Create a "Guess the time with a broken watch" puzzle

Question

I have an idea in my mind for a while but I can't finish it so I share it here, hoping you can help me.

The puzzle is based on a broken watch with only a few part of the digits working (and a big part of my problem is to dtermine which part of the digits) so it is not possible to determine exactly which number it is.
For example we can have the following green parts working only :

Here is the current state of the puzzle :

A husband and a wife are sleeping. The husband has the broken watch as desribe before and he knows which lights are working/broken. In the middle of the night, the wife wakes up her husband :
- What time is it ?
The husband looks at his watch and says :
- I can't tell you
He falls alseep but later her wife wakes him up again :
- What time is it now, I can't sleep !
- It has not changed, says the man looking at his watch, don't wake me up every 10 secondes
- But I'm sure that I have been waiting for more than 1 minute so it should have changed
- Oh I know what time it is then...

Can you be stronger than the husband and tell what time it is without looking at the watch ?

However at this current state the puzzle is not working (there is no solution) and I need your help to make it works !

Here are the potential changes to make it work :

• Find the digits so there is only one combination where there is exactly two different times with the same lights on
• Change the puzzle a bit (make him look at his watch more than twice for example)
• ... Any change that would make this puzzle.

Answer 1

You need significant changes to make an interesting problem out of this. It is impossible to determine the time from the given information because:

• If any digit does not change between the two times, then any value of that digit would work. For example, if the intended time is 1:23 to 2:03, then 1:20 to 2:00, 1:21 to 2:01, etc. would also work.
• If all digits change between the two times, then there are at least 8 times that work, because you could choose either value for each digit (not even considering the tens digit of the hour). For example, if the intended time is 1:23 to 4:56, then 1:26, 1:53, 1:56, etc. would all be the same and the answer could be any pair of them.

Answer 2

I don't really know how many broken lights you want on the watch, one can perfectly propose a time such as :

XX:X8 -> XX:X9 with only 1 broken light on the last digit

 --
|  |
 --
x  |    x : broken light
 --

Or more complex number changed such as

19:59 -> 20:00 with these broken lights

 xx    oo    oo    oo
|  o  o  o  o  x  o  o        o : on
 xx    xx    xx    xx      -- | : off
x  x  x  o  x  o  x  o
 xx    oo    oo    oo

But wait, the transition between the 9 and the 0 is containing the broken light of the change between 8 and 9, so the husband can't know from this broken light below whether it is 8->9 or 9->0 when these 2 lights are broken.

In fact, let's check every lights that change on each possible transition (with only the addition of 1 minute) :

 01    12    23    34    45    56    67    78    89    90    50    (set of +1)
 xx    xx    oo    xx    xx    oo    oo    oo    oo    oo    oo
x  o  |  o  |  o  x  o  o  x  o  |  x  x  x  o  o  o  o  o  o  x
 --    xx    oo    oo    oo    oo    xx    xx    oo    xx    xx
x  o  x  x  x  x  |  o  |  o  x  o  x  o  x  o  x  o  x  o  x  o
 xx    xx    oo    xx    xx    oo    xx    xx    oo    oo    oo 

The existence of the 8-9 gap makes many others transition ambiguous. For example, if you choose gap 6-7, 7-8 or 9-0 in the (+1) set, you don't know if it could hide a 8-9 below.

You'll have to check every A-B swap and avoid the sets that have the kind of single segment change (if possible), as they can easily getting themselves included inside others swap positions. Then you can proceed onto fixing the sets on each number (while taking the carry numbers into account) and find a unique set of lights and time.

An interesting setting would be that the man does check his watch only twice, but with a large gap of time between these checks (at least 100 minutes) :

A man is taking a nap in a foreign country after his arriving flight, because of time difference. After having slept for XXX minutes, he noticed that his broken watch displayed the same segments just before he began to sleep. He knows exactly which of his segments are malfunctioning, so he could have guessed at which time he began his slumber, and at which time he awakened.

Obviously, if many solutions have a match with the problem, we can try to narrow the possibilities by precising how many broken lights (at minimum) there are, or at around what hour did the man decide to sleep (after lunch, probably), etc...