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I used a similar technique to

Enigma 93for this one (and reused some of the code). I’m assuming the hands being “aligned” means on top of each other (which is the only way I can think of all three hands being “aligned”), and that the hands move continuously and smoothly.This Python program constructs Esu’s journey (although with a bit of analysis there is a much easier way to get the answer). This program runs in 59ms.

Solution:Esu was on the minute hand for 327 and 3/11 minutes.Analytically:

The hour and minute hands coincide 11 times in a 12 hour period. 12:00:00 is the only time all three hands are coincident. Esu is on the minute hand in the 2nd and then every other eleventh (which excludes the 1st eleventh from consideration, which is 1 minute shorter than the rest), of which there are five. So Esu is on the minute hand for 5/11 of the 12 hour period, and the answer follows with a simple calculation.