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Difficulty: MediumClocks and Calendars

A historian finds a perfectly preserved wall calendar for the year 1892. In which of the following subsequent years could this exact same calendar be used for the entire year without any discrepancies?

  1. A
    1897
  2. B
    1898
  3. 1904Answer
  4. D
    1920

Answer

The exact same calendar as 1892 can be reused in the year 1904.
For a calendar to completely match a leap year like 1892, the target year must start on the same day of the week and also be a leap year. While leap years typically repeat every 28 years, the inclusion of the non-leap century year 1900 disrupts this cycle. Counting the odd days from 1892 forward yields exactly 14 odd days by the end of 1903. This means 1904 starts on the same weekday, and because 1904 is a leap year itself, it shares the exact same calendar.

Step-by-Step Solution

1
Determine the properties required for the calendar to match.
The target year must begin on the same day of the week and must also be a leap year, just like 1892.
A leap year has 366 days. If the target year is not a leap year, the days will not match after February 28th.
2
Calculate the cumulative odd days year by year starting from 1892.
1892 (2 odd days), 1893 (1), 1894 (1), 1895 (1), 1896 (2), 1897 (1), 1898 (1), 1899 (1). The sum so far is 10 odd days.
Every standard year adds 1 odd day, while a leap year adds 2 odd days. We must find a point where the sum is a multiple of 7.
3
Evaluate the century year 1900 and continue the odd days calculation.
1900 is a standard year (1 odd day). Continuing: 1900 (1), 1901 (1), 1902 (1), 1903 (1). The total sum of odd days from 1892 to 1903 is exactly 14.
Century years are only leap years if divisible by 400. Since 1900 is not divisible by 400, it is a standard year, which breaks the usual 28-year repetition cycle.
4
Verify the candidate year 1904.
Because the odd days up to 1903 sum to 14 (a multiple of 7), 1904 starts on the same day. Since 1904 is divisible by 4 and not a century year, it is a leap year.
Satisfying both requirements (same starting day and leap year status) means the entire calendar for 1904 is identical to 1892.

Key Concept

Calendar Repetition Across Non-Leap Century Years
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Topics

Clocks and Calendars

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