Overlap

When the start-end time is not 12 o'clock (or any other overlap time), there are 11 overlaps every 12 hours = $\large\boxed{22}$ overlaps for 24 hours.

If the start-end time was 12 o'clock, for example, the overlap at 12 would occur at 00:00, 12:00, and 24:00, making 23 overlaps for 24 hours.

Approximate overlap positions for the Minute and Hour hands.

Image based on A Clock In The Mirror .

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[Edit May 31, 2017] To make this solution a bit more complete.

The equation for the angle $\angle m$ of the minute hand at any given minute $M$ and the angle $\angle h$ of the hour hand at any given hour $H$ are as follows:

Source

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Each minute is 6 degrees.

$\large\frac{360}{60} = 6$

$\angle m = 6 \times M$

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Each hour mark is 30 degrees.

$\large\frac{360}{12} = 30$

But for us to be able to calculate $\angle h$ at any given minute, we also need to account for the additional movement of the hour hand for each minute.

The hour hand moves at 0.5 degrees per minute $\large\frac{360}{(60 \times 12)}$ , and 30 degrees can be represented as $0.5 \times 60$ .

$\angle h = 0.5 (60 \times H + M)$

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So when we are considering an overlap, we are looking for when $\angle m = \angle h$

$6 \times M = 0.5 (60 \times H + M)$

$12 \times M = 60 \times H + M$

$11 \times M = 60 \times H$

$M = \large\frac{60}{11} \times H$

$M = 5.\overline{45} \times H$

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Each overlap occurs every $65.\overline{45}$ minutes.

24 hours is 1440 minutes.

$\large\frac {1440}{65.\overline{45}} = \large\boxed{22}$

The minute hand will go round the dial 24 times, but the hour hand will also complete two circuits. So, $(24 - 2) = 22$

The answer is $\boxed{22}$

The hour and minute hand cross each other every 65 minutes (because the minute hand needs to move 5 minutes extra for each hour tick).

Therefore, in 24 hours (1440 minutes) the hands cross $\lfloor \frac{1440}{65} \rfloor = 22$ times

Edit: This is approximate! I have not accounted for the additional hour-hand movement within those 5 minutes, hence why I had to use the floor function. So it might work, but let's say it's an engineer's solution, not a mathematician's :^). The approximation would not work if it was over a greater timespan of 48+ hours. See @Jonathan Quarrie 's solution above.

Overlaps occur at:

$0:00 \space\space (12:00)$

$1:05+x$

$2:10+2x$

$3:15+3x$

$....$

$11:55+11x$

But this brings it back to $12:00$

Therefore, there are 11 overlaps within 12hrs; $\boxed{22}$ for 24hrs. $$ It can also be seen that the overlaps occur every $65 + x$ mins.

The value $x$ can calculated to be $\frac{5}{11}$ mins from:

$11:55+11x = 12:00$

$11x=5$

So, overlaps occur every 65 $\frac{5}{11}$ mins, and:

$(60 \times 24)\div(65\frac{5}{11})$

$=$ $\boxed{22}$

In $t$ hours, the minute hand completes $t$ laps. In the same amount of time, the hour hand completes $t/12$ laps.

The first time the minute and hour hands overlap, the minute hand has completed one lap more than the hour hand. So we have $t=t/12+1$ . This means that the first overlap happens after $t=12/11$ hours (approximately 1:05 AM). It therefore follows that, the second time they overlap, the minute hand would have completed two more laps than the hour hand. We can conclude that, for $n$ overlaps, $t=t/12+n.$ Since we have 24 hours in a day, we can solve the above equation for $n...$

$24=24/12+n$

$24=2+n$

$n=22$

Ergo, the hands overlap 22 hours a day.

that,s a great question to think.i was thinking at first from 12.o clock. then i thought it would be 23. but when i again saw the the question ,the clue, it was a analog clock where you started from 7.o clock.so,simply simply the overlap of 12.0 clock will not be here. so it will be 23.

In every 12 hours there are 11 crossings, so in 24 hours there are 22 crossings

Starting at 12.00 it can be seen that the next overlap will be roughly at 13.00. But not precisely, because the hour hand moves at 1/12 speed of the minute hand. So the exact time would take into account 1/12 of the incremental intervals swept out by the minute hand. The length of the actual interval between overlaps is, in hours, 1 + (1/12)+(1/144)+(1/1728)...... Approximately, this is 1.090909, and 24/(1.090909) is as nearly exactly 22 as any fraction where the denominator is an irrational number.