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We can now use this knowledge to calculate which weekday the doomsday is for any year.
To do this efficiently, you must hold a few values in your short-term memory. Don’t worry if you find that tricky to start with — it gets much easier with practice.
Below is Conway’s original arithmetic, which is what the rest of this course assumes. There is also a shortcut called odd + 11 that reaches the same answer with different moves. Both are correct; they suit different memories.
| Divide by 12 | Odd + 11 | |
|---|---|---|
| What you hold | Three partial numbers, summed at the end | One running number |
| The operations | Divide by 12, remainder, divide by 4 | Add 11, halve, add 11 |
| Easiest to get wrong | Dividing the wrong number by 4 | Forgetting to take the result from 7 |
Example year: 1966.
Determine the anchor day for the date and store the number 3 in your memory bank. Then discard the century — you won’t need it again. The remaining calculations concern only the last two digits of the year.
Count how many whole 12s fit into the last two digits of the year. You can do this by dividing the number by 12 and rounding the answer down to the nearest whole number. Another way to think about it is to count how many times you can add 12 without overshooting. In our example the answer is 5, because 5 × 12 = 60 while 6 × 12 = 72, which is more than 66 — so only five 12s fit into 66. Let’s add 5 to our working memory.
Now find the remainder: the five 12s cover 60, and 66 − 60 leaves 6. That 6 is how many years you are past the last complete block of twelve. Add it to your mental tally.
Quick check
What is 66 mod 12?
One number left: how many whole 4s fit into that remainder of 6. Each one is a leap year hiding in those left-over years, and every leap day shifts the weekday pattern by one. A single 4 fits into 6, so remember 1.
Time to cash in the numbers you’ve been holding.
Sum them: the anchor day (3), the count of twelves (5), the remainder (6) and the count of fours (1) — 3 + 5 + 6 + 1 = 15. Weekdays repeat every 7 days, so throw away the sevens: 15 modulo 7.
Performing the modulo 7 operation on 15, we get 1; therefore, the Doomsday for 1966 is Monday.