The current time is between 1 o'clock and 12 o'clock in a wall clock 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 precisely a pentagonal 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 also a pentagonal number greater than 1.

Determine the current time from the information provided in the above statements.

The hour hand is on a minute mark at 0, 12, 24, 36, 48 minutes past the hour since the marks break the hours into fifths. 12 is the only pentagonal number of these. The time must be ??:12, with ?? unknown. But, starting with 3:12, the hour hand is 4, 9, 14, 19, ... marks ahead of the minute hand, and none of these are pentagonal numbers. If one wanted to add multiples of 60 marks to the differences, I don't think that would work either; I don't think any pentagonal numbers can end in 4 or 9.  

Posted by Steven Lord on 2022-01-30 09:59:25