Puzzle – 6×6 Grid: How Many Ways?

Question: 

You begin in the top left corner of a 6×6 grid, and your objective is to move to the bottom right corner. There are just two directions you can move: right or down. Both diagonal and backward movements are prohibited. How many different ways are there to get from the start to the finish?

Solution:

The problem is to count all the possible paths from the top left to the bottom right of a 6×6 grid with the constraint that from each cell you can either move only to the right or down.
we can solve this problem using two approaches:-

Approach 1

Using Combinatorics:

Three conditions of reaching at the last end

Example of some ways to reach endpoint

We can see here that the number of paths from starting left point to the right ending is not depending on the way of the path, it depends on the number of rows and columns taken to reach the end. Whenever we face such kind of problem, where we have a choice to take or a fixed number of rows or columns to be taken in grid. We can think about mathematics in those cases. Here, we are going to use a mathematical concept, called combinatorics.

Why combinatorics?

In this case of a 6×6 grid, all the paths must consist of a total of 10 moves, 5 down and 5 right, our job is to select the 5 right moves from the collection of 10 moves. we must employ a certain number of rows and columns (5 of the total 10 blocks) to travel from the left beginning to the right end.
if we choose 5 rows box then the answer is 10c5=252 and the same if we choose 5 column answer is 10c5=252.

Approach 2

Using Pascal Triangle:

 Pascal Approach

If we know the number of ways to reach the left box and an upper box of a given box, then, the number of ways to reach at the given box, we can easily visualize, it will be the sum of both because we can either reach here from the left box paths or upper box paths. As shown in the figure here, we can reach the left box in A ways and reach the upper blocks in B ways, so the total answer to reach will be A+B. 

Here, for the first row, they can only be taken from the left move, not from the upward move. So the answer is 1 for the first row and similarly, for the first column they can be only taken from the upward move, not from the left move. So the answer here is also 1, and for the remaining grid, it is calculated using the Pascal Approach which is explained before. To reach the right endpoint, we have taken the sum of (126+126), which are moves at its top and left.

Practical Approach:

Visit https://www.geeksforgeeks.org/count-possible-paths-top-left-bottom-right-nxm-matrix/ to practice and have detailed solutions using recursion and dynamic programming.