## The Rare Days

The subject doesn’t quite feel right for my occasional roundups of mathematics-themed comic strips, but I noticed this month that the bit about “what is so rare as a day in June?” is coming up … well, twice, so it’s silly to call that “a lot” just yet, but it’s coming up at all. First was back on June 10th, with Jef Mallet’s Frazz (which actually enlightened me as I didn’t know where the line came from, and yes, it’s the Lowell family that also produced Percival), and then John Rose’s Barney Google and Snuffy Smith repeated the question on the 13th.

The question made me immediately think of an installment of Walt Kelly’s Pogo, where Pogo (I believe) asked the question and Porky Pine immediately answered “any day in February”. But it got me wondering whether the question could be answered more subtly, that is, more counter-intuitively.

It’s fairly clear that for any given calendar year, a day in June is just as rare as those in April, September, and November, and a little bit more common than one in February. In Pogo this lead to a disagreement about whether days in February were more powerful than those in July, since fewer of them had to do the work of making up the whole month. I *think* Pogo and Porky hoped to settle the question by arm-wrestling for it, which isn’t a bad approach from that start.

Back to the question of rarity, though, for any continuous stretch of twelve months (I seem to be finding more excuses to revive the term “twelvemonth” lately, and couldn’t guess why) the June days are rare or common the same way they would be for a single year. And for any finite whole number of years (or twelvemonths) the same holds.

The rarity of June days gets more arbitrary — to the point of there being no correct answer — if we can take parts of years: there’s few days as rare as a day in June if you just look for the period from July 1 through the next May 31; but they’re awfully common if you look at May 30 through July 2 instead.

And yet if you had infinitely many years — project the calendar forward until there’s a year for every integer — and infinitely many days, then, June days become exactly as common as those of February or July. That’s for the same reasons there are as many even numbers as there are whole numbers, and as many prime numbers as there are even or whole numbers.

In a fit of excess I started to write “as many prime numbers that end in 9”, but I realize that I don’t actually know whether there *are* countably infinitely many such prime numbers. It seems like there should be, or at least there’s no obvious reason there shouldn’t be, but without some more details that’s not a proof. I know that, after 2 and 3, every prime number is either one less than or one more than a whole multiple of 6, which suggests there’d be something peculiar going on if, after a while, primes ending in 9 just petered out, but peculiar things are wonderful to run across.

It feels to me like this subject should be related to the question of the probability that the 13th of the month should be a Friday, although I haven’t got quite what the link should be in mind.

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