There are many ways to approach this problem.
A general approach to this problem involves creating
a step-by-step procedure for determining the day of the
week for a past date. One way to describe such a procedure
is with a flowchart. The flowchart given here finds the
day of the week for any date from 1900 to 2000 (The
Farmers Almanac, p. 113).
Another way to find the day of the week
of your birth follows: Consider the number of days in
a year. Non-leap years have 365 days. Since 365 ÷
7 gives a remainder of 1, then any particular date moves
ahead one day of the week in the following year. Going
back a year, each date moves back one day of the week.
Since leap years have 366 days, and 366 ÷ 7 gives
a remainder of 2, any date moves ahead two days of the
week in the year following a leap year.
This means that when going back in time, each date moves
back two days of the week for each leap year. (Note: If
the current year is a leap year, then only dates after
February 29 will have moved two days ahead.)
Suppose that you were born on December 31, 1986. From
the hint, you know that January 1, 2000 was a Saturday.
That means that December 31, 1999 was a Friday. The year
1986 was 13 years before 1999. The years 1988, 1992, and
1996 were all leap years. That means that the day of the
week moves backward 13 + 3, or 16 days. Since 16 ÷
7 gives a remainder of 2, you should count back two days
from Friday to get Wednesday.