The minute hand of a clock overtakes the hour hand every 65 minutes. In how many days will the minute hand gain a total of 1440 hours over the hour hand?

Introduction / Context:
This puzzle is about the relative motion of the hour and minute hands on a clock. It states that the minute hand overlaps the hour hand every 65 minutes and asks in how many days the minute hand will gain a total of 1440 hours over the hour hand. The problem requires understanding that each overlap represents the minute hand gaining one full hour mark over the hour hand, then scaling that process up to a very large total gain measured in hours and finally converting that interval into days.

Given Data / Assumptions:

- The minute hand overlaps the hour hand every 65 minutes. - Each overlap corresponds to the minute hand gaining one hour mark on the dial relative to the hour hand. - We are asked when the total gain will reach 1440 hours. - One day has 24 hours. - We consider uniform motion of clock hands with no mechanical errors.

Concept / Approach:
Between two consecutive overlaps, the minute hand gains exactly one full revolution in terms of hour marks relative to the hour hand, which is equivalent to a gain of one hour on the 12 hour dial. Because the overlaps occur regularly approximately every 65 minutes, the total time needed to gain a large number of hours is the number of overlaps multiplied by 65 minutes. Once the total minutes are found, we convert them into hours and then into days in order to match the units requested in the question. Finally we compare with the options provided.

Step-by-Step Solution:
Step 1: Each overlap of the minute hand over the hour hand corresponds to a gain of 1 hour on the dial. Step 2: The total gain required is 1440 hours. Step 3: Therefore, the minute hand must overlap the hour hand 1440 times to gain 1440 hours. Step 4: The time between overlaps is 65 minutes, so the total time in minutes is 1440 * 65. Step 5: Compute the total minutes: 1440 * 65 = 93,600 minutes. Step 6: Convert minutes to hours by dividing by 60: 93,600 / 60 = 1560 hours. Step 7: Convert hours to days by dividing by 24: 1560 / 24 = 65 days. Step 8: Thus, it will take 65 days for the minute hand to gain 1440 hours over the hour hand.

Verification / Alternative check:
We can check the logic by considering smaller numbers. If the minute hand gains 1 hour every 65 minutes, then in 24 overlaps it gains 24 hours. The time for 24 overlaps is 24 * 65 = 1560 minutes, which is 26 hours, or a little more than one day. Scaling this up, 1440 overlaps is 60 times 24 overlaps, so the total time becomes 60 * 26 hours, which is 1560 hours. Dividing by 24 hours per day again gives 65 days. The alternative reasoning matches the direct calculation, confirming that 65 days is consistent.

Why Other Options Are Wrong:
Options such as 57, 58, 60, or 61 days would correspond to fewer total hours than 1560 when multiplied by 24. That in turn would imply fewer than 1440 overlaps at 65 minutes per overlap or a gain smaller than 1440 hours. Since the arithmetic shows that 1440 overlaps require exactly 65 days, the other values do not satisfy the gain condition stated in the problem.

Common Pitfalls:
One common source of confusion is mixing up hours and minutes in the phrase gain 1440 hours and mistakenly treating 1440 as minutes, which is the number of minutes in a single day. Another pitfall is forgetting that each overlap corresponds to a gain of one hour on the dial rather than one minute. Carefully distinguishing between the gain per overlap and the time between overlaps ensures that the scaling to 1440 hours is handled correctly and that the correct number of days is obtained.

Final Answer:
The minute hand will gain a total of 1440 hours over the hour hand in 65 days.