I have a clock at home that chimes on every hour and half hour mark. On whole hour marks it chimes once for each hour, and on half hour marks it chimes only once.

One night I was wakened suddenly and realized the the clock had chimed, but I did not know how many times. As I lay awake thinking about nothing in particular I heard the clock chime once and started wondering what time it was.

What is the longest amount of time I would have to lie awake before I knew for sure what time it was?

The answer would have to be 90 minutes, from the time you were woken by the original chime.

There are three situations, the first of which is very easy to understand:

The original chime was some hour, other than midnight, and the single chime was a half hour. In this case, the next indication of time will be more than one chime, in which case you've been wondering for 60 minutes

The other two situations create the ambiguity that makes the answer greater than 60 minutes:

The original chime was 12 midnight, in which case the single chime was for 12:30, and the next chime will be a single chime also, for 1 AM.

or, The original chime was 12:30, in which case the single chime was 1, and the next chime will be the single chime for 1:30.

In both of these situations, the chimes follow this pattern in the first hour (first three rinings of the chimes):

Unknown number, 1 chime, 1 chime

This is why you must listen for 30 minutes more - if the clock chimes once, you know it has been midnight, 12:30, 1, and is now 1:30.

If the clock chimes twice, it has been 12:30, 1, 1:30, and is now 2.

Either way, the maximum amount of time it will take you to figure out the current time is 90 minutes from the first awakening. Then again, who in their right mind would stay up 90 minutes just to see what time it is?!

Posted by JoeFish on 2005-08-09 03:02:56