Over on the BoardAndCardGames beta we've been discussing a theoretical problem about the board game Carcassonne. While attempting to construct an upper bound on the maximum score we ran into the problem:
What is the maximum number of edges in a planar bipartite graph which have partitions of order 7 and 15?
By playing around on paper the answer appears to be 40, and this seems to have something to do with the Euler Characteristic and the three utilities problem, but I was wondering if there's a straight-forward proof.
Thanks!