If a valid time on a 12 hour time period has the hour hand and minute hand switched, how many times will that result in another valid time? I have found 11, which is when the hour hand points at the same place as the minute hand. I know there is much more, but I can't figure it out.
Math - Steve, Saturday, February 6, 2016 at 5:52pm
Do you mean the physical clock hands are switched? No problem. I mean, the hands still point to places on the dial, right?
If you mean the digital time has hh:mm swapped to mm:hh, then any time from
01:01 to 01:12 is ok
and so on for the other hours. As long as the minutes are from 1-12, the swapping will produce a valid hour.
Math - Andre, Saturday, February 6, 2016 at 6:01pm
physical hands
they can't point to any place on the dial, because it won't be a correct time
Math - Steve, Saturday, February 6, 2016 at 6:53pm
Hmm. I'm still unclear. What is an example where switching the hands is invalid?
For example, lets say a time is 6:30. The hour hand will point halfway between 6 and 7 and the minute hand will point at 6. If we switch the hands, then the hour hand will point exactly at 6 and the minute hand between 6 and 7. Then the time is around 6:32-6:33. But that isn't possible because the hour hand cannot be pointing exactly at 6. I think this is what the question is asking
Hmm. In that case, you need times where both hands are exactly on numbers. So, there are 11 such times, exactly on hours.
At 2:00, say if you switch hands the hour hand will be on 12 and the minute hand will be on 2, so the time is 12:10.
Now, on a real clock, the hour hand will be a bit past the 12, but does that disqualify it?
So, I'd say your idea is good -- it captures all 12 times where the hands coincide. If we also allow the fudging above, then there are 11 other times, corresponding to
1:00 --> 12:05
...
11:00 --> 12:55
But, as you say, the hour hand will always be a bit off the mark.
I think there are more than 11
Yes, you have understood the question correctly. The question is asking for how many times switching the hour and minute hands on a physical clock will result in another valid time on a 12-hour time period.
To solve this problem, let's go through the possible scenarios step by step.
First, let's consider when the hour hand is at 12. In this case, switching the hands will result in the minute hand pointing between 1 and 2, and the hour hand pointing at 12. As you mentioned, this would not be a valid time because the hour hand cannot be exactly at 12.
Next, let's consider when the hour hand is at 1. In this case, switching the hands will result in the minute hand pointing between 12 and 1, and the hour hand pointing at 1. This is a valid time since the minute hand is pointing between 12 and 1, which is the correct range for the minutes.
We can continue this analysis for each hour on the clock and find that in each case, switching the hands will result in a valid time. Therefore, for each hour from 1 to 11, there will be one valid time obtained by switching the hour and minute hands.
So, to answer the question, there will be a total of 11 valid times when switching the hour and minute hands on a physical clock in a 12-hour time period.