Mentally calculating the day of the week in 2023

Mentally calculating the day of the week will be especially easy in 2023. The five-step process discussed here reduces to three steps in 2023.

One of the steps involves leap years, and 2023 is not a leap year. Another step involves calculating and adding in the “year share,” and the year share for 2023 is zero. So the five steps reduce to these three steps.

  1. Start with a constant corresponding to the month.
  2. Add the day of the month.
  3. Take the remainder by 7.

The constants for each month are as follows.

    | January | 6 | February | 2 | March     | 2 |
    | April   | 5 | May      | 0 | June      | 3 |
    | July    | 5 | August   | 1 | September | 4 |
    | October | 6 | November | 2 | December  | 4 |

You have to come up with your own mnemonics for memorizing this table. As I commented here,

It’s typically much easier to memorize something than to come up with a mnemonic that other people would find acceptable. No matter how natural your mnemonics sound to you, they usually sounds like nonsense to anyone else.

Examples

To find the day of the week for July 4, 2023 we start with 5 for July, add 4, and get 9, which leaves a remainder of 2 when divided by 7. Days of the week are numbered starting with Sunday as 0, so 2 corresponds to Tuesday.

Pearl Harbor Day, December 7, will be on a Thursday next year because 4 + 7 leaves a remainder of 4 when divided by 7.

Doomsday

An alternative to the method above is John Conway’s “Doomsday” rule. This rule observes that that every year, the dates 4/4, 6/6, 8/8, 10/10, 12/12, 5/9, 9/5, 7/11, and 11/7 fall on the same day of the week. These dates are easy to memorize because they break down in double pairs of even digits—4/4, 6/6, 8/8, 10/10, and 12/12— and symmetric pairs of odd digits—5/9 and 9/5, 7/11 and 11/7.

For 2023 the Doomsday dates all fall on Tuesday. You can bootstrap your way to calculating other days of the week starting from these landmarks.

You can add “March 0”, i.e. the last day of February, to the list. In 2023 this is February 28 though of course it is sometimes February 29. And when March 0 is February 28, you can add February 0, i.e. January 31, to the list of Doomsday dates.

The Doomsday rule is less work up front than the rule above, but it’s more work in the long run.

Beyond 2023

In general you have to compute the year share in order to find the day of the week of a date, though in 2023 the year share is zero. This post gives multiple ways to compute the year share in your head.

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