When will the calendar be the same as 2002? I know the answer is 2013, but I don't know how to answer this mathematically, or if it is possible to do so. This is a ninth grade level algebra question my daughter received from her teacher. Any ideas? Thanks.

John

Hi John,

I think that you answered the question the same way we all did. You just count.

The answer depends on the fact that our calendar in ordinary years consists of 52 weeks plus one day (365 = 7 x 52 + 1); in leap years there are 52 weeks plus 2 days. That means, for example, that since 2002 began on Tuesday January 1, the year 2003 will begin on Wednesday, 2004 on Thursday, 2005 on SATURDAY (don't forget that 2004 is a leap year).... Just keep going until you get back to January 1 on a Tuesday of an ordinary year.

There are seven days in the week and January 1 can be on any of the seven days, but the year may be an ordinary year or a leap year. Thus there are 14 different calendars.

The number of years that it takes to get back to the same calendar depends on the year. Doing the counting gives some interesting results.

For any year, divide the year by 4 and look at the remainder. (In 2002 the remainder is 2.)

Remainder Number of years until
Jan 1 is on the same day
Number of years until
the calendar repeats
166
2611
31111
0528

This is of course not exactly correct. Things get complicated by a rule that makes ordinary those century years that are not divisible by 400 (so that 1900 and 2100 are not leap years, but 2000 is).

By the way, we have answered the question for our civil calendar. If you want to include movable Christian holidays such as Easter, the answer is quite different. Indeed, the number of possible interesting questions one can ask about calendars is boundless.

Chris, Patrick and Penny
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