Number of Digit One
This problem asks us to count the total number of digit ‘1’ appearing in all nonnegative integers less than or equal to a given number n
.
We can solve this problem by considering the contribution of each digit position to the count of ‘1’s.
Here’s a stepbystep explanation:
Divide the Number into Parts: Consider the
i
th digit from the right, dividen
into three parts:left
,curr
,right
.left
contains the digits to the left of thei
th digit.curr
contains thei
th digit.right
contains the digits to the right of thei
th digit.
Calculate the Contribution for each Digit Position:
 If
curr
is 0, the count is contributed by theleft
part and isleft * pow(10, i)
.  If
curr
is 1, the count isleft * pow(10, i) + right + 1
.  If
curr
is greater than 1, the count is(left + 1) * pow(10, i)
.
 If
Sum the Contributions: Sum all the contributions for each digit position to get the final answer.
Here’s the code implementation following these steps:


Key Takeaways:
 The problem can be solved by considering the contribution of ‘1’s at each digit position.
 Dividing the number into parts helps in calculating the contribution for different cases.
 This approach ensures that we consider all possible occurrences of the digit ‘1’ within the given range.
10 Prerequisite LeetCode Problems
“233. Number of Digit One” involves number theory and logic around how numbers are constructed. Here are 10 problem to prepare for this one:
231. Power of Two: Provides practice in understanding the properties of numbers in binary.
7. Reverse Integer: Gives practice in manipulating individual digits of a number.
8. String to Integer (atoi): Develops understanding in interpreting a number in a different format.
9. Palindrome Number: Provides practice in reading a number in both directions.
67. Add Binary: Reinforces understanding of number base conversions.
168. Excel Sheet Column Title: Practices conversion from number to a representation.
171. Excel Sheet Column Number: Inverse of the problem above; practices conversion from representation to number.
172. Factorial Trailing Zeroes: Gives practice in understanding the properties of factorial in relation to number theory.
204. Count Primes: Provides understanding of prime numbers and the Sieve of Eratosthenes, which may be useful in number theory problems.
263. Ugly Number: Provides practice in understanding number factors.
Problem Classification
Problem Statement:Given an integer n, count the total number of digit 1 appearing in all nonnegative integers less than or equal to n.
Example 1:
Input: n = 13 Output: 6
Example 2:
Input: n = 0 Output: 0
Constraints:
0 <= n <= 109
Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.
Visual Model of the Problem
How to visualize the problem statement for this problem?
Problem Restatement
Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.
Abstract Representation of the Problem
Could you help me formulate an abstract representation of this problem?
Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific realworld details?
Terminology
Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?
Problem Simplification and Explanation
Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?
Constraints
Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.
What are the key insights from analyzing the constraints?
Case Analysis
Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.
Identification of Applicable Theoretical Concepts
Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?
Problem Breakdown and Solution Methodology
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Inference of ProblemSolving Approach from the Problem Statement
How did you infer from the problem statement that this problem can be solved using ?
Stepwise Refinement
Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the highlevel solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?
Solution Approach and Analysis
Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.
Thought Process
Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.
From Brute Force to Optimal Solution
Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?
Coding Constructs
Consider the following piece of complex software code.
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?
Language Agnostic Coding Drills
Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.
Dissect the code and identify each distinct concept it contains. Remember, this process should be languageagnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problemsolving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the stepbystep process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.
Targeted Drills in Python
Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Pythonbased coding drills for each of those concepts.
Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problemspecific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.
Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.
Q&A
Similar Problems
Given the problem , identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problemsolving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.


Problem Classification
Language Agnostic Coding Drills
Working with integers: Understanding how to work with integers, including arithmetic operations and conversion to other data types.
Drill: Practice arithmetic operations with integers. Create an integer variable and perform addition, subtraction, multiplication, and division operations on it.
String conversion and manipulation: Knowing how to convert integers to strings, and manipulate strings, such as extracting individual characters or sections of a string.
Drill: Convert an integer into a string, and write a function that iterates through each character in the string.
Lists and List comprehension: Understanding of list data structure, how to create, iterate, index, slice, and modify lists. Also learn about list comprehensions, which are a powerful tool for creating and manipulating lists based on existing lists.
Drill: Create a list of numbers. Write a list comprehension that creates a new list containing the squares of all numbers in the original list.
Conditionals: Understanding how to use conditionals (if, elif, else) to control the flow of your program based on certain conditions.
Drill: Write a function that takes a number as input and prints whether the number is positive, negative or zero.
Loops: Understanding how to use loops, including for and while loops, to repeatedly execute a block of code.
Drill: Write a for loop that iterates over a list of numbers and prints each number.
Using Builtin functions: Understanding and utilizing Python’s builtin functions like
range()
,len()
, andmath.comb()
, and mathematical operations like power (**
).Drill: Use the
math.comb()
function to calculate and print the combinations of a set of elements.Function Definitions: Understanding how to define a function, its input parameters, and its return values. Learn how to call a function with the required arguments.
Drill: Write a function that takes two numbers as input and returns their sum.
When solving this problem, first you define a helper function count()
, which calculates a count based on input parameters. Then, you convert the input integer to a list of its digits, and iterate over this list. In each iteration, you calculate a count based on the digit and its position in the number, and add this count to a running total. You also update a counter variable based on the value of the digit. Finally, after all digits have been processed, you add the final count to the total and return it. This approach relies on understanding combinations and calculating power values.
Targeted Drills in Python
 Working with integers


 String conversion and manipulation


 Lists and List comprehension


 Conditionals


 Loops


 Using Builtin functions


 Function Definitions


With these concepts, you’ll be able to construct the final solution for the given problem.