## Clock Hands

Here is a cute old problem that Facebook recently reminded me of.

Puzzle.By mistake, a clock-maker made the hour hand and the minute hand on a clock exactly the same. How many times a day, you can’t tell the current time by looking at the clock? (It is implied that the hands move continuously, and you can pinpoint their exact location. Also, you are not allowed to watch how the hands move.)

Here is the solution by my son who was working on it together with my grandson.

The right way to think about it is to imagine a “shadow minute hand”, like this: Start at noon. As the true hour hand advances, the minute hand advances 12 times faster. If the true minute hand were the hour hand, there would have to be a minute hand somewhere; call that position the shadow minute hand. The shadow minute hand advances 12 times faster than the true minute hand. The situations that are potentially ambiguous are the ones where the shadow minute hand coincides with the hour hand. Since the former makes 144 circuits while the latter makes 1, they coincide 143 times. However, of those, 11 are positions where the true minute hand is also in the same place, so you can still tell the time after all. So there are 132 times where the time is ambiguous during the 12-hour period, which leads to the answer: 268.

I love the problem and gave it to my students; but, accidentally, I used CAN instead of CAN’T:

Puzzle.By mistake, a clock-maker made the hour hand and the minute hand on a clock exactly the same. How many times a day can you tell the current time by looking at the clock?

Obviously, the answer is infinitely many times. However, almost all of the students submitted the same wrong finite answer. Can you guess what it was? And can you explain to me why?

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## FG:

Here is a visual solution (count the intersections exclusing those on the main diagonal):

15 December 2021, 8:21 amhttps://www.desmos.com/calculator/9t61adx1lo

## Joseph:

Let me guess: most of your students said 22, since that’s the number of times in a day where the hour and minute hand are pointing in the same direction.

30 December 2021, 4:09 pm## tanyakh:

Yes, Joseph, you are right.

30 December 2021, 7:18 pm