- #1

- 86

- 0

## Main Question or Discussion Point

1) Suppose that the streets of a city are laid out in a grid with streets running north–south and east–west. Consider the following scheme for patrolling an area of 16 blocks by 16 blocks. An officer commences walking at the intersection in the center of the area. At the corner of each block the officer randomly elects to go north, south, east, or west. What is the probability that the officer will

a. reach the boundary of the patrol area after walking the first 8 blocks?

b. return to the starting point after walking exactly 4 blocks?

2) How many k-digit ternary sequences (sequences of 0s, 1s, and 2s) have the total number of 0s and 1s as even number of 0s and 1 even?

3) Find a recurrence relation for the number of regions n lines divide the plane into. Assume all lines are straight, no lines are parallel and no three lines intersect in the same point. For example: A single line divides the plane into two regions.

How to solve these questions?

a. reach the boundary of the patrol area after walking the first 8 blocks?

b. return to the starting point after walking exactly 4 blocks?

2) How many k-digit ternary sequences (sequences of 0s, 1s, and 2s) have the total number of 0s and 1s as even number of 0s and 1 even?

3) Find a recurrence relation for the number of regions n lines divide the plane into. Assume all lines are straight, no lines are parallel and no three lines intersect in the same point. For example: A single line divides the plane into two regions.

How to solve these questions?