The current time is 12 minutes past 3 o'clock.
EXPLANATION :
Since both the minute hand and the hour hand are situated precisely on two distinct minute marks, the current time is 12R minutes past K o’clock ( say) where 1<=K<=11 and R=0,1,2,3,4. By conditions of the problem,both (5K+R) and 5K+R-12R = 5K-11R are perfect squares with the numerical magnitude of (5K+R) and (5K-11R)being different and the value of (5K-11R) being greater than 1. Hence, (5K+R) can assume any square value from 9 to 49. Now :
* 5K+R=49 =>K=9,R=4=>5K-11R=1, which is a contradiction;
* 5K+R=36 =>K=7,R=1=>5K-11R=24, which is a contradiction ;
* 5K+R=25 gives K=5 and R=0 so that 5K-11R=25 which is a contradiction since the numerical magnitudes of (5K+R) and (5K-11R) cannot be equal; as per conditions of the problem.
* 5K+R=16 =>K=3,R=1=>5K-11R= 4=2^2, which is a perfect square different from 16 which is greater than 1;
* 5K+R=9 =>K=1,R=4=>5K-11R= -39, which is a contradiction.
# Consequently K=3 and R=1 giving 12R=12 so that the current time is 12 minutes past 3 o’clock.
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