Extreme Math
Home Study Assignment #3

Matilda likes to walk to school by a different route each day. Fortunately, she lives in a boring midwestern city where all the streets are in a rectangular grid, as shown in the figures below.

Figure 1: An obvious way to walk to school	Figure 2: A more interesting way to walk to school	Figure 3: This way doesn’t count, as it’s more than 17 blocks!

As you see, Figure 1 shows a straightforward way to walk to school: 6 blocks east, then 11 blocks north, for a total of 17 blocks.

But of course there are other possibilities. For example, Figure 2 shows a more interesting path: 3 blocks east, then 7 blocks north, then 2 blocks east, then 4 blocks north, and finally 1 block east — still for a total of 17 blocks.

Matilda would like to walk to school a different way each day. (But she never wants to walk more than 17 blocks, so the method in Figure 3 doesn’t count, since it takes 19 blocks.) She wants to know whether it’s possible to walk a different 17-block route every day for an entire school year. Your job is to figure it out for her.

As usual, you will want to start with a simpler example, since there are far too many routes to count. So here is a sequence of questions that you can explore as a way of solving Matilda’s problem:

1. Six of Matilda’s classmates live on the street where the school is located — one of them 1 block away from school, one 2 blocks, one 3 blocks, and so forth up to 6 blocks. Figure out how how ways there are for each of them to walk to school (always, of course, walking no more than the minimum distance). To simplify things, we will assume here that everyone always lives at or near a corner, not in the middle of a block. Summarize your findings for this situation by making a chart. The top row should show the number of blocks the student has to walk (1, 2, 3, 4, 5, or 6), the bottom row the number of ways for that student to get to school:
   
   
   
   
   
   
   
   
   
2. Now suppose Matilda lived somewhere on the street next to the one where the school is located — such as the east-west street just south of the school. So she could walk, say, 1 block north and 5 blocks east, or 1 block north and 3 blocks east, or some such thing, depending on where she lived. Experiment with different locations in this situation, and then summarize your findings in a chart like the one above.
   
   
   
   
   
   
   
   
   
3. Here’s another idea: By a strange coincidence, there are exactly seven of Matilda’s classmates who live exactly seven blocks from school. But they all live in different locations, as shown in Figure 4 below, where the little red circles show where they live. Make another chart, but this time use the numbers on the map for the top row. Still use the bottom row of your chart for the number of ways to get to school.

   Figure 4

   
   
   
4. Finally, see whether you have found enough of a pattern so you can answer Matilda’s original question. And — most important and most difficult of all — can you explain why your pattern works?