#### How often in years do calendars repeat with the same day-date combinations?

Let's start with the basics. There are 7 days in a week. And 365 days in a year, right? Yes, but there is a leap year with an extra day every 4 years. Except for years evenly divisible by 100 but not by 400, but we'll get to that later.

Now, let's say that this year started with a Saturday. Any number of days that is divisible by 7 after that day will also be Saturday. 365 is 7×52+1, so in exactly one common year, it will be Sunday. This means that every year starts later by one day of the week than the year before, except for leap years which start later by 2 days.

 Sat Sun Mon ... Fri Sat Sun Mon Tue ... Sat Sun Mon Tue Wed ... Mon Tue Wed Thu Fri ... 2022 2023 2024 2025 Common Common Leap Common

Now, here is where it gets complicated.

According to what we just demonstrated in the last paragraph, after every leap year cycle we advance by 5 days in the week (1+1+1+2). This means that the calendar repeats itself after 6 years (1+1+1+2+1+1), right? Let's get back to our concrete example again and see if that's true.

 0 1 2 4 5 6 7 9 10 11 12 14 15 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 Common Common Leap Common Common Common Leap Common Common Common Leap Common Common

The first cycle took 6 years but the second took 5. See what happened there? The first cycle starts at the second year of a "leap cycle" (1+1+2+1+1+1) and the second cycle starts at a leap year (2+1+1+1+2). The calendar repeats itself after a different number of years depending on which year of the leap cycle we start counting from.

• 1st year in a leap cycle : 6 years later (1+1+1+2+1+1) we end up on the 3rd year of the next cycle.
• 2nd year in a leap cycle : 6 years later (1+1+2+1+1+1) we end up on the leap year of the next cycle.
• 3rd year in a leap cycle : 11 years later (1+2+1+1+1+2+1+1+1+2+1) we end up on the 2nd year of 3 cycles later!
• leap year : 5 years later (2+1+1+1+2) we end up on the 1st year of the next cycle.

We have another cycle right here! 1st -> 3rd -> 2nd -> leap -> 1st -> 3rd -> 2nd -> leap -> etc ... So, back to our example again : starting from 2022, the calendar repeats itself after 6 years (2028), then 5 (2033), then 6 again (2039), then 11 (2050), then 6 (2056), etc ... The sum of this cycle is 28 years. This means that at any given year, 28 years later, the calendar repeats itself.

And there you have it. The answer is 28 years...

... Or is it?

There is an exception for leap years : years that are divisible by 100 and not by 400 are not leap years. For example 1700, 1800 and 1900 were not leap, but 2000 was. So with this exception, in every 400 years, there are 97 leap years instead of 100. So 97×2+303×1=497 days, which happens to be a multiple of 7! This exception breaks the 28 years rule on years like 1800, 1900, 2100, etc ... but makes a new rule : the calendar repeats itself every 400 years!

By the way, this year marks my 28th birthday. Which happens to be the same day of the week as the day I was born : a Thursday.