Under what circumstances will a strobe reveal the direction of the fan?
As long as the frequencies of the strobe and the fan are known (maybe from the handbook?) you can calculate it, except for the corner cases where you can't tell the apparent direction of the fan.
fs: strobe freq
ff: fan freq (in revolutions per time unit)
n: ammount of (evenly spaced) wings on the fan
delta: angle between wings of the fan
If the fan rotates at ff, the frequency of the wings fw is ff*n. If fw is a whole-number multiple of fs, the fan will appear to stand still.
If the positive angle change of the fan between strobe flashes is between k*delta and (k+1/2)*delta for some whole number k, the fan will appear to move in the positive direction (i.e. the same direction it actually moves). At exactly k+1/2 the fan will appear to stand still and have the double ammount of wings.
The fan moves delta in time t_wing = 1 / (ff*n)
The time between strobe flashes is t_strobe = 1 / fs
If t_strobe modulo t_wing = 0, there will be no apparent direction.
If t_strobe modulo t_wing < 1/2 t_wing, the apparent direction will be correct.
If t_strobe modulo t_wing = 1/2 t_wing, there will be no apparent direction.
If t_strobe modulo t_wing > 1/2 t_wing, the apparent direction will be the opposite of the real direction.
So by calculating t_strobe modulo t_wing and looking at the fan, you can derive the real direction except for the two corner cases.
Please correct me if I made a mistake
EDIT: modulo t_wing, not 1/2 t_wing of cause.