A small guide on calculating the day of week fairly easily
(Image ‘borrowed’ from here)
Twelve dates are all we need to remember, (and one day of week each year), the rest is simple calculus that anyone can do without paper.
1) Memorise twelve DATES that got the same weekday (four weeks apart), note the start in March, to avoid dealing with Feb 29.
Mar. 31.
Apr. 28.
May 26.
Jun. 23.
Jul. 21.
Aug. 18.
Sep. 15.
Oct. 13.
Nov. 10.
Dec. 8.
Jan. 5. (*)
Feb. 2. (*)
(*) the following year
If you prefer a formula: Floor(39.3 – m×2.6) … with m in 3 to 14.
2) Each year memorise this day of week for current year. E.g. for 2021: Wednesday, and optionally the week-number for the first: 13 in 2021
Usage:
3) Remember that January and February are handled as previous year. Count the number of days relative to one of the reference days, While larger than 7 subtract 7 (or: take the remainder of this number by division with 7), and forward the day of week that number of days. (Obviously you can count both days before and after, stepping backwards/forwards)
Example: what is the weekday of new years eve?
Either count this as 5 days before Jan. 5. And as 7-5=2, that is 2 days after the memorised day.
Or if you prefer counting forwards only start from Dec 8. It is (31-8)=23 days after, but 23 
2 (Or: 23-7-7-7=2), so 2 days after.
So based on Wednesday for 2021, that is: Friday.
Note that Mar. 2. in normal years and Mar. 1 in leap years obviously have the same weekday as Feb. 2 (as 28 0). And as Mar. 3 obvious got the same weekday as Mar 31. (31-3=28 0) the day of week of a particular day is advanced by one every normal year and by two in leap years as you most likely already knew.
If you want to calculate for a date somewhere further in the future, you thus need to add 1 per year PLUS an additional 1 per intermediate 29/2 (leap year).
Obviously subtract similar for a date in an earlier year.
If you have trouble memorising the 12, but do remember the number of days per month, moving four weeks forward is moving one month ahead and days backwards 31-28=3 respectively 30-28=2 going from a month with 31 respectively 30 days to the next.
Week numbers
Note that the dates selected for memory are deliberately taken four weeks apart, this also makes it easier to calculate date distances and work with week-numbers.
For obvious reasons Jan 4 is always week 1. So Mar 29 in normal years (Mar 28 in leap years) is thus always week 13
So looking at the weekday of March 31 (two or three days later):
- If Monday or Tuesday it is always week 14 (going back 2 or 3 days always goes to previous week)
- If Wednesday then week 13 normally and 14 in leap years (2 days back is same week, 3 days previous week)
- If Thursday to Sunday it is always week 13 (going back 2 or 3 days is always the same week)
List for some years March 31. of week-number and weekday
2020: 13, Tuesday
2021: 13, Wednesday
2022: 13, Thursday
2023: 13, Friday
2024: 13, Sunday
2025: 14, Monday
2026: 14, Tuesday
2027: 13, Wednesday
2028: 13, Friday
2029: 13, Saturday
2030: 13, Sunday
2031: 14, Monday
2032: 14, Wednesday
2033: 13, Thursday
2034: 13, Friday
2035: 13, Saturday
2036: 14, Monday
2037: 14, Tuesday
2038: 13, Wednesday
2039: 13, Thursday
2040: 13, Saturday
2041: 13, Sunday
2042: 14, Monday
2043: 14, Tuesday
2044: 13, Thursday
2045: 13, Friday
2046: 13, Saturday
2047: 13, Sunday
2048: 13, Tuesday
.
.
(and 28 years pattern repeats itself)