The 7-Pointed Magic Star Problem

#
#   Solve the 7-Pointed Magic Star Problem
#
#   Numbers 1-14 give equal sums in each of 7 lines making up
#   the star shape below.  Eg: B+C+D+E = 30, B+F+J+M = 30, etc.
#
#                 A
#
#       B     C       D    E
#
#        F                G
#     H                      I
#          J             K
#                 L
#            M         N
#
#

This is a similar problem to the 6-Pointed Magic Star Problem, below.

We now have fourteen unknowns with seven equations. Choosing seven values gives 14 x 13 x 12 x 11 x 10 x 9 x 8 (14 choices of A x 13 choices of B etc. or 17,297,280) possible solutions.

Many of these choices give values which are less than 1 or greater than 14 for some of the letters. We also must eliminate illegal combinations where some numbers are duplicated. Finally we get a total of 1,008 solutions.

Counting 7 rotations and 2 reflections as equivalent, we find that there are just (1008 / 14) or 72 unique solutions.

Here are the first few:

 N   a  b  c  d  e  f  g  h  i  j  k  l  m  n                     P V

 1:  1  2  4 10 14 13  8 12 11  6  3  7  9  5 Peak: 54 Valley: 51
 2:  1  2  5 14  9 13  7 11  8  3  4  6 12 10 Peak: 53 Valley: 52
 3:  1  2  8 14  6 11 12 10  3  4  5  9 13  7 Peak: 42 Valley: 63
 4:  1  2 10 14  4 11  6  8  9 12 13  3  5  7 Peak: 36 Valley: 69
 5:  1  2 10 14  4 13 12  6  3  8  9 11  7  5 Peak: 28 Valley: 77
 6:  1  3  5 13  9 10 12 14  4  6  7  8 11  2 Peak: 44 Valley: 61
 7:  1  3  6 14  7 12 10 11  5  2  4  8 13  9 Peak: 49 Valley: 56
 8:  1  3  8 13  6  7 12 14  4  9 10  5 11  2 Peak: 41 Valley: 64
 9:  1  3  9 14  4 12 13  8  2  5  7 11 10  6 Peak: 34 Valley: 71
10:  1  3 11 14  2 13  7  5  8 10 12  6  4  9 Peak: 32 Valley: 73
11:  1  3 11 14  2 13  9  5  6 10 12  8  4  7 Peak: 28 Valley: 77
12:  1  4  5 12  9 13  3 11 14  7  8  2  6 10 Peak: 55 Valley: 50
13:  1  4  6  8 12 10  7 13 14  5  2  3 11  9 Peak: 64 Valley: 41
14:  1  4  7 14  5 12 13 10  2  6  9 11  8  3 Peak: 33 Valley: 72
....

This time there are no solutions where the Peak or Valley sums are equal to 30.

[ Peak Sum = A + E + I + N + M + H + B, Valley Sum = D + G + K + L + J + F + C ]

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