system
Given the following post, respond as if you're commenting on Reddit:
human
How long do I have to keep a calendar until I can reuse it again?

I have a 2016 Taylor Swift calendar that is still hanging on my wall, it just occurred to me that if I leave it up long enough then eventually I'll be able to use it again. 

I love math, but I'm not fluent in date calculations. (base 7? base 365?) I'm very curious about patterns. I'd love to hear more about the how and why of this question. (don't forget about leap years!)
gpt
Because a normal year is 52 weeks plus one day (365 mod 7 = 1), weekdays normally skip forward by one day every year. So 2017 started on Sunday and 2018 will start on Monday. If all years had the same length, it would then take seven years for the calendar of any given year to "roll around".

However, leap years make the matter more complicated. They have 366 days and thus the calendar skips forward by two weekdays on a leap year. Plus, of course, you cannot reuse a leap year calendar on a non-leap year even if they start on the same weekday.

Given a non-leap year, say, 2017, you can just count years, summing the "remainder" days until you hit a number divisible by seven. So:

2017|2018|2019|*2020*|2021|2022|2023
-|-|-|-|-|-|-
0|1|2|3|*5*|6|7

And indeed by checking the calendar you see that 2023 looks like 2017. But for some years, your counting will skip from 6 to 8 due to a leap year, and you have to keep on going:

2015|*2016*|2017|2018|2019|*2020*|2021|2022|2023|*2024*|2025|2026
-|-|-|-|-|-|-|-|-|-|-|-
0|1|*3*|4|5|6|*8*|9|10|11|*13*|14

With leap years, such as 2016, the year of reuse has to be a leap year as well, and you'll have to wait longer. 28 years, to be exact! In 2044 your Taylor Swift calendar will be pretty vintage.

Final disclaimer: Years divisible by 100 are not leap years unless they're also divisible by 400. So for example around 2100 you have to take that into account. Also, these calculations only concern days of week. Lunar cycle markings, date of Easter etc. will take much longer to get in sync.