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A group of men are working and counting on a very serious prophecy, which says "Doomsday will occur in the year with a triple blue moon". When will Doomsday occur?

Note:

  • A blue moon is a second full moon in a month of the common calendar.
  • Next year (2018) is a double blue moon year.
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    $\begingroup$ ha ha..."serious" and "prophecy" in the same sentence. You made ma laugh. +1 $\endgroup$
    – Marius
    Commented Nov 15, 2017 at 7:17
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    $\begingroup$ I think we need its math and here it is willbell.com/math/mc1.htm, not sure there is pure math formula of this anywhere else. $\endgroup$
    – Oray
    Commented Nov 15, 2017 at 7:19
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    $\begingroup$ It would be interesting to take tidal acceleration into account. Maybe the doomsday should already have happened or should happen in a few million years. $\endgroup$
    – Eric Duminil
    Commented Nov 15, 2017 at 19:41
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    $\begingroup$ @scmi: +5 billion years from now; the sun will on transition to the red giant phase disrupt the moon's orbit sufficiently; about a 50 percent change you get your extra full moon just before the moon is ejected completely. $\endgroup$
    – Joshua
    Commented Nov 16, 2017 at 3:13
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    $\begingroup$ @smci: Sorry, I'm afraid I don't know enough about long term orbital mechanics to answer your question. From the wikipedia link : This geological record is consistent with these conditions 620 million years ago: the day was 21.9±0.4 hours, and there were 13.1±0.1 synodic months/year and 400±7 solar days/year. But then, it depends how you define a month... $\endgroup$
    – Eric Duminil
    Commented Nov 17, 2017 at 16:48

2 Answers 2

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If I understand the question properly, I don't think doomsday will ever happen. The period of the moon's phases is 29.53 days. To have a blue moon, you need one full moon near the start of the month and one near the end. If there are two such months in a year, then time interval between those months must be a bit less than a whole number of moon periods. The only month that is less than a moon period is February, so you can have two blue moons, one in January, one in March.

Suppose you wanted to have a third blue moon in December. The months are:
31 (28) 31 (30 31 30 31 31 30 31 30) 31
The interval April-November is a total of 245 days, or about 8.3 moon periods. This is not close enough to a whole number of periods for the blue moon to occur in December. Earlier in the year the situation is worse.
The problem is that all months other than February are only slightly longer than a moon period so you need too many months put together for those extra days to make up a whole extra moon period.

Edit: The following is a simple proof based on Nurnani Tati's excellent answer. I add it here only to make this accepted answer more complete.

A year is at most 366/29.53 = 12.39 moon periods long, so there are at most 13 full moons in a year. If there were 3 blue moons in a year, there would be 10 normal full moons. A year has 12 months so this means that there would need to be (at least) two months without a full moon, but only February is short enough for that. Therefore 3 blue moons is impossible.

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  • $\begingroup$ Totally agree with you. Used the same logic to calculate till 2101, none had three blue-moons. $\endgroup$
    – ABcDexter
    Commented Nov 15, 2017 at 9:37
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    $\begingroup$ Right! So it is the good news for the mans, doomsday will never occur !!! $\endgroup$
    – Jamal Senjaya
    Commented Nov 15, 2017 at 9:46
  • $\begingroup$ Does a calendar system( like solar/lunar/Gregorian/Egyptian/Chinese etc.) play a role here? As each system has its own number of days in their respective month notation ! $\endgroup$
    – Mea Culpa Nay
    Commented Nov 15, 2017 at 10:08
  • $\begingroup$ @MeaCulpaNay Yes. Imagine a hypothetical calendar system, where you have 40-day months and 21-day months alternatively. Then you can easily fit 2 full moons in the first month, none in the second, 2 full moons in the third, etc., thereby having 6 blue moons in 12 months. $\endgroup$
    – twisteroid ambassador
    Commented Nov 15, 2017 at 14:29
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    $\begingroup$ According to an online calculator I found, there are no years with 3 blue moons between 0 and 3000 A.D. Incidentally, based on that data, "once in a blue moon" is about "once every 2.4 years" on average. $\endgroup$
    – Engineer Toast
    Commented Nov 15, 2017 at 15:49
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Doomsday will never occur

A triple blue moon is mathematically impossible.

The lunar cycle has 29.5 days, which means there's a tad over 12 per year (365/29.5=12.37). Hence, the "extra moon" is not every so often, when the first full moon occurs in early January we get 13 in a year.

Since the cycle is 29.5 days, it is also possible for February to get "skipped", and this, in conjunction with 13 moon year, causes the double blue moon (typically Jan and March), or more rarely two doubles in a row. February is the only month short enough to get skipped, as all the other months are longer than 29.5 days.

To get a triple blue moon, you'd have to have at least 14 moons (11, 1 for each month skipping February, plus 3 more for the blue moons). To get 14, even if we start with 1 (needed for the double), and the year would need to be at least 13 more long, but that means a year having 383.5 days.

Hence, it is not possible.

i.e. We are all safe.

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    $\begingroup$ That's a nice straightforward proof. +1 $\endgroup$
    – Jaap Scherphuis
    Commented Nov 15, 2017 at 10:08
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    $\begingroup$ Nicely rigorous argument. $\endgroup$
    – Rosie F
    Commented Nov 15, 2017 at 10:09

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