The Ambiguous Clock
This is one of mine.
Imagine a clock with an hour hand and a minute hand (but no second hand). This particular clock has a rather minimalist design: You can tell the exact position of each of the hands but you cannot tell which hand is which. Most of the time you will have no problem telling the time even with this handicap. For example, at 4.30 the hour hand will point between the 4 and the 5 and the minute hand will point at the 6. The two hands could not be confused because if the hour hand was exactly on the 6 the minute hand would have to be on the 12.
The question is this: On how many occasions in a 12-hour period could you get the time wrong? Oh, by the way, you're not allowed to sit and watch to see which hand is which - we're talking about an instant view here.
| The
answer is 132 times. My solution is to imagine a clock with three hands. Colour the hands red, green and blue. The hands are geared together in such a way that for each revolution of the red hand, the green hand will go round 12 times, and for each revolution of the green hand the blue hand will go round 12 times. If the green hand is a minute hand, then the hour hand will be the red hand. But if the green hand is an hour hand, then the minute hand will be the blue hand. Back to our original clock. To get the time wrong, both hands must show a valid time whichever way round they are. If we colour one of the hands green, then the other one must be in the position of our special clock's red hand in case the green hand is the minute hand. But (on the other hand!) the other hand must also be in the position of the blue hand in case the green hand is the hour hand. So we can only get the time wrong if the blue hand coincides with the red hand. Now because of the gearing, the blue and red hands coincide 144 times in one revolution of the red hand, so there are 144 places on the clock face where the two original hands could be confused. But we have to subtract from this 12 places when all three hands coincide, because on those occasions it doesn't matter which way round the hands are. So the number of times in a 12-hour period in which this clock could be misread is 144 - 12 = 132. |
 |