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.
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.
physical hands
they can't point to any place on the dial, because it won't be a correct time
Hmm. I'm still unclear. What is an example where switching the hands is invalid?
To determine the number of times that switching the hour hand and minute hand on a valid time in a 12-hour time period results in another valid time, we can think through the problem step by step.
1. First, let's identify the valid times in a 12-hour time period. There are 12 hours on a clock face, so we have 12 possible positions for the hour hand. The minute hand can be in any of the 60 minute positions.
2. Since switching the hour and minute hands will always result in another valid time (albeit different), the total number of possible combinations is the product of the number of hour positions (12) and the number of minute positions (60). Hence, we have 12 * 60 = 720 possible combinations.
3. However, not all of these combinations will result in another valid time. For example, if the hour hand points to the "1" and the minute hand points to the "5", switching the hands would result in an invalid time of "5:1". Therefore, we need to eliminate such invalid combinations.
4. We know that switching the hands would result in another valid time when the hour hand points at the same place as the minute hand. This occurs every 12 minutes, as the minute hand moves 12 times faster than the hour hand. Hence, for every 12-minute interval (e.g., 12:00, 12:12, 12:24, etc.), switching the hands results in a valid time.
5. Now, we need to determine how many 12-minute intervals there are within a 12-hour time period. Since there are 60 minutes in an hour, and 12 minutes in each interval, there are 60 / 12 = 5 intervals within an hour.
6. As there are 12 hours in a 12-hour time period, the total number of intervals is 5 * 12 = 60 intervals.
7. However, we already accounted for the valid combinations within each interval when calculating the total number of possible combinations (720). So, we need to subtract these from the total.
8. Since we identified 1 valid combination within each of the 60 intervals, we subtract 60 from 720 to get the number of additional valid combinations resulting from switching the hands. Therefore, there are 720 - 60 = 660 additional valid combinations.
In conclusion, switching the hour and minute hands on a valid time in a 12-hour time period will result in a total of 660 additional valid times, in addition to the initial time where the hour hand points at the same place as the minute hand.