The doomsday trick

Conway's doomsday rule: a dozen dates, a little arithmetic, and the weekday of any date, in your head — the one contraption in this corner that runs on a human brain. ← Calendar Contraptions

Every other machine in this corner computes for you: type a date, receive an answer. This piece is the opposite. It is a calendar contraption designed to run on the computer between your ears — John Conway's doomsday rule, devised in 1973. Conway could answer in under two seconds, and kept in practice by having his own computer quiz him with a random date every time he logged in.

The trick stands on one observation: every year has a handful of easy-to-remember dates that all fall on the same weekday — 4/4, 6/6, 8/8, 10/10, 12/12, the last day of February, 9/5 and 7/11 (and 5/9 and 11/7 — "I work from 9 to 5 at the 7–11"), Pi Day, and January's 3rd. Conway called that shared weekday the year's doomsday. Find the doomsday in two small steps, count from the nearest one, and you are done. Worked traces below; a drill at the bottom.

The twelve dates

These twelve fall on the doomsday in every year — the entire memorization load of the trick. Even months carry their own day (4/4, 6/6, 8/8, 10/10, 12/12); the odd ones are 9-to-5 at the 7-Eleven; January and February hold the only subtlety, the "3rd in three years out of four, 4th in the fourth" rule. Conway's own summary: "'Last' in Jan and Feb, otherwise nth in even months, (n ± 4) in odd ones" — plus for odd months, where the sign matters.

Some famous dates ride along free: they are doomsdays too, every single year, and Christmas is always the day before one (December 25 is thirteen days after 12/12 — two weeks less a day).

The drill

The trick is a skill, and skills want reps. Pick a century range, take a random date, and name its weekday before you look. Streaks and records live in your browser only.

The century wheel

Which weekday a year's doomsday lands on is pure arithmetic: the century anchor plus y + ⌊y/4⌋ for the year's last two digits. Conway suggested thinking of weekdays as numbers — Noneday or Sansday (0), Oneday, Twosday, Treblesday, Foursday, Fiveday, Six-a-day — so the wheel below shows each year of the century as its number. The rhythm is visible: +1 on common years, +2 into leap years (gold rings), and the whole column repeats every 28 years.

The 400-year ledger

The Gregorian calendar repeats itself exactly every 400 years: 146,097 days, which is exactly 20,871 weeks — so the 400 years are a closed loop, and its doomsday census is the same for every such cycle. No weekday gets an equal share: doomsdays land on Monday, Wednesday and Saturday 58 times each, on Sunday and Tuesday only 56.

Why it works

The year part. A common year is 52 weeks and one day — 365 = 52 × 7 + 1 — so each common year pushes the doomsday forward one weekday, and a leap year pushes its own doomsday two (the February 29 is inside it). For a year's last two digits y, the shift is y + ⌊y/4⌋ (mod 7): one for every year, one extra for every fourth. Conway noticed the pattern almost repeats every 12 years — 12 years move it 12 + 3 = 15 ≡ 1 day, "a dozen years is but a day", as he put it in the 1973 paper — so he splits y into dozens (a = ⌊y/12⌋), remainder (b), and fours in the remainder (c = ⌊b/4⌋), and adds a + b + c. The odd + 11 method (Fong and Walters, 2010) computes the same number with no division by 4, by doubling and nudging with 11s; this page runs both and shows they agree.

The century part. A common Gregorian century holds 24 leap years, so it moves the doomsday (100 + 24) mod 7 = 5 days forward — that is, two back. A leap century, with 25 leap years, moves it six forward — one back. Three common centuries then a leap one: two back, two back, two back, one back — 7 back in total, a full week. Four centuries cancel, so the anchor depends only on c mod 4: 5·(c mod 4) mod 7 days after Tuesday — 2000s and 1600s Tuesday, 1900s Wednesday ("We-in-dis-day"), 1800s Friday, 1700s and 2100s Sunday ("twenty-one-day is Sunday"), 2200s Friday.

Putting it together. Anchor + year shift = the year's doomsday, a number 0–6. Then take your date, find the nearest of the twelve dates, and add the day difference. That is the entire machine — three steps, all of them small enough to do while someone waits for your answer.

The Julian twin

The rule works unchanged on the Julian calendar — only the anchor differs: 6·c mod 7 days after Sunday. Every Julian century has 25 leap years (the century year itself is always leap), so each century slides the anchor one weekday back, forever, with no four-century reset. The 28-year rhythm is exact there: 28 Julian years are 10,227 days = 1,461 weeks precisely. Switch the date box above to "Julian" to run your date through the Julian machine. For example, on the Julian calendar the year 1582 has Wednesday as its doomsday, so Wednesday 10 October 1582 is a doomsday and Thursday 4 October 1582 follows from it — the day before the Gregorian reform deleted ten days and made the next sunrise land on Friday 15 October (the whole story here).

A short history

Weekday formulas are old: Lewis Carroll published one in Nature on 31 March 1887 — "To Find the Day of the Week for Any Given Date" — and it was Carroll's method that Conway credited as his inspiration. Christian Zeller's congruence, first published in 1882, is the formula the machines use (this corner's pages among them). Conway's doomsday rule appeared in 1973, in the Cambridge student magazine Eureka, as "Tomorrow is the Day After Doomsday" — the same paper that coined "a dozen years is but a day". The odd + 11 refinement is from Chamberlain Fong and Michael Walters (2010), and Hirofumi Nakai published yet another doomsday formula in 2023 — people are still polishing this trick, a century and a half after Zeller's first paper and fifty years after Conway's.

Honest limits

The rule names weekdays, not history: both calendars here run proleptic across their whole range, while real countries switched between them at different times — in 1582, or 1752, or 1918, or never (the switch table lives here). A date before a country's switch is a Julian date, and the Gregorian reading of those numbers is a fiction; the toggle exists to tell the two apart. January and February keep their common-year/leap-year trap (the 3rd versus the 4th), the drill only draws Gregorian dates, and the decade 2020–2029 is easy mode — records set there are not comparable to the full-range ones.