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How long, after any chosen time, will the minute & hour hands take to overlap?
This is a variation of puzzle 2, and one could simply look through the list of all times that the hands overlap to find the appropriate answer. Here, though, we first find their starting positions before calculating how long they then take to overlap. So:

  • what is the angle between the minute & hour hands at 08:55:43?
  • how long will they take to subsequently overlap, and at what time?
To run a demo, select a time from the hrs, mins & secs below. Scroll through and click the required numbers which will turn blue.
NB:  reset with thereset buttonbutton before choosing a new time.


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Answers
  • The minute/hour hand internal angle at 08:55:43 is 66.442°.
  • They will subsequently take 53 4/11 mins to overlap at 09:49:05.
For the demo:
  • The minute/hour hand internal angle at the chosen time of 00:00:00 is °.
  • They will subsequently take 0 mins to overlap at 00:00:00.
Notes:
  • See the Notes to puzzle 2 regarding the potential errors when quoting times to the nearest second.
  • This puzzle is often asked in terms such as: "When, between 4 & 5 o'clock, do the minute and hour hands first overlap?"
    How would you answer if the time frame were 4 o'clock to 4.55? And if it were 11 o'clock to 11.55?
    See the Answer table of puzzle 2 to check your answers.
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Maths
Method 1 (see Basics):
If the minute hand is initially 'behind' the hour hand by angle A (or opposite, when A = 180), it needs to catch up by moving A degrees plus the hour hand movement over the same period.
The minute hand moves 6° and the hour hand ½° per minute, so where 'm' represents the minute hand movement required: 6m = A + ½m, hence m = A × 2/11 [see puzzle 6 to calculate A].

If the minute hand starts 'ahead' by A (or coincident, when A = 0), it must move 360 - A plus hour hand movement, i.e. 6m = (360 - A) + ½m, hence m = (360 - A) × 2/11.
These may be combined in the form m = (A + ((360 - 2A) × p)) × 2/11, where position factor 'p' is 0 for the minute hand initially behind or opposite, and 1 if ahead or coincident.
The choice of p is easy to make mentally, but see the Extra maths section of puzzle 7 for a way to calculate it.

For example, for initial time 08:55:43:
  • see puzzle 6 to find internal angle A, which is 66.442°.
  • see puzzle 7 to find p, which is 1.
  • therefore, m = (66.442 + ((360 - 132.884) × 1)) × 2/11,  = 587.116/11,  = 53 4.116/11, = 53mins 22secs†.
  • the hands will therefore overlap at 08:55:43 + 00:53:22 = 09:49:05.

Method 2 (see Basics):
First convert the initial time to seconds since midnight: T = (hrs × 3600) + (mins × 60) + secs.
As shown in puzzle 2, the hands overlap at T = 360(N - 1) × 120/11, from which N = ((11T / 120) / 360) + 1.
Substituting the initial time for T and rounding down will therefore give the last time they overlapped.
The next time therefore becomes simply T = 360 × floor (N) × 120/11.

For example, for initial time 08:55:43:
  • T = (8 × 3600) + (55 × 60) + 43,  = 32143.
  • from puzzle 2, N = ((11 × 32143 / 120) / 360) + 1, = 9.1846, so floor (N) = 9.
  • therefore, the hands will subsequently overlap at: T = 360 × 9 × 120/11,  = 35345 secs†, = 09:49:05, which is (35345 - 32143) = 3202 seconds later, i.e. 53mins 22secs.
† = to the nearest second

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