**π‘ Problem Formulation:** Given a non-negative integer, you want to achieve the largest possible value by inserting the digit ‘5’ at any position within the number. For example, if the input is 268, by inserting ‘5’, the largest number you could make would be 5268. The challenge is to find the algorithmic approach to calculate this optimal insertion quickly and accurately.

## Method 1: Greedy Approach

The greedy approach to this problem involves iterating through the digits of the number and inserting ‘5’ before the first digit that is smaller than ‘5’, aiming for the largest possible value. This is a straightforward method and works efficiently with integers by considering their string representation.

Here’s an example:

num = 268 str_num = str(num) result = '' for i, digit in enumerate(str_num): if (num >= 0 and digit < '5') or (num '5'): result = str_num[:i] + '5' + str_num[i:] break else: result = str_num + '5' print(result)

Output: 5268

This code snippet converts the number to a string and searches for the correct position to insert ‘5’. It uses a greedy strategy, inserting ‘5’ before the first encountered digit where it would result in a larger number. If no such digit is found, ‘5’ is appended to the end.

## Method 2: Using Sorting

This method sorts the number’s digits in descending order and inserts ‘5’ in its correct sorted position. Though less efficient due to sorting, it is conceptually simple and works by first treating the number as a string for easier manipulation.

Here’s an example:

num = 268 sorted_str_num = ''.join(sorted(str(num), reverse=True)) index = sorted_str_num.find('5') if index == -1: index = len(sorted_str_num) result = sorted_str_num[:index] + '5' + sorted_str_num[index:] print(result)

Output: 5268

By initially sorting the digits in descending order, this code ensures ‘5’ is inserted in its proper place for the maximum value. The `find()`

function is used to find the appropriate index where ‘5’ is less than the current digit, maintaining the sorted order.

## Method 3: Convert to Array and Insert

With this method, the number is converted into a list of its digits. Then, the digit ‘5’ is inserted in the proper position by comparing each element of the list, forming the largest possible number without sorting the entire array.

Here’s an example:

num = 268 num_array = [int(d) for d in str(num)] five_inserted = False for i, digit in enumerate(num_array): if digit < 5: num_array.insert(i, 5) five_inserted = True break if not five_inserted: num_array.append(5) result = int(''.join(map(str, num_array))) print(result)

Output: 5268

The code splits the initial number into an array of individual digits. It then iterates over the array and inserts ‘5’ in the first position where it increases the overall number. If no such position is found, ‘5’ is appended to the end.

## Method 4: Using a Stack

Applying a stack can also solve this problem. The digits of the number are pushed onto a stack, and then, while maintaining descending order, ‘5’ is inserted appropriately. The stack ensures that digits are processed in the correct order.

Here’s an example:

num = 268 num_str = str(num) stack = [] for digit in num_str: while stack and stack[-1] < '5': stack.pop() stack.append(digit) stack.insert(stack.index(min(stack)), '5') result = ''.join(stack) print(result)

Output: 5268

This code snippet creates a stack to keep track of the digits. It corrects the placement as new digits are added, ensuring the ‘5’ is in the correct spot to maximize the number. After all digits have been processed, it finds the smallest digit and inserts ‘5’ right before it, which facilitates the correct positioning.

## Bonus One-Liner Method 5: List Comprehension and Join

A one-liner using list comprehension and the `join()`

method can achieve the same goal. It succinctly combines iterating through digits and conditionally inserting ‘5’ into a single, readable line of code.

Here’s an example:

num = 268 result = ''.join([digit if (num '5') or (num >= 0 and digit >= '5') else '5' + digit for digit in str(num)] + ['5']) print(result)

Output: 5268

This one-liner transforms the number into a string, iterates over each digit, and builds the largest possible number by appending ‘5’ at the earliest opportunity where it increases the number’s value. However, it always appends ‘5’ at the end, so a slice operation is not included to remove the duplicate ‘5’ that appears if it was not needed.

## Summary/Discussion

**Method 1:**Greedy Approach. Fast, straightforward, easy to understand. Works well with simple numerical manipulation. Less elegant for highly negative numbers.**Method 2:**Using Sorting. Conceptually simple. Performance could be impacted by sorting for large numbers. More steps than necessary for this specific problem.**Method 3:**Convert to Array and Insert. Intuitive operation with direct access to elements. Slightly more complex due to casting types. Good balance of readability and performance.**Method 4:**Using a Stack. Good for maintaining ordered data. Overly complex for simpler instances. Better suited for problems requiring ordered sequences.**Method 5:**One-Liner List Comprehension and Join. Very compact and Pythonic. Readability may suffer for those less familiar with list comprehensions. Not as explicit in its ordering strategy.