The calendar for the year 1856 is to be matched with another year so that all dates fall on the same days of the week. Among the years listed below, for which year is the calendar exactly the same as that of 1856?

Introduction / Context:
Older year calendar questions require you to combine leap year rules with weekday shifts across many years. Here the reference year is 1856. We must decide which one of the given years produces the same calendar, meaning that 1 January and every other date fall on the same weekday in both years.

Given Data / Assumptions:
- Reference year = 1856. - Candidate years = 1883, 1884, 1864, 1880. - Gregorian calendar rules are used. - A leap year has 366 days, and a normal year has 365 days. - Leap year rule: divisible by 4 but not by 100, unless divisible by 400.

Concept / Approach:
For two years to share the same calendar, they must both be leap years or both be normal years, and the weekday shift from the first year to the second must be a multiple of 7. The shift for a normal year is 1 day, and for a leap year it is 2 days. We first check whether 1856 is a leap year, then compare it with each candidate year, focusing on both leap year status and whether the accumulated weekday shift is divisible by 7.

Step-by-Step Solution:
Step 1: Check if 1856 is a leap year. It is divisible by 4 and not a century year, so 1856 is a leap year. Step 2: Leap year status of candidate years: - 1883: not divisible by 4, normal year. - 1884: divisible by 4 and not by 100, leap year. - 1864: divisible by 4 and not by 100, leap year. - 1880: divisible by 4 and not by 100, leap year. Step 3: Only leap years can match 1856 in calendar pattern, so 1883 is eliminated. Step 4: Now we must find which among 1864, 1880, and 1884 has 1 January on the same weekday as 1856. Step 5: The cumulative shift in weekdays from 1856 to 1884, considering the sequence of leap and normal years, totals a multiple of 7 days. This brings the starting weekday back to the same value as in 1856. Step 6: A more direct check shows that 1856 and 1884 are both leap years starting on the same weekday, leading to identical monthly day layouts.

Verification / Alternative check:
A practical approach is to use known calendar tables or to apply a day of week calculation formula such as the standard century and year code method. Doing this for 1 January 1856 and 1 January 1884 reveals that both fall on the same weekday. Comparing the February layouts confirms that they both have 29 days, and subsequent months line up exactly, which proves that the calendars match.

Why Other Options Are Wrong:
- 1883 is not a leap year, so February and all later months misalign with the leap year structure of 1856. - 1864 and 1880 are leap years but their starting weekdays differ from that of 1856, so the complete calendars do not match. - Any mismatch in starting weekday or leap year status prevents an exact calendar repeat.

Common Pitfalls:
Students often rely on rough cycle rules such as assuming repetition every 28 years without handling century corrections or actual weekday shifts. Others compare only the leap year status and ignore starting weekdays. For precise answers, always consider both conditions, or verify using a known day of week formula or trusted calendar comparison method.

Final Answer:
The calendar for the year 1856 is the same as the calendar for 1884.