On a clock, we have hour and minute hands. Usually, if we swap their positions, we don't get a valid time. But sometimes, like at 12 o'clock, swapping them still gives us a correct time. How many such moments exist in 12 hours?
我们来看一个有趣的钟表问题。在任意时刻交换时针和分针的位置，通常不能表示正确的时间。但在某些特殊时刻，比如12点整，交换后仍然是正确的时间。我们需要找出12小时内有多少次这样的情况。
让我们建立数学模型。设时间为h小时m分钟，时针的角度是30h加0.5m度，分针的角度是6m度。当我们交换指针位置时，原来的分针角度必须能表示有效的时针位置，原来的时针角度必须能表示有效的分针位置。
通过数学分析，我们发现当交换有效时，分钟数m等于小时数h乘以720除以143。在12小时内，满足这个条件的时刻总共有22次。这些时刻包括0点0分、1点5分、2点11分等等，答案是22次。
Let's establish the mathematical model. For time h hours and m minutes, the hour hand angle is 30h plus 0.5m degrees, and the minute hand angle is 6m degrees. When we swap the hands, the original minute hand position must represent a valid hour hand position, and vice versa. This gives us the equation m equals 12h divided by 11.
Here are visual examples of valid swap times. We can see times like 1:05 and 5:01, where swapping the hands gives another valid time. The clock shows these special moments where the mathematical relationship holds true. In total, there are 22 such valid swap times in a 12-hour period.
In conclusion, using the mathematical relationship m equals 12h divided by 11, we can find all valid swap times in a 12-hour period. For each hour from 0 to 11, we get one valid time, plus the special case at 12:00. Therefore, the final answer is 22 times in 12 hours where swapping the hour and minute hands still gives a valid time.