stoutfellow: Joker (Joker)
[personal profile] stoutfellow
A couple of days ago, I posted a clock puzzle. As a reminder:
Suppose you have an ordinary, fully-functional twelve-hour clock, whose hour and minute hands are the same length. (There is no second hand.) How often will it be impossible to tell what time it is, by looking at the clock? That is, how often will it be impossible to tell which is the hour hand and which the minute?
I initially solved this problem by crank-and-grind methods, which gave the answer without explaining why that answer was correct. Later, I thought of a genuine explanation; that's what's under the cut.
The first thing to notice is that, strictly speaking, the minute hand of the clock is redundant; if you know the precise position of the hour hand, you know the position of the minute hand. So the question can be rephrased: "How often is it true that, if the hour hand were where the minute hand is, the minute hand would be where the hour hand is?"
All right: let's give the clock a third hand, shorter than the others, which is always where the minute hand would be, if the hour hand were where the minute hand is. The question is now "How often does the third hand point in the same direction as the hour hand?". Now, the relation between the hour hand and the minute hand is this: at midnight precisely, both hands point at "12", and the minute hand moves twelve times as fast as the hour hand. So, at midnight, all three hands point at "12", and the third hand moves twelve times as fast as the minute hand, or 144 times as fast as the hour hand. In the twelve hours from midnight to noon, the hour hand completes one circuit of the clock face; the third hand completes 144 of them. Hence the third hand will catch the hour hand 144 times in that time-span. Now, on some of those occasions, there's no ambiguity; the hour and minute hands are pointing in the same direction. How often does this happen? By the same reasoning, it happens twelve times in twelve hours. So there are 132 occasions when the hour and minute hands point in different directions and the third hand points in the same direction as the hour hand; or, eleven times an hour, you can't tell what time it is. The first of these occurs 5 5/11ths minutes after midnight, which looks exactly like 5/11ths of a minute after 1 AM.
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