Problem 90


Problem 90


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?


立方体数字对

在一个立方体的六个面上分别标上不同的数字(从0到9),对另一个立方体也如法炮制。将这两个立方体按不同的方向并排摆放,我们可以得到各种各样的两位数。

例如,平方数64可以通过这样摆放获得:

事实上,通过仔细地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,其中一种方式就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}。

在这个问题中,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}

但因为我们允许在摆放两位数时将6和9颠倒过来互相表示,这个例子中的两个不同的集合都可以代表拓展集{1, 2, 3, 4, 5, 6, 9}。

对这两个立方体有多少中不同的标法可以摆放出所有的平方数?