Examkin

Question

Difficulty: HardClocks and Calendars

On Wednesday, August 12, 17611761, a rare astronomical alignment was recorded in a historical journal. Exactly 200200 years later, on August 12, 19611961, the alignment was observed again. Based on the rules of the Gregorian calendar, what day of the week was August 12, 19611961?

Answer: Saturday / saturday / Sat / sat

Answer

Saturday
The total day shift over a 200-year span is the sum of standard year shifts (200 days) and the true number of leap days (48 days, which excludes the non-leap century years 1800 and 1900). A total shift of 248 days is equivalent to 3 days modulo 7, meaning the starting day of Wednesday shifts forward by 3 days to land on Saturday.

Step-by-Step Solution

1
Determine the basic shift in days for the total number of years.
The timeframe is exactly 200200 years, which provides a base shift of 200200 days.
A standard year of 365365 days shifts the day of the week by 11 day (365≡1(mod7)365 \equiv 1 \pmod 7). Therefore, each year contributes at least 11 day of shift.
2
Count the total multiples of 4 in the given year range to estimate leap years.
There are 5050 multiples of 44 between 17611761 and 19611961.
The first multiple of 44 after 17611761 is 17641764, and the last multiple of 44 before 19611961 is 19601960. The number of multiples is calculated as 1960−17644+1=50\frac{1960 - 1764}{4} + 1 = 50.
3
Apply the Gregorian calendar rules for century years to determine the true number of leap days.
The true number of leap days is 4848.
Century years are leap years only if they are divisible by 400400. The years 18001800 and 19001900 fall within this range and are divisible by 100100 but not 400400. Thus, they are standard 365365-day years. Subtracting these 22 exceptions from the initial 5050 yields 4848 true leap days.
4
Calculate the total shift in days and find its equivalent modulo 7.
The total shift is 248248 days, which is equivalent to a shift of 33 days.
The total shift equals the base shift (200200 days) plus the extra leap days (4848 days). Dividing 248248 by 77 gives 3535 with a remainder of 33. So, 248≡3(mod7)248 \equiv 3 \pmod 7.
5
Add the calculated modular shift to the original day of the week.
Wednesday + 33 days = Saturday.
Moving 33 days forward from the starting day of Wednesday (Thursday, Friday, Saturday) provides the correct day of the week for August 12, 1961.

Key Concept

Gregorian calendar leap year century rules and modular arithmetic for day shifting

Alternative Method

You can calculate the shift per century. In a 100-year span crossing a non-leap century year (like 1761-1861 or 1861-1961), there are exactly 24 leap years. Each 100-year span shifts the day by 124 days. Since 124 modulo 7 is 5, each century shifts the date by 5 days. For two centuries, the total shift is 5 + 5 = 10 days. Finally, 10 modulo 7 results in a 3-day forward shift.
Estimated Time:3m 0s
Rate this question

Topics

Clocks and Calendars

More questions from this topic