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Clock (an old favorite of mine)



This one may have been posted already but here it goes:

An extraordinary man constructed a special clock where the hour hand and minute hand were identical. It was hard to tell the time when one looked at this clock, but the old man claimed that it is usually straightforward to interpret the positions of hands. For example, at 5:00, when one hand points to 12 and the other to 5, there is only one possible interpretation: if the hour hand pointed to 12, it would not be possible for a minute hand to point to 5. On the other hand, the old man admitted, there are a few ambiguous positions that might be interpreted in two different ways. The question is: how many times - during a 12 hour period- are the hands in such an ambiguous position?

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The problem sounds familiar, but I don't think it has been posted.

Here is my approach.

Suppose we begin with a clock that has just two hands: H and M.

Now we give it a third hand, let's call it T,
and let it rotate 12 times as fast as M, or 144 times as fast as H.

Further, we align T to point (with H and M) to 12 at noon.

Let's start our 12-hour time interval just after noon.

Twelve hours later (just after midnight) H and M will have coincided 11 times, and H and T 143 times.

It is just the times when H and T coincide that the clock reading is ambiguous.

So the first answer is 143.

But at 11 of those times, when M joins the group, the clock reading is certain.

So the final answer is 132.

A very fine puzzle indeed.

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GOOD NEWS: If the clock hands only move once per minute, then this clock is good enough! There are no completely ambiguous times!

BAD NEWS: If the clock hands move every second (which is more likely), then from my calculations there are 66 pairs of times that you can not tell which it is...here are a few of them (I won't list them all):

04:31:50 06:22:35
02:56:10 11:14:40
03:41:30 08:18:25
09:54:05 10:49:30
01:55:45 11:09:35
03:31:25 06:17:35
12:55:20 11:04:35


I didn't go as far to try to calculate a CONTINUOUSLY moving hand, but that would open up a whole new set of possibilities...Now, while I agree that the old man is THEORETICALLY correct that it's pretty straightforward to tell what time it is using his clock, I, for one, will not be purchasing one of these...

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An hour hand will point directly at the hour mark only when the minute hand is pointing directly at 12. This means that every time one hand points exactly at the mark while another is not pointing at 12, the hand pointing at the mark is the minute hand. So all ambiguous times are when both hands are between the marks. Every hour there are 12 positions of the minute hand (between each pair of marks) that creates ambiguity, so 12*12=144.

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