Enigma 1311: Minimal clockspan
From New Scientist #2469, 16th October 2004
I have a clock with a sweep second hand, and I watch how the three hands – hour (H), minute (M) and second (S) – come together and spread apart. Let us call their span, at any given moment, the smallest fraction of the clock face to contain all three hands.
I ignore the hands’ thickness, so at 12.00.00 the span is zero. At 12.00.01 it is just under 1/60, because H has just past 0 (the vertical); S is in position 1/60 (that is 1/60th of the clock face from 0); and M is between them. After that the span gets bigger, but by 12.01.00 it is down again to exactly 1/60, with S at 0, M at 1/60 and, this time, H in between.
(a) Between 12.00.01 and 18.00.00, at what time is the span a minimum? (Give your answer to the nearest minute)
(b) What exact fraction is this minimum span?