2.1. Inductive Formulation of the Grid Path

Try these first

1. 1.

   How do you calculate paths to (2, 1)?

   Hint

   Consider the potential paths from left and below.

2. 2.

   What happens at the leftmost column?

   Hint

   Think about the available directions at boundaries.

3. 3.

   What is the formula to calculate Paths(i,j)?

   • Paths(i-1,j) - Paths(i,j-1)
   • Paths(i-1,j) + Paths(i,j-1)
   • Paths(i,j-1) * Paths(i,j-2)

   Hint

   Think about only possible movements on the grid.

4. 4.

   True or False: A hole on a grid contributes to the total path count.

   • True
   • False

   Hint

   Reflect on how obstacles affect pathways.

5. 5.

   Evaluate how many paths could exist for a grid of size 5x5 with holes at (2,2) and (3,3). Show your calculations.

   Hint

   Ensure that whenever a hole is reached, all paths leading into it are declared zero.

6. 6.

   If a grid has a series of holes along one row, analyze how that impacts total path calculations if starting from a top-left corner to bottom-right corner.

   Hint

   Evaluate critical paths that must navigate around those holes!