Problem #45 Difficulty: 5%

Triangular, Pentagonal, and Hexagonal

Solution Language: Java
View Solution View on Project Euler

Problem Statement

Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:

  • Triangle: T(n) = n(n+1)/2 (1, 3, 6, 10, 15, …)
  • Pentagonal: P(n) = n(3n-1)/2 (1, 5, 12, 22, 35, …)
  • Hexagonal: H(n) = n(2n-1) (1, 6, 15, 28, 45, …)

It can be verified that T(285) = P(165) = H(143) = 40755.

Find the next triangle number that is also pentagonal and hexagonal.

Approach

The solution involves:

  1. Generating hexagonal numbers (all hexagonal numbers are also triangular)
  2. For each hexagonal number, checking if it’s also pentagonal
  3. Using the pentagonal test: n = (1 + sqrt(1 + 24k))/6 must be an integer
  4. Finding the next number after 40755