Coincidences
At 12 noon (or 12 midnight) it is clear that the hour and minute hands on an analogue display coincide. We ask if this happens at other times.

• Over a 12 hour period, how often will the hour hand and the minute hand coincide in position?

To see exactly when this happens, we need to figure out the relationship between the hour and minute hands.

• Show that if the hour hand is at h (on the minute dial), and the minute hand is at m, then

m = 12h – 60k,

where k represents one of the numbers 0, 1, 2, ... , 11.

• Now, at what times exactly do the hour and minute hands coincide?

Since the minute hand makes
12 sweeps of the dial in a
12 hour period, we might expect
the minute and hour hands to coincide
12 times over that period.
However, the correct answer is 11.
For, if we start with 12:00,
the coincidence in the final
(12th) sweep is again at 12:00.

The coefficients of m and h in
m = 12h – 60k arise from the
observation that each revolution
of the minute hand corresponds to
a rotation of ‘5 minutes’ by
the hour hand.
The multiples of 60 allow for multiple revolutions of the minute hand.
Check some simple times!

To answer this question, we set
m = h in m = 12h – 60k and take
k = 0, 1, 2, ... , 11. This gives the
values of m; the corresponding h value then needs to be converted to hours.
The answers are mostly unpleasant decimals, but to the nearest minute
we obtain: 12:00, 1:05, 2:11, 3:16, 4:22, 5:27, 6:33, 7:38, 8:44, 9:49,
10:55 (and 11:60!).