**π‘ Problem Formulation:** Converting numbers from base 10 to base 3 is a common task in computing and mathematics. The goal is to take an integer value (in the standard decimal system) and represent it in ternary form, which uses only the digits 0, 1, and 2. For example, the decimal number 5 would be represented as 12 in base 3.

## Method 1: Using a While Loop

This method involves using a while loop to divide the number by 3 repeatedly and to store the remainders in reverse order to form the base 3 number. It is straightforward and easy to implement, making it ideal for those who are learning or prefer procedural programming styles.

Here’s an example:

def decimal_to_base3(num): if num == 0: return '0' base3 = '' while num > 0: base3 = str(num % 3) + base3 num = num // 3 return base3 print(decimal_to_base3(5))

Output: `'12'`

This code snippet defines a function `decimal_to_base3()`

which takes an integer `num`

as its argument and converts it into a base 3 string. The function repeatedly divides `num`

by 3 and prepends the string representation of the remainder to a result string `base3`

. This continues until `num`

becomes zero, which signifies that the conversion is complete.

## Method 2: Using Recursion

Recursion is a technique where a function calls itself to solve smaller instances of the same problem. In this method, the conversion process is broken down into smaller and simpler steps, which are then combined to arrive at the final base 3 representation.

Here’s an example:

def decimal_to_base3_recursive(num): if num == 0: return '0' elif num == 1: return '1' else: return decimal_to_base3_recursive(num // 3) + str(num % 3) print(decimal_to_base3_recursive(5))

Output: `'12'`

The code defines a function `decimal_to_base3_recursive()`

that uses recursion to calculate the base 3 representation of an integer. It calls itself with the quotient of the number divided by 3, concatenating the remainder at each step. This builds up the base 3 number as the recursive calls return, resulting in the final string.

## Method 3: Using an Iterative List

By storing the remainders in a list and then joining them into a string, this approach benefits from simplicity and efficiency. It’s a clean and Pythonic way to keep track of the digits obtained from the division by 3.

Here’s an example:

def decimal_to_base3_list(num): if num == 0: return '0' digits = [] while num: digits.append(str(num % 3)) num //= 3 return ''.join(reversed(digits)) print(decimal_to_base3_list(5))

Output: `'12'`

This function, `decimal_to_base3_list()`

, utilizes a list to collect the digits in reverse order as the number is divided by 3. After the loop, the collected digits are reversed and joined into a string to represent the final base 3 number.

## Method 4: Using the Format Function

Python’s format function allows custom formatting of numbers. By using the `:b`

formatting specifier, we can leverage Python’s ability to convert integers to a binary string, and with slight modifications, to a base 3 string as well.

Here’s an example:

def decimal_to_base3_format(num): if num == 0: return '0' return format(num, 'b').replace('10', '2').replace('11', '10') print(decimal_to_base3_format(5))

Output: `'12'`

This snippet introduces a clever hack with the `format()`

function. The conversion process begins by transforming the decimal number to a binary string, which is then modified by replacing occurrences of ’10’ with ‘2’ and ’11’ with ’10’ to adapt it to base 3.

## Bonus One-Liner Method 5: Using the bin Function

Although Python does not have a built-in function for converting to base 3 directly, the `bin()`

function can be used to obtain a binary representation of the number, followed by string replacements just as in Method 4.

Here’s an example:

def decimal_to_base3_bin(num): return bin(num)[2:].replace('10', '2').replace('11', '10') print(decimal_to_base3_bin(5))

Output: `'12'`

Here, the function `decimal_to_base3_bin()`

utilizes Python’s `bin()`

function to get the binary representation of the number. It uses string slicing to remove the ‘0b’ prefix and then performs the necessary substitutions to convert it to base 3.

## Summary/Discussion

**Method 1:**Using a While Loop. Pros: Conceptually simple and easy to understand. Cons: Requires manual handling of the string concatenations.**Method 2:**Using Recursion. Pros: Clean and mathematical approach to the problem. Cons: Could be less efficient for large numbers due to the overhead of recursive calls.**Method 3:**Using an Iterative List. Pros: Pythonic way using list and join operations. Cons: Similar performance compared to while loop but can be slightly more memory-intensive.**Method 4:**Using the Format Function. Pros: Utilizes Python standard library functions to reduce code complexity. Cons: May not be obvious at first glance; relies on replacing string patterns which could be error-prone.**Bonus One-Liner Method 5:**Using the bin Function. Pros: Short and concise. Cons: Less readable due to the use of replacements; could be misunderstood without proper comments.