You'd expect a date to fall on each of the seven weekdays about equally often. It doesn't โ
and it can't. The Gregorian calendar runs in an exact 400-year cycle, and 400 isn't
divisible by 7, so every fixed date has weekdays it prefers and weekdays it avoids. Because
the cycle is exact, these aren't estimates from a big sample โ they're the permanent,
countable truth of the calendar.
Why the calendar is exact
400 years contains exactly 146,097 days, and 146,097 รท 7 = 20,871 โ a whole number of
weeks. The leap rule (every 4th year, except centuries not divisible by 400) gives 97 leap
years per 400, so the average year is exactly 365.2425 days. Land on a whole number of
weeks and the entire mapping of dates-to-weekdays resets: the calendar for any year is
identical to the one 400 years earlier. That's what makes the counts below exact rather than
statistical.
New Year's Day is a Sunday more than a Monday
Count every 1 January across one full cycle and the weekdays don't come out even:
- Sunday, Tuesday, Friday โ 58 times each
- Wednesday, Thursday โ 57
- Monday, Saturday โ 56 times
A perfectly even split would be 57.14 per weekday, which is impossible with 400 occurrences and
7 weekdays. The gap is small โ but it never averages out, because 400 years is the whole
calendar. There's no larger sample coming.
Every date has its own favourite weekdays
The tilt isn't uniform. Where a date sits relative to February and the leap-day machinery
changes which weekdays it leans toward. US Independence Day (July 4) leans the opposite way
from New Year's โ its common weekdays are Monday, Wednesday and Saturday. And a tidy
consequence of seven-day spacing: Christmas and the New Year's Day a week later always share a
weekday (25 December + 7 days = 1 January), so their distributions are identical, just shifted
a year.
Leap day: 97, not 100 โ and rarest on a Sunday
The cycle has 97 leap years, not 100: the Gregorian rule drops three century leap days (1700,
1800 and 1900 were common years; 2000 kept its) to hold the average year at 365.2425 days. So 29
February appears just 97 times in 400 years, and even those aren't even โ it lands on a Sunday
least often and on a Monday or Wednesday most. A Sunday leap-day birthday is the rarest slice of
an already rare cake.
Your calendar comes back every 6 or 11 years
You don't have to wait 400 years to reuse a calendar. A whole year's calendar returns as soon as
two things line up again: the weekday it starts on, and whether it's a leap year. In the current
era those matches arrive in a 6-, 11- and 28-year rhythm. 2025's exact calendar returns in
2031, 2042, 2053 and 2059. So the dusty wall calendar in the drawer really is reusable โ you
just have to catch it on the right year.
Reproduce it
No dependencies, no randomness โ just the weekday of each date summed over one 400-year cycle,
using Sakamoto's algorithm:
const T = [0,3,2,5,0,3,5,1,4,6,2,4] // 0 = Sunday โฆ 6 = Saturday
function dow(y, m, d) {
if (m < 3) y -= 1
return (y + (y/4|0) - (y/100|0) + (y/400|0) + T[m-1] + d) % 7
}
// count dow(y, 1, 1) for y = 2000..2399 -> New Year's Day per weekday
Any 400 consecutive years give the same counts โ that's the whole point. To check your own dates,
the date calculator,
yearly calendar and
leap-year calculator all run in the browser.
Originally published on LK Forge.
