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dxu comments on Harry Potter and the Methods of Rationality discussion thread, March 2015, chapter 119 - Less Wrong
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3:54 is 234 seconds, which is exactly 1/400 of 26 hours, which happens to be Harry's extended sleep cycle. I have no idea if this is significant, but just throwing it out there.
It's also almost exactly 368 and 2/9 uses per siderial day, the actual period of rotation of the earth without reference to the sun.
It would've been exactly that figure about 5,300 years ago.
Based on that timing the stone was Gilgamesh's pearl
And guess who steals Gilgamesh's How-the-Old-Man-Once-Again-Becomes-A-Young-Man plant? That's right, a snake.
Ain't numerology grand?
So the Stone was created to be used no more than 400 times a day. This allows for the prediction that someone with a normal cycle should be able to use it every 1/400 of our normal days, or 216 seconds.
The limit might be the result of limited capacity rather than design.