January 16, 1997 was a Thursday. What day of the week will it be on January 4,…

To find the day of the week for a date far from a known reference date, use the 'odd days' method: each ordinary (non-leap) year contributes 1 odd day and each leap year contributes 2 odd days, and any leftover days of a partial period contribute odd days equal to their count mod 7. Adding the total odd days to the reference day (in the forward direction of the calendar) gives the day of the week for the target date.

1. Days remaining in 1997 after January 16 = 365 - 16 = 349 days = 49 weeks + 6 odd days.

2. 1998 is not a leap year, so it contributes 1 odd day; 1999 is not a leap year, so it contributes 1 odd day - together 2 odd days.

3. Days from January 1, 2000 to January 4, 2000 = 4 days = 4 odd days.

4. Total odd days = 6 + 2 + 4 = 12 = 7 + 5, so the net shift is 5 odd days.

5. Thursday shifted forward by 5 days is Tuesday.

Cross-check by stepping year to year: January 16, 1997 (Thursday) to January 16, 1998 is 365 days later (1 odd day) - Friday; to January 16, 1999 is 365 days later (1 odd day) - Saturday; to January 16, 2000 is 365 days later (1 odd day) - Sunday. Counting back from January 16 to January 4 (a 12-day shift, i.e. 5 odd days) gives Tuesday - matching the direct calculation.