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Problem 660

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Pandigital Triangles

We call an integer sided triangle $n$-pandigital if it contains one angle of $120$ degrees and, when the sides of the triangle are written in base $n$, together they use all $n$ digits of that base exactly once.

For example, the triangle $(217,248,403)$ is $9$-pandigital because it contains one angle of $120$ degrees and the sides written in base $9$ are ${261}_9,{305}_9,{487}_9$ using each of the $9$ digits of that base once.

Find the sum of the largest sides of all $n$-pandigital triangles with $9\le n\le 18$.

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全数字三角形

如果一个边长为整数的三角形有一个内角为$120$度，且其三边长在$n$进制下恰好使用了全部$n$个数字各一次，则我们称这个三角形为$n$-全数字三角形。

例如，三角形$(217,248,403)$是$9$-全数字三角形，因为它有一个内角为$120$度，而且它的三边长在$9$进制下分别是${261}_9,{305}_9,{487}_9$，恰好使用了全部$9$个数字各一次。

对于$9\le n\le 18$，求所有$n$-全数字三角形中最长边的长度之和。

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