ANSWER:
The two segments that CAN go blank without affecting one’s ability to distinguish the numbers are the bottom horizontal segment and the lower-right vertical segment. See the graphic below for details:

In the very bottom sample, you can see that if one KNOWS that both the bottom and lower-right segments are missing, it’s still possible to distinguish each number from the other.
if the middle and the upper left segments are out you would still know it’s seven.
posted by Grant on 7-29-2008 at 7:37 am
nevermind, i misread the question.
posted by grant on 7-29-2008 at 7:39 am
If the upper right segment is missing, you also can’t tell 5 from a 9.
posted by The Other Brian on 7-29-2008 at 9:04 am
The top and bottom segments.
posted by jordon on 7-29-2008 at 10:11 am
It’s not top and bottom becuase that would make the 4 and 9 indistinguishable.
posted by Joe on 7-29-2008 at 11:10 am
If the bottom and upper left segments were missing, each of the numbers would still be distinguishable.
posted by Mortal Light on 7-29-2008 at 11:49 am
This only works if that segment of the number is burnt out for all the numbers on the clock. None of the answers are relivent if one of the segments are out only on the tenth minute number because it only goes up to 5 (12:59).
posted by Annie on 7-29-2008 at 12:36 pm
Bottom and lower right
posted by Dave A. on 7-29-2008 at 1:03 pm
useless stoner knowledge
posted by luisg on 7-29-2008 at 1:16 pm
The bottom LCD and the lower right LCD.
posted by Jeffrey on 7-29-2008 at 6:47 pm