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Problem 68


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?