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

Answer 1

Calendar, in the present context, refers to the Gregorian calendar, because it is the one used most frequently, worldwide. In a Gregorian calendar, centuries are not leap years unless divisible by 400 (Eg: The year 2000 is a leap year, whereas 1900 isn’t).

As in the picture below, there can be only 14 calendar patterns - 1 to 14 (Years starting from Sunday to Saturday, either leap or non-leap year). Any calendar repeats in a minimum gap of 6 years and a maximum gap of 40 years. All the patterns 1 to 14 occur at least once in a minimum span of 25 years (Eg: 2000 - 2024) and a maximum span of 40 (Eg: 1889 - 1928).

1. Repetition in the succeeding years:

I. Calendars of non-leap years:

A. Calendars of non-leap years of the form 4n and 4n+1 repeat after 6 years. Eg: Calendar of the year 1900 repeats in 1906; that of 2017 repeats in 2023.

B. Calendars of non-leap years of the form 4n+2 and 4n+3 repeat after 11 years. 
Eg: Calendar of the year 2018 repeats in 2029; that of 2019 in 2030.

Exception: The years of the 90s of a century ending in a non-leap year do not follow this pattern. Calendar of such a year Y repeats after

(a) 12 years, if Y ∈ {90, 91, 97, 98}

Eg: Calendar of the year 1890 repeats in 1902.

(b) 6 years, if Y ∈ {93, 94, 95, 99}

Eg: Calendar of the year 1895 repeats in 1901.

II. Calendars of leap years:

Calendar of a leap year repeats after

A. 40 years, in the 70s and 80s of a century ending in a non-leap year.

Eg: Calendar of the year 1872 repeats in 1912.

B. 12 years, in the 90s of a century ending in a non-leap year.

Eg: Calendar of the year 1892 repeats in 1904.

C. 28 years, otherwise.

Eg: Calendar of the year 2020 repeats in 2048.

Note: Combining I and II, this is the order of repetition of calendars of the 90s of a century ending in a non-leap year:

12 12 12 6 6 6 12 12 12 6

2. Occurrence in the preceding years:

I. Calendars of non-leap years:

A. Calendars of non-leap years of the form 4n and 4n+3 are the same as that of the 6th preceding year. 
Eg: Calendar of the year 1900 is the same as that of 1894; that of 2019 is the same as that of 2013.

B. Calendars of non-leap years of the form 4n+1 and 4n+2 are the same as that of the 11th preceding year.
Eg: Calendar of the year 2021 is the same as that of 2010; that of 2022 is the same as that of 2011.

Exception: The first ten years of a century ending in a leap year do not follow this pattern. Calendar of such a year Y is the same as that of

(a) the 6th preceding year, if Y ∈ {01, 05, 06, 07}

Eg: Calendar of the year 1905 is the same as that of 1899.

(b) the 12th preceding year, if Y ∈ {02, 03, 09, 10}

Eg: Calendar of the year 1910 is the same as that of 1898.

II. Calendars of leap years:

Calendar of a leap year is the same as that of

A. the 40th preceding year, in the second and third decade of a century ending in a leap year. 
Eg: Calendar of the year 1916 is the same as that of 1876.

B. the 12th preceding year, in the first decade of a century ending in a leap year.
 Eg: Calendar of the year 1908 is the same as that of 1896.

C. the 28th preceding year, otherwise.
 Eg: Calendar of the year 2024 is the same as that of 1996.

Note: Combining I and II, this is the order of the immediate preceding years in which calendars of the first decade (01-10) of a century ending in a leap year, are the same:

6 12 12 12 6 6 6 12 12 12

Answer 2

Common years repeat every 6 or 11 years, depending on how leap years cause a day to be skipped, and leap years repeat every 28 years.