Confusing Number II

A confusing number is a number that becomes a different number when it’s rotated 180 degrees, and each digit of the number must be valid after rotation. For this problem, you can use a Depth-First Search (DFS) approach to generate all possible confusing numbers from 1 to n and count them.

Here’s a Python function that does just that:

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class Solution:
    def confusingNumberII(self, n: int) -> int:
        valid_numbers = {0: 0, 1: 1, 6: 9, 8: 8, 9: 6}
        count = 0
        
        def dfs(number, rotation, place_value):
            nonlocal count
            if number != rotation:
                count += 1

            for k, v in valid_numbers.items():
                next_number = number * 10 + k
                if next_number <= n and next_number != 0:
                    dfs(next_number, v * place_value + rotation, place_value * 10)

        dfs(0, 0, 1)
        return count

The dfs function is used to build numbers using only the digits in valid_numbers. It keeps track of the current number and its 180-degree rotation. If the number is not equal to its rotation, it’s considered a confusing number.

This function iterates through all possible valid numbers within the given constraints and returns the count of confusing numbers in the range [1, n].

Identifying Problem Isomorphism

“Confusing Number II” can be approximately mapped to “Strobogrammatic Number II”.

Both deal with numbers that can be rotated 180 degrees to form a valid but potentially different number. The concept of numbers that remain valid upon rotation is common between the two problems. In “Confusing Number II”, we seek to count the number of confusing numbers within a range, while in “Strobogrammatic Number II”, we generate all strobogrammatic numbers of a given length.

However, there’s a key difference in the criteria of the problems: a strobogrammatic number becomes identical to the original when rotated 180 degrees, while a confusing number becomes a different valid number.

“Strobogrammatic Number II” is simpler due to its generation-based nature, while “Confusing Number II” is more complex as it requires a counting mechanism within a specified range, and the judging criteria are a bit more nuanced. So, this is an approximate mapping based on the underlying concept, but the problems are not identical.

10 Prerequisite LeetCode Problems

The “1088. Confusing Number II” is about generating numbers under a certain limit where the number is still valid when rotated 180 degrees. The generated number may not be the same but should be valid. This problem involves recursion, depth-first search (DFS), and handling corner cases. Here are some simpler problems to prepare:

784. Letter Case Permutation: This problem can help you understand how to generate all possible combinations of a string. It can help you understand the concept of recursion and depth-first search.
46. Permutations: This problem is a classic recursion problem which helps you understand how to generate all permutations of a given list.
77. Combinations: This is another problem that helps with understanding recursion and backtracking concepts.
22. Generate Parentheses: This problem also helps with recursion and understanding how to generate all valid combinations of a certain format.
17. Letter Combinations of a Phone Number: Similar to the “Letter Case Permutation” problem, it requires generating all combinations of a given input.
79. Word Search: This problem uses a depth-first search in a grid but can be helpful to understand how to handle corner cases and stop conditions.
200. Number of Islands: This problem also uses depth-first search and can help understand the concept of visiting each possible position.
98. Validate Binary Search Tree: This problem helps with understanding how to handle limits when traversing through the recursion.
332. Reconstruct Itinerary: The problem is about constructing a path given a list of edges. DFS is used to solve this problem.
31. Next Permutation: This problem deals with generating the next bigger number which can help in understanding how to manipulate numbers.

Problem Classification

Problem Statement:A confusing number is a number that when rotated 180 degrees becomes a different number with each digit valid. We can rotate digits of a number by 180 degrees to form new digits.

When 0, 1, 6, 8, and 9 are rotated 180 degrees, they become 0, 1, 9, 8, and 6 respectively. When 2, 3, 4, 5, and 7 are rotated 180 degrees, they become invalid. Note that after rotating a number, we can ignore leading zeros.

For example, after rotating 8000, we have 0008 which is considered as just 8. Given an integer n, return the number of confusing numbers in the inclusive range [1, n].

Example 1:

Input: n = 20 Output: 6 Explanation: The confusing numbers are [6,9,10,16,18,19]. 6 converts to 9. 9 converts to 6. 10 converts to 01 which is just 1. 16 converts to 91. 18 converts to 81. 19 converts to 61.

Example 2:

Input: n = 100 Output: 19 Explanation: The confusing numbers are [6,9,10,16,18,19,60,61,66,68,80,81,86,89,90,91,98,99,100].

Constraints:

1 <= 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.

Clarification Questions

What are the clarification questions we can ask about this problem?

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 real-world 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 Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level 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

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 high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, 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 language-agnostic 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 problem-solving 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 step-by-step 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 Python-based 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 problem-specific 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.