[math-fun] Cross section coloring problem

A friend of mine has a cube, composed of several convex objects. He has to provide 2 cross sections of this cube, from left and right. What is the minimum number of colors needed so that no two shapes within a cross section have the same color? I'm thinking it is 9, based on the Earth-Mars 2-pire problem. http://www.fortunecity.com/emachines/e11/86/mpire.html (Has any progress been made on this problem? Has a map needing 10 colors been found?) How about if top, left, right cross sections must be provided? What is the minimal number of colors then? Ed Pegg Jr