Solving Project Euler Problem #45: Finding Triangular, Pentagonal, and Hexagonal Numbers
By
wenderen
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Summary
The article presents a technical walkthrough of solving Project Euler Problem #45, which involves finding numbers that are simultaneously triangular, pentagonal, and hexagonal. The author explains the mathematical properties of these number sequences, provides a Python implementation to solve the problem efficiently, and discusses the computational approach including optimization techniques and mathematical insights.
Key quotes
· 5 pulledProject Euler is a repository of fantastic computational problems that require a modest to heavy degree of mathematical thinking.
The problem asks us to find the next number after 40755 that is triangular, pentagonal, and hexagonal.
The key insight is that every hexagonal number is also a triangular number, so we only need to check for numbers that are both hexagonal and pentagonal.
We can use the quadratic formula to check if a number is pentagonal: if (1 + sqrt(1 + 24n)) / 6 is an integer, then n is pentagonal.
The solution involves generating hexagonal numbers and checking if they are also pentagonal, which is much more efficient than checking all three properties.
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