MathDB

Let

ABCD

be a rectangle consisting of unit squares. All vertices of these unit squares inside the rectangle and on its sides have been colored in four colors. Additionally, it is known that:
[*] every vertex that lies on the side

AB

has been colored in either the

1.

2.

color; [*] every vertex that lies on the side

BC

has been colored in either the

2.

3.

color; [*] every vertex that lies on the side

CD

has been colored in either the

3.

4.

color; [*] every vertex that lies on the side

DA

has been colored in either the

4.

1.

color; [*] no two neighboring vertices have been colored in

1.

and

3.

color; [*] no two neighboring vertices have been colored in

2.

and

4.

color.
Notice that the constraints imply that vertex

A

has been colored in

1.

color etc. Prove that there exists a unit square that has all vertices in different colors (in other words it has one vertex of each color).

Problem Statement