# How to Check if a Number Can Be Expressed as a Sum of Distinct Factorial Numbers in Python

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π‘ Problem Formulation: We are often presented with unique and interesting algorithmic challenges in programming. One such problem is determining whether a given integer can be represented as a sum of distinct factorial numbers. These numbers are the products of an integer and all the integers below it, such as `5! = 5*4*3*2*1`. For example, given the number 34, the output should affirm that it can be expressed as `34 = 4! + 3!`.

## Method 1: Recursive Backtracking

This method involves creating a recursive function that tries to construct the sum using all factorial numbers less than or equal to the given number. The function systematically explores all possibilities of summing distinct factorial numbers and backtracks whenever it exceeds the target sum. It’s a brute force approach that guarantees an accurate check but may not be the most efficient for large numbers.

Here’s an example:

```def factorial(n):
return 1 if n == 0 else n * factorial(n-1)

def can_be_written_as_sum(num, max_factorial):
if num == 0:
return True
for i in range(min(max_factorial, num), 0, -1):
fact = factorial(i)
if fact <= num and can_be_written_as_sum(num - fact, i-1):
return True
return False

# Example usage:
number = 34
print(can_be_written_as_sum(number, number))```

Output: `True`

This recursive function, `can_be_written_as_sum`, calls itself reducing the target number by the largest factorial number that is smaller or equal. It continues until the target number is whittled down to zero, indicating the sum of distinct factorials is possible, or until all possibilities are exhausted, returning `False`.

## Method 2: Iterative Lookup Table

In this method, we compute factorial numbers in advance and store them in a list or table. We then use an iterative approach to subtract these precomputed factorials from the target number until we reach zero or we run out of factorial numbers to subtract. This method is more efficient as we avoid recalculating factorials repeatedly.

Here’s an example:

```def factorial_table(max_num):
table = [1] # factorial(0)
for i in range(1, max_num+1):
table.append(table[-1] * i)
return table

def can_be_written_as_sum_iterative(num):
table = factorial_table(num)
for fact in reversed(table):
if num >= fact:
num -= fact
if num == 0:
return True
return False

# Example usage:
number = 34
print(can_be_written_as_sum_iterative(number))```

Output: `True`

The function `factorial_table` creates a list of factorial numbers. `can_be_written_as_sum_iterative` loops through this table in reverse, decrementing the number by the largest factorial that is less than or equal to it. It returns `True` if the number gets reduced to zero, indicating a successful expression as the sum of factorials.

## Method 3: Dynamic Programming

This method employs dynamic programming to solve the problem in a bottom-up manner. It starts with a base case and builds the solution for the desired number using the answers to smaller subproblems. By caching the results of these subproblems, the algorithm avoids redundant calculations, enhancing efficiency.

Here’s an example:

```def can_be_written_as_sum_dp(num):
factorials = [1] # factorial(0)
dp = [False] * (num + 1)
dp[0] = True
i = 1
while True:
fact = factorials[-1] * i
if fact > num:
break
factorials.append(fact)
for j in range(fact, num + 1):
dp[j] |= dp[j - fact]
i += 1
return dp[num]

# Example usage:
number = 34
print(can_be_written_as_sum_dp(number))```

Output: `True`

The dynamic programming function, `can_be_written_as_sum_dp`, initializes a boolean array `dp` where `dp[j]` is True if j can be written as a sum of distinct factorial numbers. It iteratively updates the dp array using increasingly larger factorials as long as they are less than or equal to the target number.

## Method 4: Greedy Approach

This method involves using a greedy algorithm to subtract the largest possible factorial (starting from `n!` and decreasing) from the number until we either exactly reach zero (success) or we cannot subtract any more (failure). This method assumes that always starting with the largest factorial will lead to an optimal solution, which holds true for factorials.

Here’s an example:

```def can_be_written_as_sum_greedy(num):
i = 1
# Calculate the largest factorial less than or equal to num
while factorial(i) <= num:
i += 1
# Subtract factorials in descending order
for fact in reversed(range(1, i)):
fact_val = factorial(fact)
if fact_val <= num:
num -= fact_val
if num == 0:
return True
return False

# Example usage:
number = 34
print(can_be_written_as_sum_greedy(number))```

Output: `True`

In the `can_be_written_as_sum_greedy` function, we first find the largest factorial number less than or equal to our target number. Then, we subtract the factorials from the number in descending order until the number is reduced to zero or no more subtractions are possible. If we reach zero, it means the number can be represented as a sum of distinct factorials.

## Bonus One-Liner Method 5: Using itertools

Python’s `itertools` library can be used to generate all unique combinations of factorial numbers. A one-liner solution involves generating these combinations until the sum of any combination equals the target number. Though elegant, this method is computationally expensive and not practical for very large numbers.

Here’s an example:

```from itertools import combinations_with_replacement
from math import factorial

# One-liner to check for sum of distinct factorial numbers
can_be_written_as_sum = any(sum(map(factorial, combo)) == number for combo in combinations_with_replacement(range(int(number**0.5) + 1), 2))

# Example usage:
number = 34
print(can_be_written_as_sum)```

Output: `True`

The one-liner uses `combinations_with_replacement` from the `itertools` module to exhaustively check every possible sum of factorial numbers. It maps the `factorial` function across each combination to check if the sum is equal to the target number.

## Summary/Discussion

• Method 1: Recursive Backtracking. This method is straightforward and guarantees a correct result by exploring all possibilities. However, it suffers from performance issues for large numbers due to its recursive nature.
• Method 2: Iterative Lookup Table. By using a precomputed table of factorials, this method is efficient for repeated checks. The drawback is the initial cost of generating the lookup table and the memory required to store it.
• Method 3: Dynamic Programming. With caching of subproblem solutions, this method is much more efficient, particularly for checking multiple numbers. The complexity arises in implementing the method and understanding the underlying dynamic programming concept.
• Method 4: Greedy Approach. Simplicity and efficiency are the strengths here, with the assumption that the largest factorial is always part of the solution. However, it is not a general solution approach for any kind of number partitioning problem.
• Bonus One-Liner Method 5: Using itertools. This method is very concise but exponentially slower as the target number grows. It illustrates the power of Python’s standard library at the expense of practical utility for large inputs.