Making a case for Wednesday, the 31st
5 min read . Updated: 14 Jan 2022, 01:42 AM ISTFor Fridays that are the 13th, three is the highest count possible in a given year.
For Fridays that are the 13th, three is the highest count possible in a given year.
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This column probably should appear in May. Or next January. That’s because this column is about Friday the 13th, and today is Friday the ... 14th of January.
There is indeed a Friday the 13th coming up in May; and since today is the 14th, you know that come next January, the 13th will fall on a Friday. Still, why wait till an actual Friday the 13th to read a column about Friday the 13th?
Why indeed. Might bring bad luck.
Why fuss about what’s just a date like any other? Because for no reason I can fathom, Friday the 13th has acquired something of a cult status. It’s considered unlucky, scary. There’s even a video game by that name, and there isn’t one called “Tuesday the 27th". There are even a couple of films called, respectively, “Friday the 13th" and “Friday the 13th". There’s even a word for the feeling the day supposedly spurs: “paraskevidekatriaphobia", the fear of Friday the 13th.
Yet as far as dates deemed unlucky go, there’s a case to be made, mathematically, for some others. Not Friday the 13th.
I’ll return to that. First, some things to note about our calendar, and what we learn from it about dates and the days on which they fall.
In a nonleap year, February has exactly 28 days, or 4 weeks. Thus, the first 28 days of March duplicate February for days of the week, and November also has the same layout. In a leap year, January, April and July are identical in this sense. So if February 13 is a Friday in a nonleap year, March 13 and November 13 will also be Fridays. Three such Fridays, which may make a year like that (2015, for example) seem particularly unlucky. Similarly, if a leap year starts on a Sunday, January 13, April 13 and July 13 will all be Fridays. Thus, 2012 was a particularly unlucky leap year.
Luckily, we can’t get more unlucky than that. For Fridays that are the 13th, three is the highest count possible in any year. Other years will usually have two, sometimes one, such Fridays. This year, for example, only May qualifies. Next year, January and October. In 2024, September and December.
The calendar we get all this from, that throws up unlucky days now and then, is of course known as the Gregorian Calendar. It is designed to a 400year cycle. This means that over that time, it chugs through all permutations of days and dates a year  really, sequences of years  can have. Then we start again. For example, this year started on a Saturday. So will 2422. The 400 years 20222421  call it an epoch  will be precisely duplicated by the 24222821 and the 28223221 epochs, and so on.
So if we want to calculate frequencies of daydate combinations, we need only look at a 400year stretch.
How many leap years in this epoch? You would expect 100, because a leap year comes once every four years. Except, years that end in “00" that are not divisible by 400 don’t leap, and there are three in any 400 year stretch (2100, 2200, 2300 in the next 400). So an epoch has only 97 leap years, and thus 303 nonleap years. The leap years have 366 days each, for a total of 35502 days. The nonleap years, each with 365 days, give us 110,595 days. Thus, the whole epoch has 146,097 days.
Now we can tabulate some frequencies. There are 7 days (Sunday, Wednesday and the like) and 31 dates (26th, 4th, etc), for a total of 217 combinations (Tuesday the 17th, Sunday the 6th, Friday the 13th, etc). How often do these days, dates and combinations appear in those 146,097 days?
A fairly meaningless question, no doubt. Anyway, the job has already been done, among others by a certain Magnus Bodin. He himself sees the futility of it—he refers to his effort as “funny worthless knowledge."
Still, there are some intriguing nuggets. Let’s start with an easy question: what’s the least frequent date in the epoch? The 31st: after all, there are only 7 in a year. Whereas there are 11 30ths, 11 29ths (but 12 in a leap year) and every other date appears 12 times. So in a 400year period, the 31st of a month happens 2800 times (7 x 400).
You would expect these 2,800 are evenly distributed among the seven weekdays, meaning 400 (2,800 / 7) times each. Close, but not quite. The 31st is most often a Thursday  402 times; least often on a Wednesday, just 398 times.
What about the 30th? 11 of those every year, thus 4,400 through the epoch. Again, our initial assumption would be that these are evenly distributed among the days: Monday the 30th, 629 (4,400/7) times, Thursday the 30th, 629 times, etc. But again, that’s not what happens. The 30th is a Monday 631 times, a Wednesday 631 times too: those are the most frequent. The least? Maybe you guess Tuesday, because the 31st happens least often on a Wednesday? Indeed: Tuesday the 30th appears only 626 times in 400 years.
And the 29th? 11 of those every year, except for leap years which have 12: thus a total of 4,497 29ths. Again, it’s not quite the even distribution over the seven days of the week, which would mean about 642 (4,497 / 7) occurrences. Instead, Monday and Saturday are 29ths 641 times each; Tuesday and Sunday happen 644 times.
Which leaves the other 28 dates, among them the 13th. Each happens 12 times in a year, thus 4,800 times in the epoch. So on average, you’d expect one of the daydate combinations  Saturday 21st, Thursday 18th, like that  to appear 686 (4,800 / 7) times. But again, it varies  from 684 (for example Wednesday 24th) to 688 (for example Monday 2nd).
You ask, why these variations? Why aren’t these occurences of the dates evenly distributed among the days? The short answer is that a year itself has 52 weeks and either one or two more days. Thus, one or two days of the week appear 53 times rather than the 52 of the other days. Over 400 years, that small difference manifests in the variations listed above.
But wait, what about Friday the 13th? In the 400 year epoch, it turns up ... 688 times. Meaning, it’s among those daydate combinations that appear most frequently of all. Clearly, its sinister reputation is not founded on scarcity. I mean, I would have thought its frequent appearance would have bred a certain familiarity and comfort. But no. For no reason I can fathom, Friday the 13th is bad news.
Think it’s time we changed that. Time for a scary film named “Wednesday the 31st."
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