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. or 2. color;
[*] every vertex that lies on the side BC has been colored in either the 2. or 3. color;
[*] every vertex that lies on the side CD has been colored in either the 3. or 4. color;
[*] every vertex that lies on the side DA has been colored in either the 4. or 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). rectanglecombinatoricscombinatorics unsolved