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I found that the solstices work on a 400 year cycle based on this. I can't find anything similar for the lunar cycle. Is there a formula I can use to calculate the date of the full moon?

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is that good enough for you? en.wikipedia.org/wiki/Full_moon#Formula –  DrCopyPaste Jun 13 at 13:43
    
14.5 hours is a pretty large margin of error. Easily enough to give a false positive or negative. If there isn't a better formula, I can make do, but I was hoping for something more accurate –  Bishop Jun 13 at 14:42
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Beware of media-defined notions such as "coincide". This is similar to the "alignment of planets" - they are aligned only according to some lax definition. So, regarding your coincidence, what is the margin of error? One day? And if it's one day, in what time zone? As you can see, these are complex questions in reality. It's just the profit-driven media that dumbs them down and makes them into seemingly simple soundbites. –  Florin Andrei Jun 13 at 16:12
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If you're talking about an exact coincidence, it virtually never happens. If you're willing to accept a two-week margin of error, it happens almost every year. The 400-year cycle is an artifact of the Gregorian calendar, which has no effect on when or how closely the solstice and the full moon coincide. –  Keith Thompson Jun 13 at 18:19
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@Bishop: The solstice is a single moment. For example, this year's summer solstice occurs at about 10:51 UTC on June 21. –  Keith Thompson Jun 13 at 20:18
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