#3D99F6}\text{blue}$ so that

all the three colors are used, and
any line of the plane receives at most two distinct colors?

No Yes

1 solution

Simon Kaib
Aug 17, 2019

One way to color the plane:

Color all points red.
Replace $1$ point with green.
Pick a line which contains the green point and replace all of its red points with blue.

For sure, the first $2$ requirements are met. Since the only line which contains green and blue does not contain red, there exists no line which contains all $3$ colors!

We call a triangle a rainbow triangle if its vertices have all the three colors.

Bonus: What if we add requirement that no rainbow triangle has unit area?

Brian Lie - 1 year, 9 months ago

It is hard to imagine that it is possible as all $3$ colors would lie dense in the plane and every line is packed densely with $2$ colors. This problem is beyond me. Could you please reveal the solution? I am very curious.

Simon Kaib - 1 year, 9 months ago

The answer is yes. Such coloring is contained in the proof of Monsky's theorem, which states that it is not possible to dissect a square into an odd number of triangles of equal area.

Brian Lie - 1 year, 9 months ago

Point Coloring

1 solution

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