How Many Calendars Do I Need?

Date: 01/02/2002 at 17:19:42
From: Danny Smith
Subject: Calendars - How Many Do I Need?

Dear Dr. Math,

In 1980, I began collecting calendars and I have done so every year 
since. I will cease collecting when every subsequent year can be 
served by at least one of the calendars I have already collected. 
What is the last year in which I must collect a calendar?

I hope the problem can be solved without too much trouble, and I 
can understand the solution. 

Thanks a lot and have a good day.

Your fan,
Danny Smith

Date: 01/03/2002 at 09:09:56
From: Doctor Rick
Subject: Re: Calendars - How Many Do I Need?

Hi, Danny.

Here are some hints. There are only two things that can make one 
calendar different from another: (1) the day of the week on which 
January 1 falls, and (2) whether it is a leap year or not. 

You can easily figure the day of the week on which a year starts, if 
you know the day of the week on which the previous year started, and 
whether that previous year was a leap year or not. Can you come up 
with this technique? Using this technique, you can build a table that 
will help you answer the question.

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

Date: 01/03/2002 at 12:39:13
From: Doctor Douglas
Subject: Re: Calendars - How Many do I Need??.

Hi Danny, and thanks for writing.

There are fourteen different calendars. To see why, consider the day
of the week on which January 1 falls. It can fall on a Sunday, Monday,
Tuesday, or any of the seven days of the week. You'll need seven 
different calendars to handle each of these situations.  

But there are also leap years, so you'll need seven different 
calendars to handle the seven different days of the week on which 
January 1 can fall, when the year is a leap year. These calendars are 
different from the ones above, since these have 366 days while the 
non-leap year calendars have the usual 365 days.

Since every year is either a leap year or it is not, these fourteen 
cases are all you need.

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

Date: 05/31/2003 at 06:18:30
From: Michael
Subject: Collecting calendars

Hello, 

In your response you say that there can be only fourteen different 
yearly calendars but do not answer his question of how long he must 
collect calendars. I suspect that starting from 1980 and collecting 
14 calendars up to 1993 will not get you 14 different calendars, 
because some of them might be the same.

Date: 05/31/2003 at 19:53:15
From: Doctor Ian
Subject: Re: Collecting calendars

Hi Michael,

Actually, Dr. Rick gives hints that can be used to work out a
solution, rather than the solution itself. That way, he doesn't 
spoil the fun for anyone who wants to work it out for himself. 

To expand on his hint, note that 365/7 = 52 remainder 1, so if a
non-leap year starts on Sunday, the following year will start on
Monday, and so on. If a leap year starts on Sunday, the following
year will start on Tuesday. 

So you can make a table like this:

  Year  Leap?  Starts on
  ----  -----  ---------
  1980    Y    Tuesday
  1981    N    Thursday
  1982    N    Friday
  1983    N    Saturday
  1984    Y    Sunday

and so on, until you've collected all the possible calendars. 

How do you know when you have all the possible calendars?  You could
set up the table a little differently:

              Sun   Mon   Tue   Wed   Thu   Fri   Sat
           
   Leap

   Non-leap

And now you can fill in the years in sequence:

              Sun   Mon   Tue   Wed   Thu   Fri   Sat
           
   Leap       1984        1980                    1988
                          

   Non-leap   1989        1985  1986  1981  1982  1983
                                      1987 
                   
At this point, it's pretty clear that you'll need more than 14 years.
But when you have at least one year in each of the cells, you know
you're done. 

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