There were five Sundays in February this year. Obvously it can only happen in a leap year, and 1st Februry has to be a Sunday. How often does this happen?
Once every 28 years. There are seven possible Leap Years, each starting on a different day of the week. Each successive Leap Year starts two days of the week earlier than the previous one, so the pattern repeats every seven Leap Years.
A modification to this is that century years are not Leap Years unless they are multiples of 400 - so 1900 was not a Leap Year nor will 2100 be, but 2000 was one.
Well, then Malacandra, it’s not every 28 years. It’s going to happen again in 2032, and again after that in 2060 and 2088, but then you’ve got leap years in 2092, 2096, and then not only do you miss your leap year, but you also throw off the cycle by having nine consecutive years with no leap year.
To put it another way, take a look at a Perpetual Calendar. The year 2100 is a “5” instead of a “12” (there are, as Malacandra rightly stated, only 14 patterns out there). Year 5 is always followed by Year 6 (or Year 13 in a Leap Year). So instead of having a 12 and then a 0, we have a 5 and then a 6. This means that our next leap year (2104) is a 9, and not a 10 as we’d expect.
The leap years cycle backwards: 7, 12, 10, 8, 13, 11, 9, 7 … So jumping from a 10 to a 9 means we skipped 8,13, and 11. In fact, the next 11 is actually in 2128, meaning that it took 40 years between the last one in the 21st century and the first one of the 22d.
Then you get 2156, 2184 (both regularly spaced) and then your next type 11 should be in 2224 (another gap of 40).
Type 11 years continue to happen as follows:
2252
2280
2320
2348
2376 (2400 is a leap year, so)
2404
So between 2000 and 2400, you get 4 this century, 3 the next, 3 the next, and 3 the next. The cycle repeats with 2404 (4 years after a century divisible by 400).
So, strictly speaking, it happens 13 times in 400 years, or on average, every 30.77 years. If either of us lives long enough to experience that “average” rate, I’ll meet you in Paris (if it’s still standing) and drink a toast to your health.
Note that this is only because of the coincidence that 400 Gregorian years makes 146097 days, and 146097 is exactly divisible by seven. If that were not so, the cycle would run 2800 years, and the average of 5-Sunday Februaries would indeed be one in 28. Alternatively, if the 400-year cycle had started on a different day of the week, it could have been less than one in 28, instead of more.
I don’t think this is related to Cecil’s columns, so I’m moving it to the General Questions Forum… if I’m wrong, I’ll be happy to move it back.