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
Calculus - Steve, Tuesday, February 9, 2016 at 2:19pm
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.
Calculus - Andre post for Steve, Tuesday, February 9, 2016 at 11:33pm
I think there are more than 11
the person that gave the question said there were more than 11
To solve this problem, we need to consider all the possible combinations where the hour and minute hands are switched and still represent a valid time on a 12-hour clock.
First, let's consider the 11 cases where the hour hand and minute hand coincide, i.e., they are pointing at the same place on the clock. These times occur at exactly 12 o'clock, 1 o'clock, 2 o'clock, and so on until 11 o'clock.
Now, let's consider the other cases where the hour and minute hands are switched, but they do not coincide. For this, we need to check if the hour hand points exactly at one of the numbers on the clock, considering the approximate position.
For example, let's take the time 2:00. If we switch the hands, the hour hand will be pointing at 12, and the minute hand will be pointing at 2. So the time represented by the switched hands is 12:10.
Similarly, we can calculate the other possible times where the hands are switched but still represent a valid time.
1:00 switched becomes 12:05
2:00 switched becomes 12:10
3:00 switched becomes 12:15
4:00 switched becomes 12:20
5:00 switched becomes 12:25
6:00 switched becomes 12:30
7:00 switched becomes 12:35
8:00 switched becomes 12:40
9:00 switched becomes 12:45
10:00 switched becomes 12:50
11:00 switched becomes 12:55
So, in total, we have found 11 cases where the hour and minute hands coincide and 11 cases where the hands are switched but still represent a valid time. Therefore, the answer is 22 valid times in total.
Note: The approximation of the hour hand position is necessary because the hour hand is not exactly on the numbers but a bit off the mark.
If we consider the times where both hands are exactly on numbers, there are 11 such times as mentioned before:
1:00 --> 12:05
2:00 --> 12:10
3:00 --> 12:15
4:00 --> 12:20
5:00 --> 12:25
6:00 --> 12:30
7:00 --> 12:35
8:00 --> 12:40
9:00 --> 12:45
10:00 --> 12:50
11:00 --> 12:55
However, if we consider allowing some flexibility in the position of the hour hand, we can find more valid times. For example, if we allow the hour hand to be a bit past the number, there would be additional valid times like:
1:05
2:10
3:15
4:20
5:25
6:32-6:33
7:38-7:39
8:43-8:44
9:49-9:50
10:54-10:55
11:00
In total, there would be 22 valid times where the hour hand and minute hand are switched.