The current time is between 1 o’clock and 12 o’clock in a wallclock and both the minute hand and the hour hand are situated precisely on two distinct minute marks. It is observed that the hour hand is a square number of minute marks ahead of the 12 Hour Mark . The number of minute marks by which the hour hand is ahead of the minute hand is a different perfect square greater than 1. Determine the current time from the information provided in the above statements.
For each minute mark moved by the hour hand, the minute hand moves twelve marks. The hour hand is a perfect square number of marks from 12 only at these five times:
12:12
12:48
3:12
5:00
9:48
The first two violate the restriction "the current time is between 1 o'clock and 12 o'clock"; At 5:00, the two perfect squares involved are the same, violating another restriction; At 9:48 the distance between the hour hand and minute hand is 1 mark, violating the "perfect square greater than one" restriction. The only time that fits all the requirements is 3:12.
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