Clock puzzles: easy maths

Clock facts

For historical reasons (see References) each day is divided into 24 hours, 60 minutes per hour, and 60 seconds per minute. There are thus:

24 × 60 = 1440 minutes per day.
60 × 60 = 3600 seconds per hour.
24 × 3600 = 86400 seconds per day.

On the 360° 12hr clockface the hour marks are therefore 360/12 = 30° apart, and minutes & second marks 360/60 = 6° apart, hence:

the hour hand moves 30/60 = ½° per minute, and 30/3600 = 1/120° per second.
the minute hand moves 6/60 = 1/10° per second.
the second hand moves 6° per second.

It follows that for any time in hours, minutes & seconds during a 12hr period:

the hour hand angle is (hrs × 30) + (mins / 2) + (secs / 120).
the minute hand angle is (mins × 6) + (secs / 10).
the second hand angle is secs × 6.

Methods

Although time progresses in a linear manner, the hands on the usual round 12hr analogue clock face revolve and 'wrap around' at the 12 o'clock position (at both midday & midnight). The hours, minutes or seconds that the hands indicate, as well as the angles between them, then restart at 0 (there is no 13 o'clock for example, though historic and specialist examples of round 24hr faces do exist).
This issue can be handled by using modular arithmetic (sometimes actually called 'clock arithmetic'), where the 'modulo operation' is simply the remainder of a number 'n' after division by another number 'm', the modulus. See References for further information.

In equations the modulo operation is represented by n mod m, so that on the round 12hr clockface the numbering of hours is expressed in terms of mod 12, and minutes & seconds as mod 60 (while on the 24-hour digital display hours are of course expressed in terms of mod 24).
And given 360 degrees in a circle, angles are always expressed in terms of mod 360.

Besides mod and of course +, -, × and / (or ÷), there are only two other common operations needed for these puzzles:
Absolute value = the magnitude of a number 'n' regardless of its sign, written |n|; e.g. |5| and |-5| both = 5.
Rounding = when integers without fractions/decimals are required in a calculation, floor means 'rounding down' to the integer component, ceiling means 'rounding up' to the next higher integer, while round alone means either of these: round down if the fractional component is less than ½ (.5), and otherwise round up (aka 'round-half-up').

Following mathematical convention, a number in front of a letter (e.g. 30H) in the following equations means a multiplication.
The shorthand hrs, mins, secs will often be employed for hours, minutes & seconds. This site uses two ways to solve the puzzles based on the simple facts above:

Method 1:
This is based on the angle equations:

the minute hand moves faster than the hour hand, and for any time in hrs, mins & secs the difference 'd' between their angles will be given by (mins×6 + secs/10) - (hrs×30 + mins/2 + secs/120).
To simplify matters, the seconds can be incorporated in the minutes as a fraction. So, where 'M' indicates the minutes plus secs/60 and 'H' the hour figure, the equation becomes (M×6 - (H×30 + M/2)), or more briefly:
d = 5.5M - 30H ^*.
the resulting angle could be negative, but we are usually interested in its magnitude rather than sign. Angular difference 'D' is thus expressed as |d|, so D = |5.5M - 30H| degrees^*.

Method 2:
This uses the total number of seconds passed since the starting point of midnight:

the time 'T' in seconds = (hrs × 3600) + (mins × 60) + secs.
the 'cumulative' angular difference 'C' between minute & hour hand positions increases by 1/10 - 1/120 degrees for each second passed, which reduces to 11/120, hence C = (T × 11/120) degrees.
in this case the result is always positive but will often exceed 360°, so angular difference D = (T × 11/120) mod 360 degrees.

Both methods:
Since angular differences concerning clocks conventionally refer to the narrower 'internal' angle 'A' between the hands (always 0 to +180°), values of D greater than 180 require adjustment, i.e:

for D < 180, A = D.
for D > 180, A = 360 - D.
for D = 180, either expression is valid.

The further steps required to answer each puzzle are detailed in the relevant Maths sections, though all start from these simple principles, and N indicates the Nth solution to those with a series of answers.

Due to their respective advantages depending on the particular problem, the program actually uses method 2 for puzzles 2 - 5 and method 1 for puzzles 6 - 9 (puzzle 10 is a little different).

* some sources use 30H - 5.5M instead, giving priority to the hour hand, but it makes no difference to the absolute value of D.