Second method (due to Conway)

The day of the week for the last day of February (either 2-28 or 2-29) is called doomsday.

Example 1.7.29   Doomsday for 2001 is Wednesday, for 2000 it is Tuesday, for 1999 it is Sunday, and for 1900 it is Wednesday.

A number of useful facts are known about the doomsday.

Fact 1.7.30  

• April 4, June 6, October 10, and December 12 (i.e., 4-4, 6-6, 8-8, 10-10, and 12-12) are all doomsdays (easy to remember since they involve all even numbered months $>3$).
• May 7, July 11, September 5, and November 7 (i.e., 5-9, 9-5, 7-11, 11-7) are all doomsdays (easy to remember since all odd numbered months $>3$ are involved, along with the mneumonic ``I work at the 7-11 from 9 to 5.'').
• March 0 (last day of February) is a doomsday.
• If the year is not a leap year then January 31 (1-31) is a doomsday.
• If the year is a leap year then January 32 (1-32) is a doomsday.

Fact 1.7.31   The doomsday for $19YY$ is given by adding $[YY/12]+(YY\ {\rm mod}\ 12)+ [{YY\ {\rm mod}\ 12\over 4}] \ ({\rm mod}\ 7)$ to Wednesday. For $20YY$, the procedure is the same except add to Tuesday.

Example 1.7.32   Find the day of the week of $9-14-99$ and $9-14-2001$ using the above algorithm.

Doomsday for 1999 is a Sunday. By the above fact, 9-5-99 is a doomsday (Sunday), so 9-12-99 is Sunday. Therefore, 9-14-99 is a Tuesday. Similary, 9-5-2001 is a doomsday (Wednesday), so 9-14-2001 must be a Friday.