How often does a whole calendar repeat exactly?

Question

You have a paper calendar for a leap year, say 2024. You would like to reuse it in some future year without changing a single page: the same weekday for 1 January, and the same number of days in February. Within the period 1901 to 2099, how many years must you wait before the exact same calendar comes round again? Why does an ordinary-year calendar sometimes come back sooner?

Accepted Answer

28 years for a leap-year calendar. An ordinary-year calendar can come back after 6 or 11 years, but the full pattern including leap years has period 28. The shift. From one 1 January to the next, the weekday advances by 1 in an ordinary year and 2 in a leap year. Over 4 years (three ordinary, one leap) the total shift is 1 + 1 + 1 + 2 = 5 days. Matching a leap year. Starting from a leap year, the next candidate years with the same leap status are 4, 8, 12, ... years later, with shifts of 5, 10, 15, 20, 25, 30, 35 days. The first shift that is a multiple of 7 is 35, after 7 leap cycles = 28 ye…

How often does a whole calendar repeat exactly?

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