leap year calculation

Question

In order to find leap years, why must the year be indivisible by 100 and divisible by 400? I understand why it must be divisible by 4. Please explain the algorithm.

Answer 1

The length of a year is (more or less) 365.242196 days. So we have to subtract, more or less, a quarter of a day to make it fit :

365.242196 - 0.25 = 364.992196 (by adding 1 day in 4 years) : but oops, now it's too small!! lets add a hundreth of a day (by not adding that day once in a hundred year :-))

364.992196 + 0,01 = 365.002196 (oops, a bit too big, let's add that day anyway one time in about 400 years)

365.002196 - 1/400 = 364.999696

Almost there now, just play with leapseconds now and then, and you're set.

(Note : the reason no more corrections are applied after this step is because a year also CHANGES IN LENGTH!!, that's why leapseconds are the most flexible solution, see for examlple here)

That's why i guess

Answer 2

Will it not be much better if we make one step further. Assuming every 3200 year as no leap year, the length of the year will come

364.999696 + 1/3200 = 364.999696 + .0003125 = 365.0000085

and after this the adjustment will be required after around 120000 years.

Answer 3

There's an algorithm on wikipedia to determine leap years:

function isLeapYear (year):
    if ((year modulo 4 is 0) and (year modulo 100 is not 0))
    or (year modulo 400 is 0)
        then true
    else false

There's a lot of information about this topic on the wikipedia page about leap years, inclusive information about different calendars.

Answer 4

There are on average, roughly 365.2425 days in a year at the moment (the Earth is slowing down but let's ignore that for now).

The reason we have leap years every 4 years is because that gets us to 365.25 on average [(365+365+365+366) / 4 = 365.25, 1461 days in 4 years].

The reason we don't have leap years on the 100-multiples is to get us to 365.24 `[(1461 x 25 - 1) / 100 = 365.24, 36,524 days in 100 years.

Then the reason we once again have a leap year on 400-multiples is to get us to 365.2425 [(36,524 x 4 + 1) / 400 = 365.2425, 146,097 days in 400 years].

I believe there may be another rule at 3600-multiples but I've never coded for it (Y2K was one thing but planning for one and a half thousand years into the future is not necessary in my opinion - keep in mind I've been wrong before).

So, the rules are, in decreasing priority:

• multiple of 400 is a leap year.
• multiple of 100 is not a leap year.
• multiple of 4 is a leap year.
• anything else is not a leap year.

Answer 5

In the Gregorian calendar 3 criteria must be taken into account to identify leap years:

1. The year is evenly divisible by 4;
2. If the year can be evenly divided by 100, it is NOT a leap year, unless;
3. The year is also evenly divisible by 400. Then it is a leap year. Why the year divided by 100 is not leap year

Answer 6

You really should try to google first.

Wikipedia has a explanation of leap years. The algorithm your describing is for the Proleptic Gregorian calendar.

More about the math around it can be found in the article Calendar Algorithms.