Cyclical Figurate Numbers

Problem Statement

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

• Triangle: P(3,n) = n(n+1)/2 (1, 3, 6, 10, 15, …)
• Square: P(4,n) = n² (1, 4, 9, 16, 25, …)
• Pentagonal: P(5,n) = n(3n-1)/2 (1, 5, 12, 22, 35, …)
• Hexagonal: P(6,n) = n(2n-1) (1, 6, 15, 28, 45, …)
• Heptagonal: P(7,n) = n(5n-3)/2 (1, 7, 18, 34, 55, …)
• Octagonal: P(8,n) = n(3n-2) (1, 8, 21, 40, 65, …)

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties:

1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
2. Each polygonal type: triangle (P(3,127)=8128), square (P(4,91)=8281), and pentagonal (P(5,44)=2882), is represented by a different number in the set.
3. This is the only set of 4-digit numbers with this property.

Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

Approach

The solution involves:

1. Generating all 4-digit figurate numbers for each type
2. Using backtracking to find cyclic sequences
3. Ensuring each number type appears exactly once
4. Checking that the last two digits of each number match the first two of the next
5. Finding the unique set and computing its sum