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I was snoozing downtown. A noise woke me up and I looked at my LED clock. I noted the product of the hour value and the minute value. I fell asleep. Another noise woke me up 55 minutes later. The product of the hour and the minute was the same as before.

What time did I wake up?

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up vote 12 down vote accepted

The minute value can't be the same after 55 minutes, so both the hour and the minute must have changed. The hour increased by 1 and the minute decreased by 5 (The hour couldn't have gone from 12 to 1 because the new product would be lower). If the first time was $h$ hours and $m$ minutes, then $(h+1)(m-5)=hm$. Expanding and rearranging, $m=5(h+1)$.

This has 10 solutions (possible values for the first wake-up time), which are

1:10, 2:15, 3:20, 4:25, 5:30, 6:35, 7:40, 8:45, 9:50, and 10:55.

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1  
my original solution was 3:20 - i didn't realise there were so many! – Jon Mark Perry Feb 27 at 5:22

6 additional times for a 24-hour display that goes from 00:00 to 23:59:

00:00 , 00:01 , 00:02 , 00:03 , 00:04 , 00:05 : first wakening
00:55 , 00:56 , 00:57 , 00:58 , 00:59 , 01:00 : second wakening

All other possibilities have been covered by @f''

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The hour must have changed, otherwise the products wouldn't have been equal, so $hm=(h+1)(m-5) -> m=5(h+1)$. $h$ can range from 0 to 10. If both products are zero and we aren't necessarily following this formula, the first time can range from 00:00 to 00:05. 00:05 appears twice, thus there are 11+6-1=16 solutions.

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The first time you woke up it was

12:55 AM.

The second time you woke up it was

1:50 AM.

Because

There was a power surge/outage after you woke up the first time and the clock reset to 12:00 AM.

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3  
This is a lateral thinking answer to a non-lateral thinking question. – f'' Feb 27 at 4:42
    
Meh, the asterisk threw me off. I thought it was a typo. didn't realize until seeing the other answers that it meant multiplication. – Herb Wolfe Feb 27 at 5:10
1  
I think that it is good to think outside the box. +1 – Mattios550 Feb 27 at 18:35

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