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Problem 891

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Ambiguous Clock

A round clock only has three hands: hour, minute, second. All hands look identical and move continuously. Moreover, there is no number or reference mark so that the “upright position” is unknown. The clock functions the same as a normal $12$-hour analogue clock.

Despite the inconvenient design, for most time it is possible to tell the correct time (within a $12$-hour cycle) from the clock, just by measuring accurately the angles between the hands. For example, if all three hands coincide, then the time must be 12:00:00.

Nevertheless, there are several moments where the clock shows an ambiguous reading. For example, the following moment could be either 1:30:00 or 7:30:00 (with the clock rotated ${180}^{\circ }$). Thus both 1:30:00 and 7:30:00 are ambiguous moments.

Note that even if two hands perfectly coincide, we can still see them as two distinct hands in the same position. Thus for example 3:00:00 and 9:00:00 are not ambiguous moments.

How many ambiguous moments are there within a $12$-hour cycle?

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不确定的钟

一个圆形时钟上有三根指针：时针、分针和秒针。所有指针看上去完全相同，且连续地移动。另外，时钟上没有数字或参考标记，所以无法判断时钟是否处于“竖直状态”。除此之外，这个时钟的功能与类似的、普通的$12$小时制时钟完全相同。

尽管设计上有所不便，但对于大多数时刻，只要准确测量指针之间的角度，就可以从时钟上读出正确的时间（在一个$12$小时周期内）。例如，如果三根指针重合，那么时间一定是12:00:00。

然而，在特定时刻，时钟会给出不确定的读数。例如，下图中的时刻既可能是1:30:00，也可能是7:30:00（将时钟旋转${180}^{\circ }$）。因此，1:30:00和7:30:00都是不确定的时刻。

注意，即使两根指针完全重合，仍然可以看出它们是同一位置的两根不同指针。因此，如3:00:00和9:00:00这样的时刻不是不确定的时刻。

在一个$12$小时周期内，有多少个不确定的时刻？

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