## What will we cover in this tutorial?

Given an input of a maze, how can we solve it?

That is – given an input like this the following.

############# S # # # # # ## # # # # ## ### # # # # ######### # # # # E #############

How can we find a path through the maze starting from **S** and ending at **E**.

############# S.#.....#...# #.#. ##...#.# #.#. ## ###.# #... # .# ######### #.# # #.E #############

We will create a program that can do that in Python.

## Step 1: Reading and representing the Maze

An good representation of the maze makes it easier to understand the code. Hence, the first part is to figure out how to represent the maze.

To clarify, imagine that this maze.

v

Was represented by one long string, like this.

maze = "#############S # # ## # ## # ## # ## ### ## # ########## # ## # E#############"

While this is possible, if you also know the dimensions (8 rows and 13 columns), it makes things difficult to navigate in. Say, you are somewhere in the maze, and you want to see what is right above you. How do you do that?

Yes, you can do it, but it is complex.

Now image you have the following representation.

maze = [['#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#'], ['S', ' ', '#', ' ', ' ', ' ', ' ', ' ', '#', ' ', ' ', ' ', '#'], ['#', ' ', '#', ' ', ' ', '#', '#', ' ', ' ', ' ', '#', ' ', '#'], ['#', ' ', '#', ' ', ' ', '#', '#', ' ', '#', '#', '#', ' ', '#'], ['#', ' ', ' ', ' ', ' ', ' ', ' ', ' ', '#', ' ', ' ', ' ', '#'], ['#', '#', '#', '#', '#', '#', '#', '#', '#', ' ', '#', ' ', '#'], ['#', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', '#', ' ', 'E'], ['#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#']]

This will make it easier to navigate in the maze.

Then **maze[1][0]** represents row 1 (2nd row) and column 0 (1st column), which is **S**.

If we assume that the maze is represented in a file, then we can read that file and convert it with the following code.

def load_maze(file_name): f = open(file_name) maze = f.read() f.close() return maze def convert_maze(maze): converted_maze = [] lines = maze.splitlines() for line in lines: converted_maze.append(list(line)) return converted_maze maze = load_maze("maze.txt") maze = convert_maze(maze) print(maze)

## Step 2: Creating a print function of the maze

As you probably noticed, the print statement of the maze (**print(maze)**) did not do a very good job.

[['#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#'], ['S', ' ', '#', ' ', ' ', ' ', ' ', ' ', '#', ' ', ' ', ' ', '#'], ['#', ' ', '#', ' ', ' ', '#', '#', ' ', ' ', ' ', '#', ' ', '#'], ['#', ' ', '#', ' ', ' ', '#', '#', ' ', '#', '#', '#', ' ', '#'], ['#', ' ', ' ', ' ', ' ', ' ', ' ', ' ', '#', ' ', ' ', ' ', '#'], ['#', '#', '#', '#', '#', '#', '#', '#', '#', ' ', '#', ' ', '#'], ['#', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', ' ', '#', ' ', 'E'], ['#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#', '#']]

In order to be able to follow what happens in the code later on, it would be nice with a better representation of the code.

def print_maze(maze): for row in maze: for item in row: print(item, end='') print() print_maze(maze)

This will print the maze in a more human readable way.

############# S # # # # # ## # # # # ## ### # # # # ######### # # # # E #############

## Step 3: Finding the starting point of the maze

First thing first. We need to find where we enter the maze. This is where the **S** is located.

We could have some assumptions here, say, that the starting point is always located on the outside borders of the maze. To make it more general, we will assume that the starting point, **S,** can be located anywhere in the maze.

This makes the algorithm to find it easy, but of course not effective in worst case, as we need to check all locations in the maze.

def find_start(maze): for row in range(len(maze)): for col in range(len(maze[0])): if maze[row][col] == 'S': return row, col start = find_start(maze) print(start)

Which will go through all rows and columns one by one and check if it is the starting point. When found, it will return it immediately.

Notice, that we do assume that we have a maze with at least one row, also that the starting point exists. If not, it will not return anything.

## Step 4: Find if there is a path in the maze

This is where a stack comes in handy. If you are new to what a stack is, then check out this tutorial.

For the purpose here, we are not looking for performance, we just need the concept of a stack.

To see if there is a way through the maze, you need to check all paths. As some leads to the same points, we need to keep track of where we have been. If we reach a dead end, where we have been, we need to backtrack to the last point with options, and check them out.

Wow. That was a lot of talking. How can we understand that a bit better.

Let’s make a simple maze and see it.

### S E ###

When you are at start, **S**, you have (in theory) 4 possible ways to go. **Up, Down, Right, Left. **In the above example you can only go **Right**. Let’s do that, and mark the position we just left.

### [email protected] ###

Now we have marked where we have been with **X** and where we are with **@**. Then we can check all possible ways. **Up, Down, Right, Left.** Again we can only go **Right**, as **Left** we have already been there.

Luckily, going Right is the exit **E** and we are done.

Now that was a simple maze. Now how does a stack come into the picture? Well, we used it secretly and in a very simple case.

So let’s try a bit more complex case.

##### S # E # # # # # #####

Now let’s also have a stack with the positions we can visit. We will use the coordinate system as the representation of the list of lists we use to represent the maze. That means that the upper left corner is (0, 0) and the lower right corner is (5, 5) in this case.

The starting point is (1 , 0).

Now, let’s have a stack in the picture and put the staring point on that stack.

##### S # E # # # # # ##### stack -> (1, 0)

When we enter we look for points on the stack. Yes, we have one point (1, 0). Then we pop it off and check all possible positions. **Up, Down, Right, Left.** Only one is possible, **Left**, which is (1, 1). Updating where we been, then it looks like this.

##### X # E # # # # # ##### stack -> (1, 1)

Notice, that the stack changed and we marked the first point with **X**.

We do the same. Pop of the stack and get (1, 1) and check **Up, Down, Right, Left.** Only down is possible, which leaves us in (2, 1). Now the picture looks like this.

##### XX# E # # # # # ##### stack -> (2, 1)

Then we do the same. We pop of the stack and get (2, 1) and check **Up, Down, Right, Left**. Now we can go both **Down** and **Right**, which is (3, 1) and (2, 2), respectively. Hence, we push both of them on the stack.

##### XX# E #X # # # # (3, 1) ##### stack -> (2, 2)

Depending on the order we put thins on stack, it will look like that.

Now we pop of the top element (3, 1) and check **Up, Down, Right, Left**. But there are no way to go from there, it is a dead end. Then we mark that spot and do not push anything on the stack.

##### XX# E #X # #X# # ##### stack -> (2, 2)

Now we pop (2, 2) of the stack, as this is the first possibility we have to go another way in the maze. Then check **Up, Down, Right, Left**. We can only go **Right**. After marking pushing we have.

##### XX# E #XX # #X# # ##### stack -> (3, 2)

Now we can both go up and down and add that to the stack.

##### XX# E #XXX# #X# # (3, 2) ##### stack -> (3, 4)

Now we pop (3, 2) of the stack. Then we check all and see we can only go **Right**.

##### XX#XE #XXX# #X# # (4, 2) ##### stack -> (3, 4)

Then we pop (4, 2) of the stack and see it is the exit, **E**. That is just what we needed.

The stack keeps track on all the possible choices we have along the way. If we reach a dead end (also, if we have visited all around us and marked it by X), we can look at the last possible choice we skipped, which is on the top of the stack.

That actually means, that we do not need to move backwards along the road we went, we can go directly to the point on the stack. We know we can reach that point, as we have already visited a neighboring point, which is connected. Also, as we mark everything along the way, we do not go in endless loops. We only visit every point at most once.

Now, if there was no way to the exit, **E**, this will eventually terminate when there is no points on the stack.

## Step 5: Implementing the algorithm

The above can be made into a simple algorithm to determine whether there is a way through the maze.

The pseudo code for the algorithm could be something like this.

stack.push(start) while stack is not empty location = stack.pop() if location is marked continue to next location in loop if location is marked as exit - return True (we found the exit) for loc as Up, Down, Left, Right: if loc is valid and not marked add to stack return False (if all locations are tried (stack empty) there is not way to the exit)

Now let’s turn that into code.

# This is only a helper function to see if we have a valid positino. def is_valid_position(maze, pos_r, pos_c): if pos_r < 0 or pos_c < 0: return False if pos_r >= len(maze) or pos_c >= len(maze[0]): return False if maze[pos_r][pos_c] in ' E': return True return False def solve_maze(maze, start): # We use a Python list as a stack - then we have push operations as append, and pop as pop. stack = [] # Add the entry point (as a tuple) stack.append(start) # Go through the stack as long as there are elements while len(stack) > 0: pos_r, pos_c = stack.pop() if maze[pos_r][pos_c] == 'E': print("GOAL") return True if maze[pos_r][pos_c] == 'X': # Already visited continue # Mark position as visited maze[pos_r][pos_c] = 'X' # Check for all possible positions and add if possible if is_valid_position(maze, pos_r - 1, pos_c): stack.append((pos_r - 1, pos_c)) if is_valid_position(maze, pos_r + 1, pos_c): stack.append((pos_r + 1, pos_c)) if is_valid_position(maze, pos_r, pos_c - 1): stack.append((pos_r, pos_c - 1)) if is_valid_position(maze, pos_r, pos_c + 1): stack.append((pos_r, pos_c + 1)) # We didn't find a path, hence we do not need to return the path return False

## The full code

Here we present the full code. It prints out the state through the processing. It expects a file called **maze.txt** with the maze in it. See below for a possible file content.

def load_maze(file_name): f = open(file_name) maze = f.read() f.close() return maze def convert_maze(maze): converted_maze = [] lines = maze.splitlines() for line in lines: converted_maze.append(list(line)) return converted_maze def print_maze(maze): for row in maze: for item in row: print(item, end='') print() def find_start(maze): for row in range(len(maze)): for col in range(len(maze[0])): if maze[row][col] == 'S': return row, col from time import sleep # This is only a helper function to see if we have a valid positino. def is_valid_position(maze, pos_r, pos_c): if pos_r < 0 or pos_c < 0: return False if pos_r >= len(maze) or pos_c >= len(maze[0]): return False if maze[pos_r][pos_c] in ' E': return True return False def solve_maze(maze, start): # We use a Python list as a stack - then we have push operations as append, and pop as pop. stack = [] # Add the entry point (as a tuple) stack.append(start) # Go through the stack as long as there are elements while len(stack) > 0: pos_r, pos_c = stack.pop() print("Current position", pos_r, pos_c) if maze[pos_r][pos_c] == 'E': print("GOAL") return True if maze[pos_r][pos_c] == 'X': # Already visited continue # Mark position as visited maze[pos_r][pos_c] = 'X' # Check for all possible positions and add if possible if is_valid_position(maze, pos_r - 1, pos_c): stack.append((pos_r - 1, pos_c)) if is_valid_position(maze, pos_r + 1, pos_c): stack.append((pos_r + 1, pos_c)) if is_valid_position(maze, pos_r, pos_c - 1): stack.append((pos_r, pos_c - 1)) if is_valid_position(maze, pos_r, pos_c + 1): stack.append((pos_r, pos_c + 1)) # To follow the maze print('Stack:' , stack) print_maze(maze) # We didn't find a path, hence we do not need to return the path return False maze = load_maze("maze.txt") maze = convert_maze(maze) print_maze(maze) start = find_start(maze) print(start) print(solve_maze(maze, start))

Then for some possible content of file **maze.txt** needed by the program above. To make the above code work, save the content below in a file called **maze.txt** in the same folder from where you run the code.

############# S # # # # # ## # # # # ## ### # # # # ######### # # # # E #############

## Further work

The above code only answers if there is a path from **S** to **E** in the maze. It would also be interesting to print a route out. Further improvements could be finding the shortest path.