TAXI-STAND

Record the date and the title 'Shirts and Pants' on a clean page in your journal (write your name at the top of the page if you are sharing or not using your own journal).

A taxi driver is given a specific territory of a town, shown below.

One night, the driver is only called out three times; each time, she travels out from the taxi stand and picks up a passenger at one of the intersections (indicated by a coloured dot on the image below).

To pass the time when everything is quiet, she considers all the possible routes she could have taken to each pick-up point and wonders if she could have chosen a shorter route.

On the image below, imagine that the horizontaial and vertical lines (grid) are roads, and that you must stay on the lines.

The problem is to find the shortest route to three specific points (coloured dots) on the grid and to determine the number of shortest routes to each point.

Solve the problem for yourself and develop a way to convince others that your solution is correct.

All trips start/originate at the taxi stand in the upper-left-hand corner of the grid (See: Image 1.)

1. What is the shortest route from the taxi stand to each of three different destination points?
2. How do you know it is the shortest route?
3. Is there more than one shortest route to each point?
   1. If not, why not?
   2. If so, how many?

Find away to convince others that your answer is correct.

This is a variation of the Towers (Pascal's Triangle) problems.

Using Powell’s et al. (2003) notation to denote coordinates on the taxicab grid, (n,r) indicates a point n blocks away from the taxi stand and r blocks to the right. So the blue dot is at (5,1), the red dot is at (7,4), and the green dot is at (10,6).

Taking the shortest route means going in two directions only (down and to the right).

Finding the number of shortest paths from the taxi stand (0,0) to any point (n,r) involves:

• The number of ways to select r segments of one kind of movement in a path that includes two kinds of movements;
  • For example, the number of shortest paths to (n,r) is C(n,r).