Problem 68
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?
魔力五边形环
考虑下面这个“魔力”三角形环，在其中填入$1$至$6$这$6$个数，每条线上的三个数加起来都是$9$。
从最外侧结点所填的数最小的线（在这个例子中是$4,3,2$）开始，按顺时针方向，每个解都能被唯一表述。例如，上面这个解可以记作解集：$4,3,2; 6,2,1; 5,1,3$。
将环填满后，每条线上的总和一共有四种可能：$9$、$10$、$11$和$12$。总共有$8$种填法：
| 总和 | 解集 |
|---|---|
| $9$ | $4,2,3; 5,3,1; 6,1,2$ |
| $9$ | $4,3,2; 6,2,1; 5,1,3$ |
| $10$ | $2,3,5; 4,5,1; 6,1,3$ |
| $10$ | $2,5,3; 6,3,1; 4,1,5$ |
| $11$ | $1,4,6; 3,6,2; 5,2,4$ |
| $11$ | $1,6,4; 5,4,2; 3,2,6$ |
| $12$ | $1,5,6; 2,6,4; 3,4,5$ |
| $12$ | $1,6,5; 3,5,4; 2,4,6$ |
把解集中的数字连接起来，可以构造一个$9$位数字串；对于三角形环来说，最大的数字串是$432621513$。
在如下的“魔力”五边形环中，在其中填入$1$至$10$这$10$个数，根据不同的填写方式，可以组成$16$位或$17$位数字串。在“魔力”五边形环中，最大的$16$位数字串是多少？