Calendar Repetition Rule

A calendar year repeats itself if the total number of days elapsed between January 1 of the base year and January 1 of the target year is exactly divisible by 7. This ensures that the day of the week for each date aligns perfectly.

Leap Year Identification

Understanding leap years is crucial. A year is a leap year if it is divisible by 4, except for end-of-century years, which must be divisible by 400. We examine the years around 1733:

• 1733: Common year (not divisible by 4).
• 1734: Common year.
• 1735: Common year.
• 1736: Leap year (divisible by 4).
• 1737: Common year.
• 1738: Common year.
• 1739: Common year.

Calculating Calendar Match

The number of days in a common year is $365$, and in a leap year is $366$. We calculate the total days elapsed from January 1, 1733, to January 1 of the potential matching years. We check the total days modulo 7.

• Days in a common year: $365 \equiv 1 \pmod 7$.
• Days in a leap year: $366 \equiv 2 \pmod 7$.

We track the cumulative days modulo 7:

Conclusion on Calendar Match

The total number of days between January 1, 1733, and January 1, 1739, is $2191$. Since $2191$ is perfectly divisible by 7 ($2191 \pmod 7 = 0$), the calendar sequence for 1739 repeats the calendar sequence for 1733 exactly.