# 5 Optimal Ways to Arrange N Rooks on a Chessboard in Python

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π‘ Problem Formulation: The task at hand is to determine the number of ways to place `n` rooks on an `n x n` chessboard such that none of the rooks can attack each other. In other words, we require that no two rooks share the same row or column. Given an integer `n`, the desired output is a single integer representing the count of distinct valid arrangements.

## Method 1: Factorial Calculation

This method leverages the mathematical fact that the number of ways to arrange `n` rooks is the same as the number of permutations of `n` items, which is the factorial of `n` (written as `n!`). We use Python’s built-in library `math` to find the factorial.

Here’s an example:

```import math

def number_of_ways(n):
return math.factorial(n)

# Example usage:
arrangements = number_of_ways(4)
print("Number of arrangements for 4 rooks:", arrangements)
```

Output: Number of arrangements for 4 rooks: 24

By calling `math.factorial(n)`, this code snippet calculates the number of ways to uniquely arrange `n` rooks on a chessboard. It is both efficient and concise, utilizing Python’s optimized factorial function.

## Method 2: Iterative Factorial Implementation

If for some educational or constraint reasons you can’t use the `math` library, you can implement factorial calculation iteratively. This method manually computes the factorial by iteratively multiplying integers from 1 up to `n`.

Here’s an example:

```def number_of_ways(n):
result = 1
for i in range(1, n + 1):
result *= i
return result

# Example usage:
arrangements = number_of_ways(4)
print("Number of arrangements for 4 rooks:", arrangements)
```

Output: Number of arrangements for 4 rooks: 24

This code defines a function `number_of_ways` that calculates the total number of non-attacking arrangements of rooks by multiplying integers from 1 to `n`. It’s simple and requires no external libraries, making it a portable solution.

## Method 3: Recursive Factorial Implementation

Another way to compute the factorial is through recursion. This method calls itself with decreasing values of `n` until it reaches the base case of `n` equals 1.

Here’s an example:

```def number_of_ways(n):
if n == 1:
return 1
else:
return n * number_of_ways(n - 1)

# Example usage:
arrangements = number_of_ways(4)
print("Number of arrangements for 4 rooks:", arrangements)
```

Output: Number of arrangements for 4 rooks: 24

In this recursive function `number_of_ways`, the base case returns 1. If `n` is greater than 1, the function calls itself with `n - 1`, multiplying the return values. It’s elegant, but not as memory efficient for large `n` due to the call stack limit.

## Method 4: Dynamic Programming

Dynamic programming can optimize the recursive factorial calculation by storing the intermediate results, thus avoiding redundant computation. This approach uses a bottom-up technique.

Here’s an example:

```def number_of_ways(n):
dp = [1] * (n + 1)
for i in range(2, n + 1):
dp[i] = dp[i-1] * i
return dp[n]

# Example usage:
arrangements = number_of_ways(4)
print("Number of arrangements for 4 rooks:", arrangements)
```

Output: Number of arrangements for 4 rooks: 24

The function `number_of_ways` uses a list `dp` to save computed factorials, calculating each number only once. It is a more efficient algorithm for very large `n` as compared to the recursive approach.

## Bonus One-Liner Method 5: Lambda with Reduce

For fans of functional programming, this one-liner uses the `functools.reduce` function together with a lambda to compute the factorial. The `reduce` function applies the lambda cumulatively to the items of the given range, effectively calculating the factorial.

Here’s an example:

```from functools import reduce

number_of_ways = lambda n: reduce(lambda x, y: x * y, range(1, n+1))

# Example usage:
arrangements = number_of_ways(4)
print("Number of arrangements for 4 rooks:", arrangements)
```

Output: Number of arrangements for 4 rooks: 24

In this approach, the lambda expression in conjunction with `reduce` from the `functools` module creates a compact factorial function. It’s a neat one-liner, but potentially less readable to those unfamiliar with functional programming tenets.

## Summary/Discussion

• Method 1: Factorial Calculation. The most straightforward and performance-efficient method using Python’s standard library.
• Method 2: Iterative Factorial Implementation. Provides a clear algorithmic understanding without library dependencies, good for educational purposes.
• Method 3: Recursive Factorial Implementation. Elegant, but less practical for large `n` due to potential stack overflow.
• Method 4: Dynamic Programming. Excellent option for large `n`, optimizing on the recursive method by storing intermediate results.
• Method 5: Lambda with Reduce. Compact and functional, great for one-liners but may sacrifice some readability.