For which one of the following years is the calendar exactly the same as the calendar of the year 1856?

Introduction / Context:
Calendar repetition questions ask you to identify another year whose dates and days of the week line up exactly with a given year. This is an important concept in aptitude exams because it brings together ideas about leap years, ordinary years, and how the pattern of days repeats after certain intervals.

Given Data / Assumptions:

• The reference year is 1856.
• We are given five candidate years: 1883, 1884, 1864, 1880, and 1872.
• We must determine which one has exactly the same calendar as 1856.
• We use standard Gregorian leap year rules: a year is leap if it is divisible by 4 and not a century year, or divisible by 400.

Concept / Approach:
For two years to have the same calendar, two conditions must hold:

• Both must be either leap years or non-leap (ordinary) years.
• The day of the week on 1st January must be the same in both years, which is controlled by the total number of days (and thus odd days) in the years between them.

A leap year has 366 days, giving 2 odd days, while a non-leap year has 365 days, giving 1 odd day. The pattern of calendars repeats when the total odd days between the two years is a multiple of 7, and the leap year pattern matches.

Step-by-Step Solution:
Step 1: Check whether 1856 is a leap year. Since 1856 is divisible by 4 and is not a century year, it is a leap year. Step 2: Among the options, identify which years are leap years. 1864 and 1884 are divisible by 4 and are not century years, so both are leap years. 1880 and 1883 also need to be tested: 1880 is divisible by 4 (leap year) and 1883 is not (ordinary year). 1872 is divisible by 4 and is also a leap year. Step 3: Now, to have the same calendar as 1856, we need a leap year where the total number of odd days between 1856 and that year is a multiple of 7. Step 4: Using standard calendar tables or known results, we find that the calendar of 1856 exactly matches the calendar of 1884. Step 5: Hence, 1884 is the year whose calendar is identical to that of 1856.

Verification / Alternative check:
A practical verification method is to compare the day of the week for 1st January and key dates (like 1st March, 1st May, etc.) of 1856 and 1884 using known calendar tools or formulas. When this comparison is done, every date aligns in both year-month grids. No such perfect match exists with 1864, 1880, 1883 or 1872, confirming that 1884 alone shares the same calendar as 1856.

Why Other Options Are Wrong:
1883: This is not a leap year, so its leap-year pattern does not match 1856, and the calendars cannot be identical.
1864: Although it is a leap year, the total odd days between 1856 and 1864 do not sum to a multiple of 7, so the days of the week do not align exactly.
1880: This is a leap year, but again the accumulated odd days between 1856 and 1880 do not give the same weekday alignment, so the calendar is not identical.
1872: Also a leap year, but the odd-day count between 1856 and 1872 does not produce an exact 7k shift; hence the weekdays do not match month by month.

Common Pitfalls:
Many students think that simply matching leap-year status is enough for two years to share the same calendar; this is not true. The sequence and count of odd days between the years also matter. Another mistake is to assume that calendars repeat every fixed set of years like 28 automatically; century years and leap year exceptions complicate this pattern. It is also easy to misapply the leap year rule, especially for century years, but here all candidate years are relatively straightforward.

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