Do the hands of a clock ever point 120 degrees apart? This is a question I have been having for a long time, but for almost as long I have known that there is no exact solution to this problem.
However, today I found the solution to a related problem: When do the hands of a clock approximately point 120 degrees apart?
TL;DR: At 05:49:09 and 06:10:51
This can be formulated as a lattice reduction problem, here shown in Sage:
x = 60*60*12 # number of possible positions for the hour hand
a = x/3 # one third
W = x*x*x # just a huge number weight for LLL to work
# First three columns: hand positions
# Last two: weight and elapsed time in seconds
# h m s w t
M = [[ x, 12*x, 12*60*x, 0, 1], # one second tick
[x*x, 0, 0, 0, 0], # hour hand modulus
[ 0, x*x, 0, 0, 0], # min hand modulus
[ 0, 0, x*x, 0, 0], # sec hand modulus
[ x, x, x, 0, 0], # rotation does not matter
[ 0, a*x, -a*x, W, 0]] # equilateral position to be reduced
print Matrix(M).LLL()
This gives the answer in the last column of the row with the weight: 20949 seconds before/after midnight/noon. This is at 05:49:09 and 06:10:51. And indeed, at this times the angles between the hands are nearly 120 degrees but not perfectly so.
