If you can only move in the directions of the arrows
(right and down, not up or left),
how many paths are there from A to B?
This is called Taxicab Geometry because it mirrors
the real-life situation of a car, or taxicab, navigating
one-way streets in a place like Manhattan, New York City.
Step 1:
We realize that to reach point B,
we must first reach point C or D.
So the number of ways to reach B =
SUM of (ways to reach C) + (ways to reach D).
And so it is for each point: the number of ways
to reach it = the sum of the ways to reach the one to the left
and the one above.
Step 2:
Return to A and work forward.
There is just one way to go to the next intersection
in either direction, and then 1 + 1 = 2 ways to
reach the first point diagonally opposite A.
Then, similarly, we can start filling in the rest of the points,
as shown here.
Step 3:
and in full here
If we rotate the grid 45 degrees,
it starts to look more familiar: Pascal's Triangle.
And in general, the number of paths to a point in the grid
is (width + length) choose (width) or
(width + length) choose (length),
which are always the same.
That is illustrated by any given path, such as this one.
The total number of steps is (length + width).
We can number them consecutively from 1 to (length+width).
Then we choose "width" of those to be the verticals
(2,5,6,8,10 in the example),
leaving "length" of them to be the horizontals (1,3,4,7,9,11).
