Different Yearly Calendars

Date: 09/30/2001 at 15:11:34
From: Marcie
Subject: Limited number of yearly calendars

My teacher gave us an intresting problem to think about and I don't 
really know where to start looking. The question is as follows. How 
many different possible calenders are there? How many years does it 
take before each calendar is used at least once? Start counting with 
the year 2001. 

I thank you in advance for your help.

Date: 09/30/2001 at 19:23:23
From: Doctor Jodi
Subject: Re: Limited number of yearly calendars

Hi Marcie,

The first question - how many different possible calendars are there - 
is not  too difficult. Each year starts on Sunday, Monday, Tuesday, 
...., Saturday. (So you would think that there are 7 different 
possible calendars.)

But in fact, there are leap years, too. And each leap year can start 
on a different day of the week. So there are 14 different possible 
calendars.

Now, a leap year only happens every four years. Since we aren't going 
to be at the turn of the millenium anytime soon, we can figure that a 
leap year happens EVERY four years.

This means that in order for each of the calendars to be used, we must 
go through at least 14 different leap years. Since the next leap year 
is in 2004, we can't possibly use each calendar once before 2028. 

Will we really use all the calendars once by then? Notice that there 
are 365 days in a regular year and 366 days in a leap year. Since 
365 = 52 weeks + 1 day, each year advances one day. So since 2001 
starts on a Monday, 2002 starts on a Tuesday. In a leap year, we 
advance two days. So since 2004 starts on a Thursday, then 2005 starts 
on a Saturday. Does this make sense?

So we can make a chart of the day each year starts on to check to see 
if we use each. We'll put a star before each leap year to remind us 
to skip two days next year.

 2001 M     2006 Sun     2011 Sat     *2016 Fri    2021 Fri
 2002 T     2007 Mon    *2012 Sun      2017 Sun    2022 Sat
 2003 W    *2008 Tues    2013 Tues     2018 Mon    2023 Sun
*2004 Th    2009 Th      2014 Weds     2019 Tues  *2024 Mon
 2005 Sat   2010 F       2015 Thurs   *2020 Weds   2025 Weds

 2026 Thurs
 2027 Fri
*2028 Sat

You can check that we have a Sun-Sat both starred (leap-years) and
non-starred (non-leap years).

So starting from 2001, we only need 28 years, until 2028, to use each
calendar once.

Does this make sense? Write back if you have more questions!

- Doctor Jodi, The Math Forum
  http://mathforum.org/dr.math/