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Latest revision as of 03:49, 9 October 2019

Problem Statement

This problem explores an extension of the concept of a triangular number, generated by the formula \dfrac{n(n+1)}{2}, to other shapes.

Exploring triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers - numbers that are generated according to particular formulae:

P_{3,n} = \dfrac{n(n+1)}{2}

P_{4,n} = n^2

P_{5,n} = \dfrac{n(3n-1)}{2}

P_{6,n} = n(2n-1)

P_{7,n} = \dfrac{n(5n-3)}{2}

P_{8,n} = n(3n-2)


Solution Technique


Our solution technique is to generate a graph (for this, we use the Guava library).

We wish to find the sum of the ordered set of six cyclic 4-digit numbers for which each polygonal type, triangle/square/pentagonal/hexaongal,/heptagonal,octagonal, is represented by a different permutation of the digits (maintaining original order).

To do this, we create a graph, with each possible connection between a prefix and a suffix marked with an edge.

This results in a 6-partite graph, and we seek a path, a cycle, that passes through all 6 partitions.