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/ |
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