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Problem 90

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Cube Digit Pairs

Each of the six faces on a cube has a different digit ($0$ to $9$) written on it; the same is done to a second cube. By placing the two cubes side-by-side in different positions we can form a variety of $2$-digit numbers.

For example, the square number $64$ could be formed:

In fact, by carefully choosing the digits on both cubes it is possible to display all of the square numbers below one-hundred: $01$, $04$, $09$, $16$, $25$, $36$, $49$, $64$, and $81$.

For example, one way this can be achieved is by placing $\{0,5,6,7,8,9\}$ on one cube and $\{1,2,3,4,8,9\}$ on the other cube.

However, for this problem we shall allow the $6$ or $9$ to be turned upside-down so that an arrangement like $\{0,5,6,7,8,9\}$ and $\{1,2,3,4,6,7\}$ allows for all nine square numbers to be displayed; otherwise it would be impossible to obtain $09$.

In determining a distinct arrangement we are interested in the digits on each cube, not the order.

$\{1,2,3,4,5,6\}$ is equivalent to $\{3,6,4,1,2,5\}$

$\{1,2,3,4,5,6\}$ is distinct from $\{1,2,3,4,5,9\}$

But because we are allowing $6$ and $9$ to be reversed, the two distinct sets in the last example both represent the extended set $\{1,2,3,4,5,6,9\}$ for the purpose of forming $2$-digit numbers.

How many distinct arrangements of the two cubes allow for all of the square numbers to be displayed?

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立方体数字对

考虑两个立方体，每个立方体的六个面上标有六个不同的、$0$至$9$之间的数字。将这两个立方体并排摆放，我们就可以得到一系列两位数。

例如，平方数$64$可以通过这样摆放获得：

事实上，通过精心选择两个立方体上的数字，我们可以摆放出所有小于$100$的平方数：$01$、$04$、$09$、$16$、$25$、$36$、$49$、$64$和$81$。

例如，其中一种方式是，在一个立方体上标上$\{0,5,6,7,8,9\}$，在另一个立方体上标上$\{1,2,3,4,8,9\}$。

在本题中，我们允许将标有$6$或$9$的面颠倒过来互相表示，只有这样，如$\{0,5,6,7,8,9\}$和$\{1,2,3,4,6,7\}$这样本来无法表示$09$的标法，才能够摆放出全部九个平方数。

在考虑什么是不同的标法时，我们关注的是立方体上有哪些数字，而不关心它们的顺序。

$\{1,2,3,4,5,6\}$等价于$\{3,6,4,1,2,5\}$

$\{1,2,3,4,5,6\}$不同于$\{1,2,3,4,5,9\}$

但因为我们允许将$6$和$9$颠倒过来互相表示，后一个例子中的两种不同标法都可以拓展为$\{1,2,3,4,5,6,9\}$。

对这两个立方体，有多少种不同的标法，可以摆放出所有九个平方数？

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