Clock puzzles #9

When, after any chosen time, do the minute & hour hands first reach any chosen angle?

This is a generalisation of puzzles 5 - 8, and concerns the full range of times and angles. So:

what is the angle between the minute & hour hands at 14:06:29?
after how long, and at what time, do they subsequently make an 84° internal angle?

To run a demo, select time and angle from the lists below.
Scroll through and click the required numbers which will turn blue.
NB: reset with the reset button

button before choosing a new time &/or angle.

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Answers

Click here to reveal the maths: show

Maths

One approach would be to produce a list of all times with the chosen angle, as in puzzle 5, then find the occasion that follows the initial time. Otherwise, to calculate from scratch:

Method 1 (see Basics):
The minute hand may be 'behind' or 'ahead' of the hour hand at the initial time by angle A, or else opposite (where A = 180) or coincident (where A = 0) [see puzzle 6 to calculate A].
Also, the chosen angle 'alpha' (α) may be less than, equal to, or greater than A. Ignoring any concurrent hour hand movement for the moment, the angle that the minute hand must move - call it 'delta', Δ - is therefore:

when minute hand initially behind or opposite (see diagrams 1 & 2):
If α < A, Δ = (A - α).
If α >= A, Δ = (A + α).
when initially ahead or coincident (see diagrams 3 & 4):
If α <= A, Δ = (360 - (A + α)).
If α > A, Δ = (α - A).

However, the hour hand moves ½° for each minute hand increment of 6°, so where 'm' is the total minute hand movement required, 6m = Δ + ½m, i.e. m = Δ × 2/11.
For example, for α = 84° and initial time 14:06:29:

the minute hand is initially behind, and A = 24.342° (so α > A).
in this case is Δ = (A + α), so m = (24.342 + 84) × 2/11, = 216.684 / 11, = 19 7.684/11, = 19mins 42secs^†.
the hands thus reach 84° at 14:06:29 + 00:19:42 = 14:26:11.

Method 2 (see Basics):
First convert the initial time to seconds since midnight: T = (hrs × 3600) + (mins × 60) + secs.
Using the appropriate expression for 'delta' given the initial time and chosen angle, as shown for Method 1, the angle is therefore reached at T + (Δ × 120/11).

For example, for α = 84° and initial time 14:06:29, when internal angle A = 24.342 [see puzzle 6 to calculate A]:

initial T = (14 × 3600) + (6 × 60) + 29 = 50789.
in this case is Δ = (A + α), hence the hands reach 84° at 50789 + ((24.342 + 84) × 120/11), = 51971 secs^†, = 14:26:11.

Both methods require mental judgements to be made regarding the appropriate expression for Δ.
See below for a way to calculate it.

Extra maths

The four possible equations for Δ may be combined as Δ = (360 × x) + (A × y) + (α × z), where x = 0 or 1, and y & z = -1 or +1, depending on the initial minute vs. hour hand position, and relative size of initial & final angles.
To determine the required values of x, y & z, the following tests can be applied:

Test p (for initial minute hand position):
See puzzle 7 to derive p, where it is shown that p = 0 when the minute hand is behind or opposite the hour hand, and 1 when ahead or coincident.
Test q (for α < A, = A, or > A):
(α - A) is negative when α < A, zero when α = A (including α = A = 0), and positive when α > A.
Remember A is always between 0 & +180°, hence q = round ((α - A + 180) / 360) gives 0 when α < A, but 1 when α >= A (including α = A = 0).
Test r (for α = A):
r = ceiling (|α - A| / 180) will give 0 when α = A, but 1 otherwise.

This table shows the initial minute vs. hour hand position, relative initial & final angles, and the results of tests p, q & r:

position	behind / opposite			ahead / coincident
α vs. A	<	=	>	<	=	>
x to be	0	0	0	+1	+1	0
y to be	+1	+1	+1	-1	-1	-1
z to be	-1	+1	+1	-1	-1	+1
test p	0	0	0	+1	+1	+1
test q	0	+1	+1	0	+1	+1
test r	+1	0	+1	+1	0	+1

Confirm for yourself, given all the information above, that the required values of x, y & z in the formula for Δ can then be found by the following equations:

x = p × ((q + r) mod 2).
y = 1 - 2p.
z = (2q - 1) - 2p(1 - r).

Although this procedure works fine and is in fact used by the program, can you find a less complicated way? See the 'challenge' on the Home page.

^† = to the nearest second

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