Question number 19. How many different paths are there between the points A and B, only traveling along the edges in the directions of the arrows shown? There are three possible ways to begin. We can move to the right. We can move up. Either of those moves will remain in the front face here, or we can move across away from the point A and switch faces. We'll consider each of those separately and enumerate all the possibilities. So I made some copies of this diagram. Let's begin by listing all the paths we would obtain by moving up from A and seeing this vertex here first. So from here, we can move across and end up at the far vertex, and once we are there, we have really no other choice but to move across to B, so that will give us just one path. If we instead moved to the right and saw this vertex second, then we have two choices. We can leave the face, end up over here, and then at B, or we can move to the right, end up over here, and then at B. So that's two possibilities, and this gives us a total of three ways of finding B if we move up from A. Okay, next, let's look at what happens when we go to the right and remain in the same face. If we go and see this vertex first, we can either go up as our first choice. We will see this vertex, and we've been here already. From here, there are two ways to reach B. We can then choose to stay in the same face, move to the right over here, and then again, we can find B in two different ways by going first up and then across, or across and then up. So two possibilities here. And finally, let's look at what happens if we go across to that vertex. We can either continue up and then to B, or we can continue across and then to B. So two more choices, and we see that going to the right gives us six paths from A to B. Finally, the option we haven't considered is what happens if we begin at A and then move away from A to the opposite face. So see this vertex first. We have the option of going to the right and seeing this vertex second, which we have visited already. There are two ways to finish. Or we can go up and see this vertex, and there is only one way to finish from here. So that tells us that moving away from A in this direction gives us three possibilities, and we have a total of 12 different ways to reach B from A.

path enumeration

distinct paths

directional movement

combinatorial analysis

vertex navigation