I whipped up another program - this time the Babylonian Royal Game of Ur. My game doesn't have errorchecking for moves yet and it doesn't even care where you place your pieces but it's still playable with a friend next to you. I'm still thinking about what rules I should code into it so, for now, enjoy.
The following image shows what I think makes the most historical sense about the gamepath. Since the related Egyptian 20-Squares game is an obvious variant of this Royal Game of Ur and shows that the 2x3 square portion of the Babylonian game has merely been "unfolded" into a straight line (or rather the Royal Game of Ur has been "folded" into its shape from the 20-Squares game, depending on whichever came first), it helps to narrow down the possibilities of gamepaths considerably. It seems rather safe to conclude the gamepath below. More on this later but essentially, the first four pieces are a safe zone that aren't shared with the opponent and beyond this the pieces proceed cautiously down symmetrically opposed paths to their respective end points where the pieces are then taken off the board as in the Egyptian game Sínat.