Run and find out, but guess first
Vox.com has a post on calendar patterns. Yuri Victor noticed that, this year, February is a nice rectangular shape on a calendar (as happens whenever it starts on Sunday in a non-leap year), and wondered how often this happened. This is the sort of question where you can easily find out the answer, so he did:
I decided to see if this occurs often so I wrote some code and found out it happens more than I thought.In the past 100 years, there have been 11 Februaries that make a rectangle.
He also noticed that February 13th would be Friday when this happened and wondered how often we got a Friday 13th:
Friday the 13ths also happen more than I thought. In the past 100 years there have been 171 Friday the 13ths, which means there is one to two a year.
This is a Good Thing. We want journalists wondering about patterns and looking up data to check them. We don’t want them being required to call an expert in calendars to give a quote. It’s also a Good Thing that he tells us his expectations were wrong
It would be even better, though, if he’d tried to work out a quantitative guess and tell us. The simplest guess would be that, in the long run, February 1 is a Sunday as often as any other day, and that the 13th of a month is a Friday as often as any other day. These are natural guesses because there’s no special reason the year or a particular month should start on a particular day of the week.
In 100 years there are 1200 months, and 1200/7 is 171.4, so it looks as though Friday 13th happens in almost exactly 1/7 of months. In the past 100 years there are 75 Februaries with 28 days, and 75/7 is 10.7, so 28-day Februaries begin on Sunday almost exactly 1/7 of the time.
You wouldn’t always expect the simplest possible explanation to hold. For example, the date of Passover is set based on the solar and lunar calendars, in a 19-year cycle. Since 7 doesn’t divide 19, you’d expect either that the days of the week didn’t divide up equally or that they took a long time (requiring lots of leap years) to do so.
Thomas Lumley (@tslumley) is Professor of Biostatistics at the University of Auckland. His research interests include semiparametric models, survey sampling, statistical computing, foundations of statistics, and whatever methodological problems his medical collaborators come up with. He also blogs at Biased and Inefficient See all posts by Thomas Lumley »
This reminds me of that awful “moneybags” myth that I’ve seen show up on Facebook every time a 31 day month starts on a Friday, claiming that only once every [ridiculously large number] years does a month have 5 Fridays, Saturdays, and Sundays.
It doesn’t take a lot of time to figure out why that must be wrong, along the same lines as your reasoning in this post. Once again, it happens on average every 7 years, although the pattern is somewhat complicated by the presence of a leap year.
http://www.snopes.com/inboxer/trivia/fivedays.asp
9 years ago
I was wondering what the rarest calendar phenomenon is. The best I can come up with is a year divisible by 100 that is a leap year and starts on a Sunday. I would have thought that we get one of those every 2800 years on average (400 x 7).
But no. It turns out that the answer is never. 1 January is always a Saturday. 1 January 2000 was a Saturday. 1 January 2400 is a Saturday. 1 January 2800 is a Saturday.
The explanation is simple – the number of days between those consecutive dates is always 146097 which is a multiple of 7.
I’m still stumped for the the rarest calendar phenomenon that isn’t infinity.
9 years ago
Since the calendar repeats every 400 years the longest repeating calendar pattern is 400 years. To beat this you have to include Easter or some other festival that depends on, say, the moon. The best one is Easter.
I looked at the next 1000000 years and calculated the date of Easter Sunday using Knuth’s algorithm. The rarest date for Easter Sunday is ANZAC Day and it falls on this day about every 590 years. That’s interestingly rare in its own right.
What’s even more interesting is that Easter Sunday falls on this late date only in leap years. You would expect that the extra day in February would on average make Easter occur one day earlier (e.g. the 12th instead of the 13th) but apparently it makes it later.
Now if we extend the requirement to “year divisible by 100 that is a leap year and has Easter Sunday on ANZAC Day” we might expect one every 400*590 = 236000 years. But we get zero in the next 1000000000 years (we would expect about 4000). The reason is that the Easter calculation (which was derived in the 1600s) is based on the Metonic cycle and not the actual moon cycle. So the 400 and 590 cycles aren’t independent so we can’t multiply the days.
Sometime in the future the Church might have to adjust the date of Easter since the Metonic cycle won’t match the actual moon. Then we might get Easter Sunday on ANZAC day in a leap year divisible by 100. Till then, tough.
9 years ago
There have been serious proposals to fix up the date of Easter, either by making it the second Sunday in April, or by basing it on the actual equinox and actual full moon.
Making the changes would take agreement across all the major Eastern and Western Christian denominations, so it’s not going to happen soon.
9 years ago