Tanya,
First we'll set the conditions into definitions:
Def. 1 ( F = 1st ) || ( F = 6th )
Def. 2 J < Q
Def. 3 QR
Def. 4 ( G == 3rd ) ? Q = 5th

Now the question states: From the sequences where G and H are fartest apart which factory is inspected on day 1.

1 2 3 4 5 6
G - - - - H
H - - - - G
This is fartest apart, however it conflicts with def. 1, so we'll implement def.1
F G - - - H
F H - - - G
G - - - H F
H - - - G F
Though we don't have any definition which conflicts directly, we still should determine whether they can be made into a valid sequence. As you can see we have three consecutive free slots and characters J, Q and R and applying def. 2 and 3 we see that they remain to this order J Q R
F G J Q R H >>> ok
F H J Q R G >>> ok
G J Q R H F >>> ok
H J Q R G F >>> ok
Thus the first day inspection can be F, G or H while G and H are as far apart as possible.

Niels