If January 1st, 2000, was Saturday, what day of the week was January 1st, 1999?

Concept

The weekday of a fixed calendar date moves forward by the number of "odd days" in the years crossed. An odd day is the remainder of the day-count divided by 7. An ordinary year of 365 days has 365 mod 7 = 1 odd day; a leap year of 366 days has 2 odd days. Crossing from an earlier date to a later one ADDS odd days; going from a later date back to an earlier one SUBTRACTS them.

Application

1. The two dates are 1 January 1999 and 1 January 2000. The span between them is exactly the year 1999.

2. 1999 is not divisible by 4, so it is an ordinary year of 365 days, giving 365 mod 7 = 1 odd day.

3. Therefore 1 January 2000 falls one weekday AFTER 1 January 1999.

4. We are given 1 January 2000 = Saturday and must go back to the earlier date, so we subtract the odd day: one weekday before Saturday is Friday.

Cross-check

Move forward again: Friday + 1 odd day = Saturday, which matches the given 1 January 2000. The result is consistent, so 1 January 1999 was Friday.