Different Yearly CalendarsDate: 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/ |
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