Magic 5-gon ring

Consider the following "magic" \( 3 \)-gon ring, filled with the numbers \( 1 \) to \( 6 \), and each line adding to nine.

Working clockwise, and starting from the group of three with the numerically lowest external node (\( 4 \),\( 3 \),\( 2 \) in this example), each solution can be described uniquely. For example, the above solution can be described by the set: \( 4 \),\( 3 \),\( 2 \); \( 6 \),\( 2 \),\( 1 \); \( 5 \),\( 1 \),\( 3 \).

It is possible to complete the ring with four different totals: \( 9 \), \( 10 \), \( 11 \), and \( 12 \). There are eight solutions in total.

Total	Solution Set
9	4,2,3; 5,3,1; 6,1,2
9	4,3,2; 6,2,1; 5,1,3
10	2,3,5; 4,5,1; 6,1,3
10	2,5,3; 6,3,1; 4,1,5
11	1,4,6; 3,6,2; 5,2,4
11	1,6,4; 5,4,2; 3,2,6
12	1,5,6; 2,6,4; 3,4,5
12	1,6,5; 3,5,4; 2,4,6

By concatenating each group it is possible to form \( 9 \)-digit strings; the maximum string for a \( 3 \)-gon ring is \( 432621513 \).

Using the numbers \( 1 \) to \( 10 \), and depending on arrangements, it is possible to form \( 16 \)- and \( 17 \)-digit strings. What is the maximum 16-digit string for a "magic" \( 5 \)-gon ring?