29th February of the year 2000 was Tuesday. After this date how many t

29th February of the year 2000 was Tuesday. After this date how many times 29th February falls on Tuesday in the whole century ?

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UPSC CAPF – 2015

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February 29th occurs only in leap years. The day of the week for a given date repeats every 400 years exactly (due to the Gregorian calendar rules for leap years), but for specific dates like Feb 29th in leap years, the cycle of days repeats every 28 years (7 days * 4 years per cycle = 28 years, modified by the 100/400 year rule which is relevant over longer periods, but within a single century like 2001-2099, the 28-year cycle holds because 2100 is not a leap year and 2000 was a leap year).
Feb 29, 2000 was a Tuesday.
Leap years after 2000 in that century are 2004, 2008, …, 2096.
The day shifts by 5 days (3 normal years + 1 leap year = 365*3 + 366 = 1461 days; 1461 mod 7 = 5) every 4 years.
Starting from Tuesday (in 2000, reference):
2004: Tuesday + 5 days = Sunday
2008: Sunday + 5 days = Friday
2012: Friday + 5 days = Wednesday
2016: Wednesday + 5 days = Monday
2020: Monday + 5 days = Saturday
2024: Saturday + 5 days = Thursday
2028: Thursday + 5 days = Tuesday (1st occurrence after 2000)
The cycle of days for Feb 29 is Tuesday, Sunday, Friday, Wednesday, Monday, Saturday, Thursday, Tuesday…
This repeats every 7 leap years, which is 28 years.
The leap years after 2000 where Feb 29 is Tuesday will be 2000 + 28n.
For n=1: 2000 + 28 = 2028.
For n=2: 2000 + 56 = 2056.
For n=3: 2000 + 84 = 2084.
For n=4: 2000 + 112 = 2112 (outside the century 2001-2100 or 2001-2099).
Thus, Feb 29 falls on Tuesday after 2000 in the years 2028, 2056, and 2084. This is a total of 3 times.

For any given date (except possibly those affected by the 100/400 year rule transitions), the day of the week repeats every 28 years. The year 2000 was a leap year (divisible by 400), and 2100 is not a leap year (divisible by 100 but not 400).

The Gregorian calendar rule states that a year is a leap year if it is divisible by 4, unless it is divisible by 100 but not by 400. This rule causes the full calendar cycle (including day of the week for a given date) to repeat every 400 years. However, for Feb 29 within a period that doesn’t cross a non-leap century year boundary (like 2100), the 28-year sub-cycle for the day of the week holds.

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