I have solved the puzzle by hand (no calculator)
Code:
To reduce the permutations, I force c < g, and ignore the edges a, b, h, i (for now)
This reduce permutations to at most 5! / 2 = 60
4 s = 45 + c + e + g
= (45 + 1+2+3) to (45 + 7+8+9)
= 51 to 69
= 52, 56, 60, 64, or 68
--> s = 13, 14, 15, 16, 17
To simplify further, as in tic-tac-toe, try center first (impossible e bracketed)
For s = 13:
c + e + g = 7 = [1] + [2] + 4
1(8)4(7)2 <-- ok
For s = 14:
c + e + g = 11
= [1] + [2] + 8
= [1] + [3] + 7
= 1 + 4 + 6
= 2 + 3 + 6
= 2 + 4 + 5
4(9)1(7)6 <-- ok
3 2(6)6
4(8)2(7)5 <-- ok
2(9)3(5)6 <-- ok
1 4(4)6
2 4(5)5
2 5(5)4
1 6(4)4
2(6)6 3
1(6)7(4)3 <-- ok
1(5)8(4)2 <-- ok
For s = 15, c + e + g = 15
But s = 15 also imply c + d + e = 15, which imply d = g, thus s != 15
For s = 16: c + e + g = 19
= 2 + [8] + [9]
= 3 + [7] + [9]
= 4 + 6 + 9
= 4 + 7 + 8
= 5 + 6 + 8
8(6)2(5)9 <-- ok
7(6)3(4)9 <-- ok
6(6)4 9
7 4(4)8
6(5)5 8
4(6)6 9
5(5)6 8
4(5)7(1)8 <-- ok
4(4)8 7
5(3)8(2)6 <-- ok
4(3)9(1)6 <-- ok
For s = 17: c + e + g = 23 = 6 + [8] + [9]
8(3)6(2)9 <-- ok
Add back missing digits to confirm above 12 cases. All confirmed !
Each solution actually represent 8 solutions, by head swap, tail swap and reverse the digits.
SUM SOLUTIONS
=== =========
13 391847256
14 284917635
14 194827536
14 482935617
14 581674329
14 671584239
16 178625934
16 187634925
16 394571826
16 475382619
16 574391628
17 458362917