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Problem 68

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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?

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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？

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