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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.
This step involves finding the remainder when the last two digits of the year are divided by 12. The modulo operation, as previously discussed, helps in dealing with cycles, such as the 7-day week cycle in the Doomsday rule. For the year 1966, we take the last two digits, which are 66, and perform a modulo 12 operation. When you divide 66 by 12, the remainder is 6. Therefore, the modulo 12 of 66 is 6. This number is important as it tells us how many years we are ‘over’ after fitting in as many complete cycles of 12 years as possible. Add 6 to your mental tally.
Quick check
What is 66 mod 12?
The final number to keep in mind is the count of fours within our modulo 12 result. Since we have determined that our modulo 12 number is 6, we now need to calculate how many full sets of four can be found in this number. This step is essential because the leap-year cycle, which comes round every four years, shifts the weekday pattern. For the number 6, there is only one full set of four, as 4 can be subtracted from 6 once, leaving a remainder of 2. So, we will remember the number 1 for this step. This count of fours represents the leap-year adjustments needed for our calculation.
Now, it’s time to combine the numbers we’ve collected and use the modulo 7 operation to determine the Doomsday of the year 1966. This step is crucial because it ties together all the individual calculations we’ve done so far.
Start by summing the numbers from our previous steps: the anchor day (3), the count of twelves (5), the modulo 12 result (6) and the count of fours (1). In our case, this adds up to 3 + 5 + 6 + 1 = 15. Now, to find the Doomsday of the year, we apply the modulo 7 operation to this sum. This is done because there are 7 days in a week, and this operation helps us to cycle back to the correct weekday.
Performing the modulo 7 operation on 15, we get 1; therefore, the Doomsday for 1966 is Monday.