Coloring problems are positions where you do not know whether the depicted units are Black or White. You have to find out the colors, knowing that the position is legal.
This kind of problems has been invented by A. K. Kniest who published an article on the topic in Diagramme und Figuren, 1964. It has been much developed since the 80's.
Prize, Isreal Ring Tourney, 1966/71
0+9. Color the pieces. Last move?
Here one of the two Ks is in double check by Rd8 and Qc6. The only possible explanation is that the last move was -1.c7xd8=R+. Thus we conclude that Rd8, Qc6 and Kd6 are White; Kc8 is black. Then further impossible checks are avoided by setting Ne8 and Rf6 White; Pb7 Black. A black Pb7 entails that Ba8 is promoted, so that Ba8 and Pa7 White.
Personally, I am a great lover of this stipulation. I don't know why. Perhaps it is because these coloring problems can be quite complex and still (almost) only require inventory and balance.
Because I am such a fan, the WWW RetroCorner includes a special page where all its coloring problems are gathered!
Some problems ask you to color the squares of a white grid. Really, you are asked to rotate the board so that you obtain a legal position.