Solutions were submitted by Yan Fridman, Bill Webb, Emanuele Macri, Loranzo Pozzoli, Massimo Brigonone, Trevor Green.

Let's write  q_h, q_m, q_s for the angles that the hour, minute and second hands make, measured in radians clockwise from 12:00.  In terms of t, the number of seconds since 12:00, these angles are given by

Let's assume that, as we look clockwise around the clock, we see first the hour hand, then the minute hand and the second hand.  For the angle between the hour hand and the minute hand to be 120⁰, we must have

for n a positive integer, 0 £ n £ 12.  Similarly, looking at the angle between the minute hand and the second hand, we must have

for m a positive integer, 0 £ n £ 60.  Substituting the equations above, solving for t and equating the two gives

.

m =,

an equation which does not yield an integer for 0 £ n £ 12.
 

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