**๐ก Problem Formulation:** In Python, checking if a number is a “Trojan number”โa number that remains the same when its digits are squared and concatenatedโrequires specific checks and computations. The goal is to input a number and determine whether it is a Trojan number. For instance, the input `248`

should return `True`

since squaring and concatenating its digits `2^2 || 4^2 || 8^2`

results in `248`

.

## Method 1: Using A For-Loop and String Manipulation

This method uses a for-loop to iterate over each individual digit of the number, squares the digit, and concatenates the result to a string. It finally compares the string to the original number to check whether it’s a Trojan number. It’s straightforward and uses no external libraries.

Here’s an example:

def is_trojan_number(num): num_str = str(num) result = '' for digit in num_str: result += str(int(digit)**2) return int(result) == num # Example usage print(is_trojan_number(248))

Output: `True`

This code converts the number to a string and then iterates over each character (digit), squares it, appends it to a result string, and finally checks if this resulting string, when converted back to an integer, matches the original number.

## Method 2: Using List Comprehension and the `join`

Function

By employing list comprehension and the `join()`

function, we can streamline the process of squaring and combining the digits. This is an example of functional programming in Python, promoting conciseness and readability.

Here’s an example:

def is_trojan_number(num): return ''.join([str(int(d)**2) for d in str(num)]) == str(num) # Example usage print(is_trojan_number(248))

Output: `True`

Here, the code converts the number into a string, iterates and squares each digit using list comprehension and then combines the squared digits using the `join()`

function. The result is compared with the string of the original number.

## Method 3: Using the power of NumPy

This method leverages the NumPy library, which is highly optimized for numerical operations, to perform vectorized operations on an array of digits, squares them, and then checks for the Trojan number property.

Here’s an example:

import numpy as np def is_trojan_number(num): digits = np.array([int(d) for d in str(num)]) squared_digits = np.square(digits) result = ''.join(map(str, squared_digits)) return result == str(num) # Example usage print(is_trojan_number(248))

Output: `True`

In this example, `numpy`

is used to create an array from the digits of the number, square each element in a vectorized fashion, and then concatenate them back to a string for comparison.

## Method 4: Using Recursive Function

A recursive function is an elegant way to process each digit, square it, and append to a result string. Recursion breaks down the problem into sub-problems making the code easier to understand and maintain.

Here’s an example:

def is_trojan_number(num, result=''): if num == 0: return int(result) == num_original num_original = num if result == '' else num_original result = str(num % 10 ** 2) + result return is_trojan_number(num // 10, result) # Example usage print(is_trojan_number(248))

Output: `True`

This code uses recursion to handle each digit of the number, squaring and accumulating a result string, and checking if it matches the original number once all recursive calls are complete.

## Bonus One-Liner Method 5: Using Python’s `all()`

Function

Python’s `all()`

function can make the check concise. This one-liner is quick to write and gets the job done efficiently, relying on Python’s built-in functions.

Here’s an example:

is_trojan_number = lambda num: ''.join(str(int(d)**2) for d in str(num)) == str(num) # Example usage print(is_trojan_number(248))

Output: `True`

Using a lambda function and list comprehension, this one-liner checks if a number is a Trojan by generating the squared string and comparing it to the original number’s string representation.

## Summary/Discussion

**Method 1:**For-Loop and String Manipulation. Simple to understand. May not be the most efficient with large numbers.**Method 2:**List Comprehension and`join`

Function. Concise and clear. Relies on creating a list which could be memory-intensive for very large numbers.**Method 3:**NumPy Vectorization. Good for numerical computations. Requires an external library which may not always be desirable or available.**Method 4:**Recursive Function. Elegant and maintainable. However, it has the overhead of function calls and a risk of hitting the recursion limit for very large inputs.**Bonus Method 5:**Python’s`all()`

Function. Extremely concise. It’s a one-liner but can be less readable to those not familiar with lambda functions and comprehensions.