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.

TotalSolution Set
94,2,3; 5,3,1; 6,1,2
94,3,2; 6,2,1; 5,1,3
102,3,5; 4,5,1; 6,1,3
102,5,3; 6,3,1; 4,1,5
111,4,6; 3,6,2; 5,2,4
111,6,4; 5,4,2; 3,2,6
121,5,6; 2,6,4; 3,4,5
121,6,5; 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?

The “magic” 5-gon ring question, which requires understanding of both combinatorial principles and careful implementation of programming logic, is a traditional problem from Project Euler (Problem 68 to be precise). Here’s a solution outline:

We’re trying to find the maximum 16-digit string for the 5-gon ring, meaning that the sum of each group of lines (comprising the “nodes”) has to be equal. This also implies that their average sum is equal to the sum of values from 1 to 10 divided by 5 – giving us a target per line sum of 21.

One insight we can take advantage of is that, since the sum has to be the same in each line and we target the maximum total, 10 has to be in an external node. If it’s not, we cannot form a 16 digit number.

Then, we can simply construct all such permutations that hold the rule, taking into account the order. But we can make it easier by defining the problem as follows:
Let’s keep the external nodes in a decreasing order (10 to 6) and try to permute the remaining 5 numbers (1 to 5) in the inner nodes, making sure that all line sums are equal.

Writing a piece of code would be the most effective solution to implement this strategy as going through all combinations manually isn’t feasible.

Using these principles, and sorting the solutions in descending order, the largest 16-digit string for a 5-gon ring summing to 21 is:
6531031914842725.

Please note: the rules of this problem and even the solution strategy itself require deep understanding of permutation principles, combinatorics as well as programming. It’s a sophisticated mix of mathematics and coding. The explanation above is simplified as far as possible!

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