exupero's blog

Calculating anchor days

The day of the week doomsdays fall on is often called the "anchor day". I don't have much to add to Wikipedia's instructions for finding it. For a long time I used Conway's original formula,

It's not hard to perform mentally. The hardest part, what's shown in parentheses, only requires finding the largest multiple of 12 smaller than y, the remainder, and whether the remainder is less than 4, between 4 and 7, or greater than 8. Given that y is at most two digits, you only need elementary school multiplication.

But as I consider teaching the doomsday algorithm to kids, the odd + 11 method will be easier for them to learn since it replaces the parenthesized expression above with odd/even checks, addition of 11, and halving, all kid-friendly operations. The full calculation is:

  1. If y is odd, add 11.
  2. Halve the result.
  3. If the result is odd, add 11.
  4. Find the difference to the next highest multiple of 7.
  5. Add the century anchor day.

I've found it's easiest to remember the century anchor day, rather than trying to calculate it. For the Gregorian calendar, Wikipedia gives the formula

In practice, however, the Gregorian calendar has only been used for the last five centuries (fewer in non-Catholic nations), so it's not difficult simply to memorize what centuries had which anchors. Additionally, Gregorian anchor days cycle through only four anchors that always decrease: 5, 3, 2, and 0 (that is, Friday, Wednesday, Tuesday, and Sunday). One can work the sequence forward and back without doing any math: the 1900s have the anchor 3, the 1800s 5, the 2000s 2.

Finding century anchors under the Julian calendar can be done with

though, not being a historian, I don't think I've ever had cause to find the day of the week for a date under the Julian calendar.

In the next post we'll put the pieces together to calculate a date's day of the week.