posted 19 years ago
assuming continuous movement:
position of second hand (360 degrees in 60 seconds) = t * 360 / 60 = t * 6;
position of minute hand (360 degrees in 60 * 60 seconds) = t * 360 / (60 * 60) = t / 10;
position of hour hand (360 degrees in 60 * 60 * 12 seconds)= t * 360 / (60 * 60 * 12) = t / 120;
hands meet when
(t*6) % 360 = (t / 10) % 360 = (t / 120) % 360;
or
(t*6) = (t / 10) - a * 360 = (t / 120) - b * 360;
where a, b are integers
(t*6) = (t / 10) - a * 360 ; => 59 t + 3600 a = 0;
(t*6) = (t / 120) - b * 360; => 719 t + 43200 b = 0;
gives two equations that we can eliminate t in:
42421 t + 2588400 a = 0
42421 t + 2548800 b = 0
2588400 a - 2548800 b = 0
so
a = (2548800 / 2588400) b;
=>
a = (708 / 719) b;
so b increments in multiples of 719, and a in 708
back to:
59 t + 3600 a = 0;
gives
t = (708 * 3600 / 59) k, k = {0,1,2...};
t = 43200 k;
every 43200 seconds, or 12 hours.