Using the doomsday to find the weekday

Once you have the year’s doomsday, every date in that year is within counting distance: find the nearest doomsday date, count the gap, and step off from a weekday you already know.

For instance, let’s use the year 1966, for which we’ve calculated the doomsday as Monday. To find out what weekday a specific date fell on, like July 20, 1966, you first identify the closest doomsday date. In July the doomsday date is July 11 (which we know is a Monday). July 20 is 9 days after July 11.

To find the weekday of July 20, add 9 days to the doomsday. Since 7 days make a week, and we’re dealing with days of the week, you can reduce 9 by multiples of 7. In this case, 9 minus 7 leaves 2. So July 20 is 2 days after the doomsday (Monday). Adding 2 days to Monday brings us to Wednesday. Therefore July 20, 1966 was a Wednesday.

That is the whole algorithm, end to end — every date reduces to its nearest doomsday plus a small count. What remains is making each step automatic, which is what the practice games are for.