But, will any year's calendar repeat again after some years? Will all dates and days in a year match exactly with that of another year?

Question

Homi · Accepted Answer

We can find when a calendar repeats by looking at the 'extra days' each year has beyond full weeks.

Step 1 — What are 'extra days'?

A normal year has 365 days. We divide 365 by 7 to find the leftover days. $365 \div 7$ $= 52 \text{ with } 1 \text{ left over}$

$\boxed{1 \text{ extra day}}$ So, a normal year has 1 extra day. This means the day of the week shifts by one day each year. For example, if January 1st is a Monday, next year it will be a Tuesday.

A leap year has 366 days. We divide 366 by 7 to find the leftover days. $366 \div 7$ $= 52 \text{ with } 2 \text{ left over}$

$\boxed{2 \text{ extra days}}$ So, a leap year has 2 extra days. This means the day of the week shifts by two days each year. Leap years happen every 4 years. For example, 2000, 2004, 2008, 2012, 2016, 2020, 2024 are leap years. A year is a leap year if it is divisible by 4. But years divisible by 100 are not leap years, unless they are also divisible by 400. For example, 1900 was not a leap year. But 2000 was a leap year.

Step 2 — When does a calendar repeat exactly?

A calendar repeats exactly when two things happen. First, the total extra days must add up to a multiple of 7. This makes sure the days of the week match. Second, the repeating year must be the same type. This means both years must be normal years. Or both years must be leap years.

Step 3 — When does a normal year calendar repeat?

Let us start with a normal year, like 2001. 2001 (Normal): adds 1 extra day. 2002 (Normal): adds 1 extra day. Total = $1+1 = 2$ . 2003 (Normal): adds 1 extra day. Total = $2+1 = 3$ . 2004 (Leap): adds 2 extra days. Total = $3+2 = 5$ . 2005 (Normal): adds 1 extra day. Total = $5+1 = 6$ . 2006 (Normal): adds 1 extra day. Total = $6+1 = 7$ . The total extra days is 7. This is a multiple of 7. 2001 is a normal year. 2007 is also a normal year. So, the calendar of 2001 repeats in 2007. This is after 6 years.

Diagram 1

Sometimes it takes longer for a normal year calendar to repeat. Let us try 2003 (Normal). 2003 (N): +1 2004 (L): +2. Total = $1+2 = 3$ . 2005 (N): +1. Total = $3+1 = 4$ . 2006 (N): +1. Total = $4+1 = 5$ . 2007 (N): +1. Total = $5+1 = 6$ . 2008 (L): +2. Total = $6+2 = 8$ . 2009 (N): +1. Total = $8+1 = 9$ . 2010 (N): +1. Total = $9+1 = 10$ . 2011 (N): +1. Total = $10+1 = 11$ . 2012 (L): +2. Total = $11+2 = 13$ . 2013 (N): +1. Total = $13+1 = 14$ . The total extra days is 14. This is a multiple of 7. 2003 is a normal year. 2014 is also a normal year. So, the calendar of 2003 repeats in 2014. This is after 11 years. Normal year calendars can repeat after 6, 11, or 28 years.

Step 4 — When does a leap year calendar repeat?

Let us start with a leap year, like 2004. 2004 (Leap): adds 2 extra days. 2005 (Normal): adds 1. Total = $2+1 = 3$ . 2006 (Normal): adds 1. Total = $3+1 = 4$ . 2007 (Normal): adds 1. Total = $4+1 = 5$ . 2008 (Leap): adds 2. Total = $5+2 = 7$ . The total extra days is 7. This is a multiple of 7. But 2004 is a leap year. 2009 (the year after 2008) is a normal year. So, the calendar of 2004 does not repeat in 2009. We need to find another leap year. The pattern of extra days (1, 1, 1, 2) repeats every 4 years. Each 4 years adds about 5 extra days. We need the total extra days to be a multiple of 7. We also need the final year to be a leap year. This happens after 28 years. In 28 years, there are 7 leap years and 21 normal years. Total extra days = $(21 \times 1) + (7 \times 2)$ $= 21 + 14$ $= 35$ The total extra days is 35. This is a multiple of 7. If we start with a leap year like 2004, then 28 years later is 2032. 2032 is also a leap year. So, the calendar of 2004 repeats in 2032. This is after 28 years. Leap year calendars always repeat after 28 years.

Answer

(i) Yes, any year's calendar will repeat after some years. (ii) Yes, all dates and days in a year will match exactly with that of another year. This happens when the total extra days add up to a multiple of 7, AND both years are the same type (both normal or both leap). (iii) A normal year calendar repeats after 6, 11, or 28 years. A leap year calendar repeats after 28 years.

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