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While sitting in the waiting hall a wall clock caught my attention. I realise there are only 15 minutes left for the conversation that will determine my path in life. I then took a glance at my watch. I realise they show different times. In my watch it is X past X, however in the wall clock it is Y to Y. It occured to me that after a 1000 minutes there will be 555 minutes between these two. What is X+Y?

PS: Both of these are 12 hour analog clocks.

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    $\begingroup$ I'm not sure I understand... After 1000 minutes, won't the difference be the same as now? $\endgroup$ – stack reader May 19 '17 at 4:58
  • $\begingroup$ Usually practice is to accept the first answer that is correct. $\endgroup$ – boboquack May 19 '17 at 8:32
  • $\begingroup$ @boboquack I know but the way was nice in the second one, however if that causes any problem I can undo. $\endgroup$ – KeithN May 19 '17 at 8:35
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The difference between the two times is 555 minutes, which is 9 hours 15 minutes.

Looking only at the minute hand, there must therefore be a 15 or 45 minute difference between them. The minute hand difference between X past _ and Y to _ is X+Y, so X+Y=15 or X+Y=45. The latter would lead to X>12 or Y>12 which is not possible since they should also represent one of the hours of a clock.

Therefore X+Y=15.

This is all the question asks for, but we can actually work out the time completely.

Now look at the whole hours. There is a difference of 9 hours between them, so you would expect |Y-X|=9 or |Y-X|=3. I ignored the minutes here, and doing so could make the real difference one hour more or less. However, from the previous equation the sum X+Y is odd, so the difference |Y-X| must also be odd, and cannot be off by one.
Solving the two equations, we get 4 potential solutions:
y=3, x=12
y=12, x=3
y=6, x=9
y=9, x=6

This gives the times:
3 to 3, and 12 past 12: 2:57, 12:12
12 to 12, and 3 past 3: 11:48, 3:03
6 to 6, and 9 past 9: 5:54, 9:09
9 to 9, and 6 past 6: 8:51, 6:06

The second and third of these don't have a difference of 9:15 but of 8:45, so are invalid.

1000 minutes is equal to 16 hours and 40 minutes, or +4:40. The two potential solutions are then
Y to Y X past X After 1000 minutes
2:57 12:12 : 7:37 4:52
8:51 6:06 : 1:31 10:46

The second of these can be considered to have a difference of 9:15 without adding/subtracting 12 hours. So the best solution is:
9 to 9 and 6 past 6

Nevertheless, the solution 3 to 3 and 12 past 12 is valid too if you allow the difference to span across the 12 hour mark.

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The answer is:

X+Y=15.
X=6 and Y=9, or as clock times, 6 after 6 (6:06), and 9 to 9 (8:51).

6:06 + 1000 minutes (16h 40m) is 10:46.
8:51 + 1000 minutes is 1:31.
555 minutes (9h 15m) separates 1:31 and 10:46 (1+9=10, 31+15=46).

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