Inspired by Hexagonal Tortoise Problem, we treat the problem of the magic hexagon.
See Puzzle 927.  Prime solutions to HTP

According to the wiki, "A magic hexagon of order n is an arrangement of numbers in a centered hexagonal pattern with n cells on each edge,
in such a way that the numbers in each row, in all three directions, sum to the same magic constant M."
Arsen Zahray discovered these order 4.
See Magic hexagon wiki

Magic constant=111 and start number=3.
The order 4 hexagon starts with 3 and ends with 39.

The numbers in the hexagon are consecutive, and run from a to a+36.
Sum=37(2a+36)/2=37(a+18).
Hence magic constant M=Sum/7=37(a+18)/7.
Since M needs to be integer, a+18=0 Mod 7.
Then a=3 Mod 7.
If we take a=3, then M=111.

We searced the solutions for the order 4 hexagon starts with 3, there were many solutions.
We show only 4 solutions below.
Though we searced the solutions for the order 4 hexagon starts with 10, no solution was found.

               
                           M=111                                         M=111

               
                           M=111                                         M=111