Day of the Week for January 1, 2078
January 1, 2024, was a Monday. What day of the week is January 1, 2078?
Express your answer as a number $0$-$6$, where $0$ = Sunday, $1$ = Monday, ..., $6$ = Saturday.
Hints
- A non-leap year has $365 = 52 \times 7 + 1$ days, so the day shifts forward by 1. A leap year has 366 days, shifting by 2.
- Count the leap years between 2024 and 2077 inclusive. Remember that a year divisible by 4 is a leap year (unless it is a century year not divisible by 400).
- The total shift is (number of years) + (number of leap years), reduced modulo 7. Add this to Monday's index.
Worked Solution
How to Think About It: The day of the week shifts forward by 1 each year (since $365 = 52 \times 7 + 1$), except after a leap year, when it shifts forward by 2 (since $366 = 52 \times 7 + 2$). So the total shift over 54 years is the number of years plus the number of leap years in that span, all mod 7.
Quick Estimate: From 2024 to 2078 is 54 years. Roughly $54/4 \approx 13$ or $14$ leap years in there. Total shift is around $54 + 14 = 68$ days. $68/7 = 9$ remainder $5$. So we shift 5 days forward from Monday, landing on Saturday ($6$). Let me verify the leap year count to be sure.
Approach: Count exact leap years, compute total day shift mod 7.
Formal Solution:
From January 1, 2024 to January 1, 2078 is exactly 54 years.
Leap years in the range 2024-2077 (these are the years whose extra day affects the Jan 1 shift for the following year; equivalently, these are the leap years we "pass through"): $$2024, 2028, 2032, 2036, 2040, 2044, 2048, 2052, 2056, 2060, 2064, 2068, 2072, 2076$$
That is $14$ leap years. (None of these are century years, so no Gregorian correction needed.)
Total day shift: $$\Delta = 54 + 14 = 68 \text{ days forward}$$
Reduce mod 7: $$68 = 9 \times 7 + 5$$
So the day shifts forward by 5 from Monday: - Monday + 1 = Tuesday - Monday + 2 = Wednesday - Monday + 3 = Thursday - Monday + 4 = Friday - Monday + 5 = Saturday
Answer: January 1, 2078 is a Saturday, which corresponds to $\boxed{6}$.
Intuition
Calendar arithmetic boils down to modular arithmetic. The key insight is that you only need to track two quantities: the number of years elapsed (each contributes +1 day mod 7) and the number of leap years in the range (each contributes an additional +1 day). Everything else cancels out because 52 full weeks is exactly 364 days.
This type of problem shows up in interviews to test mental math discipline -- can you systematically enumerate the leap years without making an off-by-one error? The common mistake is miscounting the leap years at the boundaries. A quick sanity check: in any 54-year span starting from a leap year, you expect about $54/4 = 13.5$ leap years, so 13 or 14 is reasonable. Counting confirms 14.