Determining the day of the week for any day of the year.

You don't have to remember them all, but the more rules you remember, the faster you can determine the week day:

To avoid leap-year complications, the "year" runs from March through February.
August 1 always defines the year's special day (e.g. in 2020 August 1 is Saturday).
The last day of the preceding February is also the special day.

Special days occur on most even numbered months on the same numbered day: 04/04, 06/06, 08/08, 10/10, 12/12.
05/09 and 09/05 are also special days.

Some historical or religious events occur on the special day, such as May 2 (Blue Switch Day), June 6 (D-Day), October 31 (Hallowe'en), and December 26 (Boxing Day).
Some national holidays are always on the special day, such as February 6 (New Zealand), May 9 (Russia, Europe), June 6 (Sweden), July 4 (USA), August 1 (Switzerland), and October 3 (Germany).
Some famous people have birthdays on the special day, such as John Quincy Adams, Napoleon Bonaparte, Mao ZeDong, Prince Charles, Ray Bradbury, Freddie Mercury, Walt Disney, Sandra Oh, and Albert Einstein.
Some of your friends and relatives will have birthdays, anniversaries, etc. on the special day.

Months with the same colour always have the same day numbering:

[+1] March 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	[+4] April 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30	[+6] May 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	[+2] June 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
[+4] July 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	[0] August 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	[+3] September 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30	[+5] October 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
[+1] November 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30	[+3] December 2020 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	[+6] January 2021 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	[+2] February 2021 Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Notice that the first and last rows are the same except for the second column. The first row's second column is shifted one position left and its third row is shifted one position right to produce the second row.

January and February of the next calendar year are the same as May and June.
March and November are the same, with the special day being an exact multiple of 7.
April and July are the same.
September and December are the same.

It's usually enough to remember the special days for this year, next year, and last year. But to calculate the special day for any year from 1900 through 2099:

Subtract 1900 from the year.
Increase the result by itself divided by 4 (ignoring any remainder).
Divide the result by 7 and take the remainder.
The special day is that many days after Wednesday.
E.g. 2020: 2020-1900=120; 120/4=30; (120+30)%7=3; Wednesday+3=Saturday.
E.g. 1963: 1963-1900=63; 63/4=15; (63+15)%7=1; Wednesday+1=Thursday.