Suppose you want to create a path between each number on Pascal’s Triangle. For this exercise, suppose the only moves allowed are to go down one row either to the left or to the right.
We will code the path by using bit strings. In particular, a 0 will be used for each move downward to the left, and a 1 for each move downward to the right. So, for example, consider the first five rows of Pascal’s Triangle below, and the path shown between the top number 1 (labelled START) and the left-most 3.
This path involves starting at the top 1 labelled START and first going down and to the left (code with a 0), then down to the left again (code with another 0), and finally down to the right (code with a 1). Hence, this path would be coded with binary string 001. This code is then recorded at the ending location on the triangle.
For Option #1, complete the following tasks based on the coding scheme described above:
- Determine if there are additional paths between the START (number 1 at the top) and end point (leftmost 3). If so, describe them in words, by tracing the path on the triangle, and as binary strings. Then record all such binary strings at the ending location.
- Find and record at least 5 paths using binary strings between the START location and numbers in rows 4 and 5. Compare the binary strings for each number and discuss why those similarities and differences might exist. Also, explain any connections you notice between the number of possible paths ending at each location and the corresponding entry of Pascal’s Triangle.
- Discuss what information about an endpoint and the paths leading to it can be gathered from the length of its binary strings and the number of 1s in them.
- Explain the addition rule of Pascal’s Triangle in your own words in terms of the path coding scheme you worked with in this assignment.
- Note the type of symmetry that Pascal’s Triangle has and explain it in terms of the paths.
