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Clumsy Factorial - Solution & Explanation

MediumMathStackSimulation20 min readAsked at: Amazon, Microsoft, Meta +1
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Problem Statement

The factorial of a positive integer n is the product of all positive integers less than or equal to n.

  • For example, factorial(10) = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.

We make a clumsy factorial using the integers in decreasing order by swapping out the multiply operations for a fixed rotation of operations with multiply '*', divide '/', add '+', and subtract '-' in this order.

  • For example, clumsy(10) = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1.

However, these operations are still applied using the usual order of operations of arithmetic. We do all multiplication and division steps before any addition or subtraction steps, and multiplication and division steps are processed left to right.

Additionally, the division that we use is floor division such that 10 * 9 / 8 = 90 / 8 = 11.

Given an integer n, return the clumsy factorial of n.

 

Example 1:

Input: n = 4
Output: 7
Explanation: 7 = 4 * 3 / 2 + 1

Example 2:

Input: n = 10
Output: 12
Explanation: 12 = 10 * 9 / 8 + 7 - 6 * 5 / 4 + 3 - 2 * 1

 

Constraints:

  • 1 <= n <= 104

Approach Overview

Problem Overview: Clumsy factorial modifies the traditional factorial by applying operations in a repeating order: *, /, +, -. Starting from N, you process numbers down to 1. Multiplication and division have higher precedence than addition and subtraction, so the evaluation order matters. The challenge is simulating these operations correctly while respecting operator precedence.

Approach 1: Stack-Based Simulation (O(n) time, O(n) space)

This method directly simulates the expression evaluation using a stack. Push the first value N onto the stack, then iterate from N-1 down to 1 while cycling through the operations *, /, +, and -. For multiplication and division, pop the top element, apply the operation with the current number, and push the result back. For addition, push the number. For subtraction, push the negative value so the final sum automatically handles it. After processing all numbers, sum the stack to produce the result. This works because the stack naturally handles precedence for multiplication and division before addition and subtraction. The approach is straightforward and mirrors how expression evaluation works in many stack-based parsers.

Approach 2: Direct Arithmetic Pattern (O(n) time, O(1) space)

The sequence of operations creates a predictable pattern in the result. Instead of storing intermediate values, you can track the current term and accumulate results as you iterate from N to 1. Multiplication and division are applied immediately to the current term, while addition and subtraction finalize the previous term and start a new one. Another common optimization uses the observation that clumsy factorial results follow a repeating pattern depending on N % 4 for larger values. By leveraging this arithmetic behavior, you compute the final value with constant extra memory. This approach fits well when the goal is minimizing memory usage or when recognizing patterns in math-based problems.

Both techniques rely on straightforward iteration and controlled operator application, essentially simulating the evaluation of an arithmetic expression. The stack approach emphasizes correctness and clarity, while the arithmetic method focuses on reducing auxiliary space and exploiting patterns that emerge from the operator cycle. Problems involving expression evaluation or sequential operations often appear in simulation tasks.

Recommended for interviews: The stack-based approach is the most expected solution. It clearly demonstrates understanding of operator precedence and expression evaluation. Interviewers can easily follow the logic because every operation is simulated step by step. After presenting that solution, mentioning the O(1) arithmetic pattern optimization shows deeper insight into the mathematical structure of the problem.

Approach 1: Stack-Based Approach

This approach uses a stack data structure to store intermediate results. We iterate over the numbers from n to 1, applying the operations in the sequence *, /, +, -. At each step, we push results onto a stack and pop them for further calculations. This helps in maintaining the correct order of operations as multiplication and division have higher precedence.

The C solution uses a dynamic array as a stack to compute the clumsy factorial. We iterate over each operator, applying it to the current number and maintaining results on the stack. Finally, we sum all elements in the stack to get the result.

Code

CC++JavaPythonC#JavaScript

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Java

Python

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JavaScript

Complexity

Time complexity: O(n), as we iterate through each number once.
Space complexity: O(n), due to the stack storing results.

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Approach 2: Direct Arithmetic Approach

This approach leverages direct arithmetic manipulation without extra data structures like a stack, utilizing the order of operations directly within a loop. This is useful when maintaining space efficiency.

The C solution computes the clumsy factorial using nested conditional arithmetic to handle cycles of operations without using extra data structures.
The computation handles blocks of four operations and accounts for dangling operators when n becomes low.

Code

CC++JavaPythonC#JavaScript

C

C++

Java

Python

C#

JavaScript

Complexity

Time complexity: O(n)
Space complexity: O(1)

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Approach 3: Stack + Simulation

The calculation process of clumsy factorial can be seen as a simulation of a stack.

We define a stack stk, initially we push n into the stack, and define a variable k to represent the current operator, initially k = 0.

Then we start from n-1, enumerate x, and decide how to handle x based on the current value of k:

  • When k = 0, it represents a multiplication operation, we pop the top element of the stack, multiply it by x, and then push it back into the stack;
  • When k = 1, it represents a division operation, we pop the top element of the stack, divide it by x, take the integer part, and then push it back into the stack;
  • When k = 2, it represents an addition operation, we directly push x into the stack;
  • When k = 3, it represents a subtraction operation, we push -x into the stack.

Next, we update k = (k + 1) \mod 4.

Finally, the sum of the elements in the stack is the answer.

The time complexity is O(n), and the space complexity is O(n). Where n is the integer N given in the problem.

Code

PythonJavaC++GoTypeScript

Python

Java

C++

Go

TypeScript

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Complexity Comparison

ApproachComplexity
Stack-Based Approach

Time complexity: O(n), as we iterate through each number once.
Space complexity: O(n), due to the stack storing results.

Direct Arithmetic Approach

Time complexity: O(n)
Space complexity: O(1)

Stack + Simulation—

Detailed Complexity Analysis

ApproachTimeSpaceWhen to Use
Stack-Based SimulationO(n)O(n)Best for clarity and interviews when demonstrating operator precedence handling
Direct Arithmetic PatternO(n)O(1)Useful when minimizing memory usage or leveraging mathematical patterns

Video Solution

1006. (Medium) Clumsy Factorial - Daily Leetcode (Day 87) • yeetcode • 1,980 views views

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Frequently Asked Questions

Is Clumsy Factorial easy or hard?
Clumsy Factorial is generally rated as a medium difficulty problem. The main challenge is correctly handling operator precedence while applying operations in a fixed repeating order. Once the stack simulation idea is clear, the implementation becomes straightforward.
Clumsy Factorial Python/Java solution
Python and Java implementations typically use a stack or list structure. Iterate from N to 1, apply operations in a repeating cycle, and update the stack accordingly. Python often uses a list as a stack, while Java implementations commonly use Stack or ArrayDeque.
How to solve Clumsy Factorial in O(n)?
Iterate from N down to 1 while cycling through the operations *, /, +, -. Use a stack to apply multiplication and division immediately to the top element, and push positive or negative numbers for addition and subtraction. After the loop, sum all stack elements to produce the final result. The iteration ensures linear O(n) time.
What is the best approach for Clumsy Factorial?
The stack-based simulation is the most reliable approach. It processes numbers from N to 1 while cycling through the operations *, /, +, and -. Multiplication and division are applied to the top of the stack, while addition and subtraction push new values. This method runs in O(n) time and O(n) space and closely mirrors how arithmetic expressions are evaluated.
Is Clumsy Factorial asked at Google/Amazon/Meta?
Clumsy Factorial represents a typical medium-level problem involving arithmetic simulation and stack usage. Variants of expression evaluation and stack-based operator handling appear in interviews at companies like Amazon, Google, and Meta, especially for mid-level algorithm rounds.
What data structure is used in Clumsy Factorial?
The most common solution uses a stack. The stack stores intermediate values so multiplication and division can be applied before addition and subtraction, preserving correct operator precedence during the simulation.
What is the time complexity of Clumsy Factorial?
Both common solutions run in O(n) time because each number from N down to 1 is processed once. The stack-based approach uses O(n) extra space to store intermediate values, while the optimized arithmetic approach reduces space complexity to O(1).

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