How many years’ worth of calendars would need to be kept so that they can be used in the same order over and over again for the same number of years? The calendars should always be valid, meaning that the individual dates always have the correct days of the week. (The years change, of course, and because holidays are not always consistent, we can assume that they are not marked on the calendars.) Furthermore, we assume that there will be no calendar reforms.
A year is a leap year if its year number is divisible by 400. The years 1600 and 2000 were therefore leap years. Any other year is only a leap year if its year number is divisible by 4 but not by 100. Consequently, 1896 and 1904 were leap years, but 1900 was a common year. The leap year cycle therefore repeats every 400 years. In such a period of 400 years, there are 97 leap years and 303 common years, or 97 × 366 + 303 × 365 = 146,097 days, which works out to 20,871 weeks. Because 400 years consist of a whole number of weeks, after 400 years, both the sequence of leap years and common years and the sequence of weekdays repeats itself. So you have to keep 400 years of calendars to use again and again in the same order.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission.