Calendar Math

One of my favorite questions to ask my statistics students is “What is a leap year?”  Many of them get a general idea, but don’t actually know. 

1. How many different calendars would you need to represent all possible years — accounting for all day and date combinations? (Don’t forget about leap years!)
2. Now that we have all the calendars we could possibly need, it’d be nice to know how often we’re using them. When is the next time we’ll use the 2015 calendar?
3. What is the smallest total number of years that will pass between using the same non-leap-year calendar twice?
4. What is the largest?
5. What is the smallest total number of years that will pass between using a leap year calendar twice?
6. What is the largest?

Every calendar can be completely defined by the day of the week on which the first day (Jan 1) and the last day (Dec 31) falls.  Using Su, M, T, W, R, F, and Sa for the days of the week, we can represent our 2015 calendar as (R,R) since the first and last day of the 2015 calendar were both Thursdays.

The 2016 calendar will be (F,Sa) because the extra day during this leap year will shift the last day forward one.  The 2017 calendar will then be (Su,Su).  There are 7 calendars in which both days are the same, and you can quickly see that there are also 7 calendars in which the the first day is one day of the week, while the last day is the next day of the week.  This gives us a total of 14 possible calendars.  It doesn’t take much analysis to notice that we do indeed use every one of those 14 possible calendars.

If we follow the pattern from 2017 for a while, we find that 2023 will be another (Su,Su) calendar, suggesting the potential answer to #3 being 2023-2017=6.

With 2023=(Su,Su), 2024=(M,T), and 2025=(W,W), we find that 2026=(R,R) will be the next time we use our 2015 calendar, a difference of 11 years, which gives our answer to #2 and suggests a potential answer to #4.

When we explore leap years for a while, we see 2016=(F,Sa), 2020=(W,R), 2024=(M,T), 2028=(Sa,Su), 2032=(R,F), 2036=(T,W), 2040=(Su,M), which brings us back to the 2016 calendar in 2044, a difference in 28 years.

If one were not careful, and forgot about skipping a leap year 3 out of every 4 centuries, we may think that the answer to #5 and #6 were the same number 28, being both the smallest and largest number of years that will pass between using a leap year calendar twice.  

So, let’s move up closer to 2100.  We’ll take the leap year 2032=(R,F) that I analyzed above and add 28 years to it twice to get us to 2088=(R,F).  Let’s observe the next several years beyond 2088.

The interesting thing that happens here is that we go 8 years between leap years from 2096 to 2104.  What does this do to the calendar?  It allows us to see a (T,W) leap year calendar in 12 years rather than the usual 28, and the (Su,M) calendar of 2096 again in 2108.