I was lying in bed last night, and opened my eyes to look at my digital clock to see what time it was. Being very lazy, I didn't lift up my head, and noticed that the clock was eclipsed by the edge of the bed, so that I could only see the top 4 segments of each of the digits. This was a bad thing as it got my mind racing, thinking about these problems:

If all you can see are the top 4 segments of each digit, how often can you still know exactly what time it is? How long will you have to wait in each case for the digits to change so that you DO know the time for sure? (Assume the '6' and '9' are of the design that uses 6 segments each, rather than five.)

A corollary is this: If you can see the whole display, but each digit has one segment (the same one for each digit) burnt out, will you still be able to tell what time it is with absolute certainty?

Warning: While I've had some fun thinking about this, I haven't actually worked out the solution to any of these questions.

Note: There is actually a set of design notes somewhere on the web that talks about a 'fail-safe' for critical LED displays, and gives advice on how to lay out the digits so they will be most recognizable even if a segment burns out.

I thought I was the only one that ever wanted to know this. My clock is across the room, for obvious reasons. Though the display is LARGE, but I, too, sometimes, have no view of the whole clock. Nor do I really want to sit up.
You will know what time it is when the hour is at any number, but exact time will be in 15 min increments, I think.

Let's take a few things into consideration regarding the fact that we can only see the top four segments:

1. The leftmost spot is either off, or will feature the top half of a 1, depending on the time. This is clearly distinguishable.

2. If a 1 is showing in the leftmost position, the position next to it (assuming a 12-hour clock), can only be a 0, 1 or 2. Each of which, feature a different combination of lights in the top four digits. So, you'll always be able to distinguish 10, 11 and 12 o'clock from each other.

3. If no 1 is in the left most position, the number in the 2nd position must be a 1-9. The top four segments of 2 and 3 are indistinguishable from each other, as are 5 and 6, and 8 and 9. You can, however, clearly distinguish 1, 4, 7 and 0 (which, of course, won't be in the 2nd position without a 1 in the leftmost position).

From this, we know that we can only clearly distinguish the following hours:

1
4
7
10
11
12

So, what about the minutes?

4. The third position can only contain the numbers 0 through 5. 0, 1 and 4 are the only digits that are distinguishable here.

5. The rightmost position, of course, can contain any digit from 0 through 9, as stated before. Though, you can only clearly distinguish 0, 1, 4 and 7, given only the top four segments.

This gives us 6 possible hours times 3 possible digits for the tens digit of the minutes times 4 possible digits for the ones digit of the minutes. So, there are 6*3*4, or 72, possible times during the day where you could distinguish the numbers, correct?

Nope! We need to take into consideration the same times in both AM and PM. I'm assuming here that you can either clearly distinguish AM and PM just from the course of the day, or it can be seen by indication on the clock itself.

According to this, there should be 144 times during a single day at which you can clearly determine the time, given only the top four segments!

The times of day that can be clearly determined are 1:00, 1:01, 1:04, 1:07, 1:10, 1:11, 1:14, 1:17, 1:40, 1:41, 1:44, 1:47, 4:00, 4:01, 4:04, 4:07, 4:10, 4:11, 4:14, 4:17, 4:40, 4:41, 4:44, 4:47, 7:00, 7:01, 7:04, 7:07, 7:10, 7:11, 7:14, 7:17, 7:40, 7:41, 7:44, 7:47, 10:00, 10:01, 10:04, 10:07, 10:10, 10:11, 10:14, 10:17, 10:40, 10:41, 10:44, 10:47, 11:00, 11:01, 11:04, 11:07, 11:10, 11:11, 11:14, 11:17, 11:40, 11:41, 11:44, 11:47, 12:00, 12:01, 12:04, 12:07, 12:10, 12:11, 12:14, 12:17, 12:40, 12:41, 12:44 and 12:47 (AM and PM).

I should have known that you'd be the one to apply some 'grey matter' to this one, Scott!

So, the longest gaps where you will not know exactly what time it is will be 2 hours and 13 minutes, which will happen between 1:47 and 4:00, 4:47 and 7:00, and 7:47 and 10:00. Right?

Of course, this ignores TRANSITIONS where you will be able to figure out the time by which segments change, but that's a whole other question...

WOW, Scot, that took you less than 24 hours. I assume you did not watch the clock the whole time, then?