23.2. Validity of Paths in a Square Grid

In question 2 you are given a square grid where you have the coordinate (0, 0) and you want to go to the coordinate (n, n) and you have cells here and the only movements which are allowed to you is that you can at a time you can either go one cell either to the right from the current cell or to the top of the current cell and you have to find out the number of valid paths.

In this grid, you start at the bottom-left corner at (0, 0) and your goal is to reach the top-right corner at (n, n). The only movements allowed are to go right (R) or up (T), which means you can move one cell at a time either towards the right or towards the top. Each valid path from the start to the end must consist of exactly 'n' right movements and 'n' up movements. Students can visualize this as moving along a grid of squares where each step either increases the x-coordinate (go right) or the y-coordinate (go up).

Think of navigating a city where you can only travel north or east. If you are at a street corner at (0, 0) and want to reach the corner at (n, n), you can only go north to move up into the next block or east to move to the next street. This restriction gives you a clear way to consider how many unique routes you can take to reach your destination.

For instance: one valid path could be that I go top from the current cell and then again I go top and then again I go top then I go top and then I do right, right, right and right. Whereas I can take a path where I go right and top, right and top, top, top, right and right.

Here, examples of valid paths illustrate how we can reach the destination (n, n). The first example shows a path of all upwards movements followed by all right movements. The second example mixes these movements. Both paths adhere to the rule of not going left or down, making them valid. Students need to recognize that any combination of 'n' rights and 'n' ups forms a valid path as long as no other movements are made.

Imagine a game where you are traversing a board that only allows moving up or right. When you play the game, if you try to move in other directions, like down or left, you lose a turn. So, every time you move either upwards to a new row or to the right to a new column, you are effectively choosing pathways through this grid, similar to finding shortcuts in a maze.

Whereas I cannot do the following. I am not allowed to do the following that I go right and right and then again come back left and then top and then bottom.

An invalid path would occur if a movement breaks the rules, such as moving left or down. Such moves would take you away from your goal, violating the constraints set for valid paths in the square grid. It emphasizes the importance of adhering strictly to the allowed movements. Therefore, only paths that maintain a rightward or upward trajectory towards (n, n) are considered valid.

Consider solving a puzzle where each piece can only be moved forward or up on the board. If you mistakenly try to move a piece backward, that violates the game rules. Just as in the puzzle, in the grid pathfinding task, any backward or downward movement disallows the path from being valid, similar to unable to retrace your steps in a linear path.

So, for solving this or finding the number of valid paths I do the following so imagine this is the set of all your valid paths and you have another set which denote the set of all strings of length 2n. And the 2n length string has equal number of R symbols and T symbols.

To find the number of valid paths, we can represent each path as a string composed of 'R' for right movements and 'T' for top movements. For a grid size of 'n', a valid path would require exactly 'n' rights and 'n' ups, making the string length total 2n. Establishing a bijection between valid paths and these strings allows us to calculate the number of valid paths in an organized manner.

Think of a dance routine where every right (R) movement is a step to the right and every top (T) movement is a step up. Each complete performance routine can be represented as a sequence of these moves. By counting how many times you can do both steps in succession while following the choreography (the grid rules), we understand how many unique ways there are to complete the dance.

And we know that the cardinality of the set of all strings of length 2n which has n number of R symbols and n number of T symbols is C(2n, n).

The number of valid paths is equivalent to the number of ways we can arrange 'n' R's and 'n' T's in a sequence of length '2n'. This mathematical expression is calculated using combinations, represented by C(2n, n), which essentially means choosing 'n' positions for R's out of '2n' total positions. This gives us the total number of unique sequences (or paths) we can take.

Imagine you have 10 different colored marbles, and you want to arrange 5 red and 5 blue marbles. The number of unique arrangements of these marbles can be calculated in a similar way to the above paths. Thus, arranging different colors in various orders helps you visualize how many routes you can take consistently while following path rules.

--