SOLUTION: A clockmaker makes a 12-hour clock but with the hour and minute hands identical. An ambiguous time on this clock is one where you cannot tell what time it is, since the exact posit

Algebra ->  Finance -> SOLUTION: A clockmaker makes a 12-hour clock but with the hour and minute hands identical. An ambiguous time on this clock is one where you cannot tell what time it is, since the exact posit      Log On


   

Question 1182848: A clockmaker makes a 12-hour clock but with the hour and minute hands identical. An ambiguous time on this clock is one where you cannot tell what time it is, since the exact position of the two hands occurs twice in a 12-hour cycle.
For instance, the clock can show 7:23 pm and 4:37 pm so both of these times are ambiguous.
However, 12:00pm is not ambiguous, since both hands are together.
How many ambiguous times happen in the 12 hours from midday to midnight?
Found 2 solutions by Boreal, ikleyn:
Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
There would be 11 of them, one after each hour, and the eleventh would occur at midnight.

Answer by ikleyn(52852) About Me  (Show Source):
You can put this solution on YOUR website!
.
A clockmaker makes a 12-hour clock but with the hour and minute hands identical.
An ambiguous time on this clock is one where you cannot tell what time it is,
since the exact position of the two hands occurs twice in a 12-hour cycle.
For instance, the clock can show 7:23 pm and 4:37 pm so both of these times are ambiguous.
However, 12:00pm is not ambiguous, since both hands are together.
How many ambiguous times happen in the 12 hours from midday to midnight?
~~~~~~~~~~~~~~~~~ 


If I do understand the problem and the question correctly, ambiguous moments are those 
when the minute hand and the hour hand are symmetric about the vertical line 12:00 - 6:00.


In 12 hours from midday to midnight, these moments happen when the hour hand is between  

   1)  12:00 and 1:00;     2)  11:00 and 12:00

   3)   1:00 and 2:00;     4)  10:00 and 11:00

   5)   2:00 and 3:00;     6)   9:00 and 10:00

   7)   3:00 and 4:00;     8)   8:00 and  9:00

   9)   4:00 and 5:00;    10)   7:00 and  8:00

   11)   5:00 and 6:00;   12)   6:00 and  7:00


So, if I presented it correctly, it happens 12 times, if the midday and midnight are excluded  ("strictly between"),

and  12 + 2 = 14 times, if the midday and midnight are included.

The problem does not ask to determine these time moments  PRECISELY,  so I do not compute them.
It  ONLY  asks  " how many such moments do happen ",  and  I answered it.


Strictly between midday and midnight,  it happens one time each hour.


////////////


By the way, there is another set of ambiguous configurations, when both hands are in one straight line with opposite directions.

It adds another 11 time moments between the midday and midnight, when you can not distinct / (determine) the correct time.