Why the 28 year day-date repetition rule doesn't always follow?

Question

I was studying the pattern of calendar and found out that the years repeat same day-date pattern over a period of 28 years but the pattern broke if we compare 1752 and 1780.(By day-date pattern i mean for example in year 2020 1st february is on saturday and in 2048 1st february is also on saturday.)

I found out this was because the days got reduced when we shifted from julian to gregorian calendar.

but the same thing will happen if we compare 2076 and 2104 years.

I want to ask if there is a pattern followed by these calendar or they keep changing their day - date pattern randomly.

Answer 1

The repetition of the weekdays every $28$ years is based on the assumption that every fourth year is a leap year. This assumption is, however, broken for years such as $2100$ (which is not a leap year). More precisely, since the date we switched to Gregorian calendar, and until the date we switch to something else:

• Every year divisible by $4$ is a leap year,
• Except for the years divisible by $100$, which are not,
• Except for the years divisible by $400$, which are leap years again.

This means that the pattern will break again around years $2200$ and $2300$, but not around $2400$, and will break again around year $2500, 2600, 2700$ (but not $2800$) etc.

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It also means that the pattern will repeat every $400$ years. Namely in $400$ years you have $100-3=97$ leap years and $303$ "ordinary" years, and so the number of days is:

$$303\times 365+97\times 366\equiv 303\times 1+97\times 2=497\equiv 0\pmod 7$$

(as $365\equiv 1\pmod 7$), so the number of days in $400$ consecutive years happens to be divisible by $7$.