What is the maximum number of years I have to wait till my birthday falls again on a Sunday?

Question

Suppose my birthday fell on a Sunday. What is the maximum number of years I may have to wait till my birthday falls on a Sunday again?

My intuition says 11 years:

• At the earliest, the next Sunday is in 6 years time. Reason: before I can encounter another Sunday, there must be at least one leap year. (7 - 1 = 6).

• Suppose by bad luck, 6 years later is a leap year (my birthday would skip Sunday and fall on a Monday). From this point, before I can encounter the next Sunday, there must be another leap year. There are two leap years involved, so the next Sunday is in 5 years time. (7 - 1 - 1 = 5)

• 6 + 5 = 11

Is 11 the correct answer? If yes, I would appreciate an explanation more rigorous than the above. If no, please teach.

I am also looking for exceptional cases where the above reasoning could be faulty.

Answer 1

Assuming that one is not born on February 29, then the following possibilities can occur:

• The day of the week repeats after 5 years. For example, March 1, 2015, and March 1, 2020, are both Sundays.
• The day of the week repeats after 6 years. For example, April 1, 2012, and April 1, 2018, are both Sundays.
• The day of the week repeats after 7 years. This case can only occur for dates from March 1 in a year ending with "96" to February 28 the following year ending with "97". For example, March 1, 1896, and March 1, 1903, are both Sundays, because 1900 was not a leap year.
• The day of the week repeats after 11 years. For example, October 1, 2006, and October 1, 2017, are both Sundays.
• The day of the week repeats after 12 years. This case can only occur for dates within the last decade of a century. For example, October 1, 2090, and October 1, 2102, are both Sundays, with the skipped leap year here being 2100.

Hence, the maximum is 12 years (for dates other than February 29). Of course, one could replace Sunday with any other day of the week and the same possible gaps between occurrences would still occur.

For February 29, the day of the week can only repeat after 12, 28, or 40 years. The 12-year gap can only occur for leap years ending with "92" or "96", while the 40-year gap can only occur for leap years ending with "72", "76", "80", "84", or "88".

Hence, the maximum for February 29 is 40 years. For example, February 29, 2088, and February 29, 2128, are both Sundays, with the skipped leap year here again being 2100.

By the way, it turns out that the 40-year gap can occur for leap years with any dominical letter:

• AG: 1888 and 1928
• BA: 1684 and 1724, 1780 and 1820, or 1876 and 1916
• CB: 1672 and 1712
• DC: 1688 and 1728, 1784 and 1824, or 1880 and 1920
• ED: 1676 and 1716, or 1772 and 1812
• FE: 1788 and 1828, or 1884 and 1924
• GF: 1680 and 1720, 1776 and 1816, or 1872 and 1912

Answer 2

$365$ leaves a remainder of $1$ when divided by $7$. Therefore, without accounting for leap years, your next birthday ($365$ days after your previous birthday) will shift by one weekday: for example, Sunday becomes Monday.

This year, $2020$, is a leap year. Now there are two cases: if your birthday is after February $29$, the leap day does not add an extra day between this birthday and your next birthday, so the weekday will only be shifted by $1$ instead. However, if your birthday is before February the $29$th, the leap day will be added before your next birthday, and your birthday in the following year will be shifted by $2$ weekdays.

The years $2021, 2022, 2023$ are common years. For these years, we only need to shift by $1$ weekday, with the exception of $2$ weekdays as explained in the previous paragraph.

The next year, $2024$ is a leap year. Apply the same logic as the year $2020$ again.

Repeat this logic until you get to a Sunday again.

The answers are below (spoilers!):

Birthday before February $29$th: in $5$ years, $2025$
Birthday after February $29$th: in $6$ years, $2026$

---

Surprising answer ahead:

The maximum is actually $12$ years! This uses the fact that $2100$ is not a leap year in the Gregorian calendar.

Take the date $17$ March $2097$, which is a Sunday. Because $2100$ is not a leap year, the date is shifted back $1$ day, so the $17$th of March $2103$ is not a Sunday, but a Saturday. The next Sunday is the $17$th of March $2109$.