Qn 2396. Strictly Palindromic Number. Is there a number exists return false solved the problem, beats 100% in time.
Qn 2396: Strictly Palindromic Number - Does Such a Number Exist?
In the realm of algorithmic challenges, LeetCode’s problem 2396 presents an intriguing question: “Is there a strictly palindromic number?” The answer, upon careful analysis, is a definitive false. This post delves into the problem’s nuances, provides insights on its resolution, and discusses the implications of its constraints.
Problem Overview
The task at hand is to determine whether a strictly palindromic number exists. A number is strictly palindromic if it is palindromic in every base from 2 to ( n-2 ) for a given integer ( n ). The challenge lies in understanding the properties of palindromic numbers across different bases.
Key Insight
The crux of solving this problem lies in recognizing that:
- For ( n \geq 4 ), any number ( n ) cannot be strictly palindromic.
- The reasoning stems from the fact that any number ( n ) in base ( n-1 ) is represented as “11”, which is inherently palindromic. However, it fails to satisfy the condition for at least one base (specifically, base 2).
Proof Outline
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Base Representation: Any number ( n ) can be represented in various bases. In base ( n-1 ), ( n ) translates to “11”.
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Base 2 Representation: As ( n ) increases, the representation of ( n ) in base 2 becomes increasingly complex. However, it cannot maintain strict palindromic properties across all bases from 2 to ( n-2 ).
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Conclusion: Thus, for any integer ( n \geq 4 ), there cannot exist a strictly palindromic number. Therefore, the answer to the problem is false.
Time Complexity
The solution to this problem is essentially constant time, ( O(1) ), as it does not depend on the size of ( n ). The answer can be derived through basic reasoning and does not require iteration or recursion.
Space Complexity
The space complexity is also ( O(1) ), as we are not utilizing any additional data structures that grow with input size. The solution requires a fixed amount of space for computation.
Community Insights
Top Comments
- The Hints on the Question Give the Proof: The hints provided in the problem statement are instrumental for understanding the underlying principles. They guide readers towards the realization that strictly palindromic numbers do not exist for ( n \geq 4 ), thus reinforcing the conclusion.
Conclusion
The exploration of strictly palindromic numbers reveals an interesting facet of number theory and base representation. Although the problem might initially appear complex, a closer examination of the properties of palindromic numbers across different bases simplifies the challenge. The definitive conclusion is that no strictly palindromic number exists, and this insight can be gleaned from logical reasoning rather than extensive computation.
Feel free to share your thoughts or questions regarding this problem in the comments below. Let’s foster a discussion on the fascinating world of number theory and algorithmic problem-solving!