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Problem B and Solution
Gregorian Calendars

Problem

The Gregorian Calendar is used in most parts of the world today. In order to keep this calendar in synch with the solar year (the time for the Earth to complete one orbit around the sun), it has leap years with an extra day in February. Leap years generally occur every four years, in years that are divisible by \(4\). However, years divisible by \(100\) are excluded, UNLESS they are divisible by \(400\). For example, the year \(2000\) was a leap year, but \(1900\) was not.

The year \(2025\) has a different calendar than the year \(2024\) since it started on a different day of the week. January \(1\), \(2024\) was on a Monday, while January \(1\), \(2025\) was on a Wednesday.

  1. January \(1\), \(2023\) was on a Sunday. Explain why January \(1\), \(2024\) was one day of the week later than in \(2023\), while January \(1\), \(2025\) was two days of the week later than in \(2024\).

  2. January \(1\), \(1992\) and January \(1\), \(2025\) were both on a Wednesday. Did \(1992\) have the same yearly calendar as \(2025\)?

  3. How many different yearly calendars are there in total? Two yearly calendars are considered the same if each date occurred on the same day of the week.

Calendar of February 2024, which has 29 days. February 1, 2024 falls on a Thursday and February 29, 2024 falls on a Thursday.

Solution

  1. Since the year \(2023\) had \(365\) days, and a week has \(7\) days, there were \(365\div 7=52\,\frac{1}{7}\) weeks in \(2023\). That is, there were \(52\) full weeks plus \(1\) day. Thus, all of the next year’s dates fall one day of the week later than the previous year. This is why January \(1\), \(2024\) was one day of the week later than January \(1\), \(2023\).

    However, \(2024\) was a leap year with \(366\) days, so it had \(52\) full weeks plus \(2\) days. Thus, January \(1\), 2025 was two days later in the week (a Wednesday) than January \(1\), \(2024\) (a Monday).

  2. Despite the fact that January \(1\) in \(1992\) and in \(2025\) were both on a Wednesday, the years won’t have the same yearly calendar because \(1992\) was a leap year. This is since \(1992\) is divisible by \(4\), but not \(100\).

  3. There are \(7\) possible weekdays on which January \(1\) can occur. For each of these possibilities, there is a calendar that includes February \(29\) and a calendar that does not. Thus, there are \(7\times 2 = 14\) different yearly calendars.