Cyclical figurate numbers
Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:
\[ \begin{align} & Triangle && P_{3,n} = n(n+1)/2 && 1, 3, 6, 10, 15, ... \\ & Square && P_{4,n} = n^2 && 1, 4, 9, 16, 25, ... \\ & Pentagonal && P_{5,n} = n(3n-1)/2 && 1, 5, 12, 22, 35, ... \\ & Hexagonal && P_{6,n} = n(2n-1) && 1, 6, 15, 28, 45, ... \\ & Heptagonal && P_{7,n} = n(5n-3)/2 && 1, 7, 18, 34, 55, ... \\ & Octagonal && P_{8,n} = n(3n-2) && 1, 8, 21, 40, 65, ... \end{align} \]
The ordered set of three \( 4 \)-digit numbers: \( 8128 \), \( 2882 \), \( 8281 \), has three interesting properties.
- 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).
- 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.
- 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.