Thanks for the response to this puzzle. I got correct answers from three people – Ashutosh Diwakar, Rajeev Deshpande and Pallav Pandey in that order. Well done!
I am also taking the liberty of replicating Ashotosh’s solution which I thought was very well explained. If you want to use a video for children, you can access it on the following link:
The answer is 27 minutes and 41 seconds past the hour after 6 pm.
In one hour or 60 minutes of duration, hour hand is shifted or advanced by 5 minutes.
So, let’s suppose after 6 o’clock, it takes another x minutes to reach the condition mentioned in question i.e. reflection should show the same timing.
In 60 min., hour hand is advanced by = 5 min.
In x min., hour hand will be advanced by = (5/60)*x min.
Now, at this time w.r.t. 6 o’clock, minute hand will be back by = (30-x) min.
From the question, the time by which the hour hand is advanced should be equal to the time by which minute hand is back:
=> (5/60)*x = (30-x)
=> x = (27+9/13)min.
Therefore, (27+9/13)min. after 6 o’clock, the reflection in the mirror will show the same timing.