Zeller Congruence — Day of Week

Christian Zeller published a closed formula in the 1880s that returns the weekday of a date in a single expression, with no table and no loop. It still shows up in code because it is short and uses nothing but integer division — no floating point, no date library. The page breaks out each term, which helps both for checking an implementation and for seeing where every number comes from.

The trick that makes the formula work is treating January and February as months 13 and 14 of the previous year. That moves the variable-length month to the end of the year, so the leap day stops shifting every month after it. The same device appears in nearly every serious calendar algorithm, including the ones that convert to Julian day.

Note that the result starts on Saturday: the formula returns zero for Saturday, one for Sunday, and so on. It is a convention that catches out anyone expecting Sunday as zero, and the most common cause of a seven-day error in hastily copied implementations. The page also handles the Julian calendar, which changes two terms of the expression, and rejects dates that do not exist, such as 31 February.