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How often, and when, are the minute & hour hands directly opposite each other?
In common with the first puzzle it's easy to assume that opposed positions occur once each hour, at say 'half-past-twelve', 'quarter-to-three', 'twenty-past-ten' etc., making 24 times in total.
Again, however, advancement of the hour hand must be taken into account.

The first time the hands can be directly opposite (i.e. at 180°) will therefore be a little after the minute hand has made half a revolution.
Thereafter the minute hand must make a further full revolution plus that moved by the hour hand before they can again be opposite. So:

  • when are the hands first opposite eachother?
  • at what time interval thereafter will they become opposed?
  • when would the 11th occasion be?
  • how many occasions does that make during the day?
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Answers
  • The first occurrence is at 0hr 32 8/11 mins (00:32:44).
  • The time interval thereafter is 1hr 5 5/11 mins, as for overlapping hands (puzzle 2).
  • The 11th occasion is at 11:27:16.
  • The total occurrences per 24hrs is 22.
Times that the hands are opposite:

Notes:
  • See the Notes to puzzle 2 regarding the potential errors involved when quoting times to the nearest second.
  • The hour, minute & second hands can obviously never all be opposite each other, but are in line twice, at 06:00:00 and 18:00:00.
  • If the question is instead:  "When, after any chosen time, will the chosen angle between them be 180°?", see puzzle 9.
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Maths
One approach is to produce a list of all the times the hands overlap as in puzzle 2, then find the half-way points between them. Otherwise, to calculate from scratch:

Method 1 (see Basics):
When opposite each other the minute/hour hand angular difference D is 180°, and being neither less nor greater than 180°, A = |5.5M - 30H| = 180, from which each occurrence will be at M = (30H + 180) × 2/11, where M is the minutes after each hour mark H. Therefore:
  • during the first hour (H = 0):  M = ((30 × 0) + 180) × 2/11, = 360/11, = 32 8/11, or 32mins 44secs†.
    The time is thus 0hr + (32mins 44secs) = 00:32:44.
  • during the second hour (H = 1):  M = ((30 × 1) + 180) × 2/11, = 420/11, = 38 2/11, or 38mins 11secs†.
    The time is thus 1hr + (38mins 11secs) = 01:38:11 .
  • during the third hour (H = 2):  M = ((30 × 2) + 180) × 2/11, = 480/11, = 43 7/11, or 43mins 38secs†.
    The time is thus 2hr + (43mins 38secs) = 02:43:38.
As for puzzle 2, the recurrent interval IR (calculated in this case from H = 1 & 2) is 65 5/11 mins. To find the Nth occurrence, however, the initial interval 'I1' (from H = 0) must be included, so the time will be given by I1 + ((N - 1) × IR) mins.
So, for N = 11:  the time is 32 8/11 + (10 × 65 5/11) = 687 3/11 mins, = 11hrs 27mins 16secs† (11:27:16).

Such positions continue to occur until H = 5, when M = (180 + (30 × 5)) × 2/11 = 60 mins, a whole hour, the time thus being 5 + 1 hrs or 06:00:00. The hands cannot therefore oppose during the 5th hour of either half day, totalling 22 occasions.
This can again be found from mins per day / IR: total N = 1440 / 65 5/11, = 22.
Not including the initial interval is balanced by the need to exclude the 23rd occasion which would be at 24:32:44 (during the next day).

NB:  Again, take care to use the precise figures of minutes and fractions of a minute, rather than times rounded to the nearest second.

Method 2 (see Basics):
The first opposing position occurs when the cumulative angular difference C amounts to 180°, and thereafter at intervals of 360° and multiples thereof. Hence:
  • for N = 1:  C = 180.
  • for N = 2:  C = 180 + (360 × 1) = 540.
  • for N = 3:  C = 180 + (360 × 2) = 900 etc...
  • it follows that C = 180 + (360 × (N - 1)), and since C = T × 11/120, T = (180 + (360 × (N - 1))) × 120/11 secs.
So, for N = 11:  T = (180 + (360 × 10)) × 120/11 = 453600/11, = 41236 secs† = 11:27:16.

As shown in puzzle 2, by substituting 86400 for T, solving for N but subtracting 1 (to avoid including a 23rd occasion after 24:00:00), the total occurrences during the day can be predicted: total N = ((86400 × 11/120) - 180) / 360, = 22.

† = to the nearest second

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