PyPE solution to PE problem 61

    ProjectEuler.net problem 61
    ===========================

    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)=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, ...

    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.

    Solution
    --------

    This is a brute-force solution. We note that the six 4-digit numbers can be
    represented by six 2-digit numbers:

        ab bc cd de ef fa

    We loop through all possible values of a..f, checking along the way that
    each new number generated is figurate (to any of the specified polygons).
    For each of the resulting combinations, we generate the full cyclic set
    of 4-digit numbers and make sure that all s-gons are represented by unique
    elements in the cyclic set. If so, return the sum of the set.

    Answer
    ------

    28684