John Horton Conway (1937–2020) came up with an algorithm in 1973 for mentally calculating what day of the week a date falls on. His method, which he called the “Doomsday rule” starts from the observation that every year, the dates 4/4, 6/6, 8/8, 10/10, 12/12, 5/9, 9/5, 7/11, and 11/7 fall on the same day of the week , what Conway called the “doomsday” of that year. That’s Monday this year.
Once you know the doomsday for a year you can bootstrap your way to finding the day of the week for any date that year. Finding the doomsday is a little complicated.
Conway had his computer set up so that it would quiz him with random dates every time he logged in.
Recently I’ve been thinking about mental exercise rituals, similar to Conway having his computer quiz him on dates. Some people play video games or solve Rubik’s cubes or recite poetry.
Curiously, what some people do for a mental warm-up others do for a mental cool-down, such as mentally reviewing something they’ve memorized as a way to fall asleep.
What are some mental exercise rituals that you’ve done or heard of other people doing? Please leave examples in the comments.
More on Conway
The drawing at the top of the page is a sketch of Conway by Simon Frazer. The strange thing coming out of Conway’s head in the sketch is Alexander’s horned sphere, a famous example from topology. Despite appearances, the boundary of Alexander’s horned sphere is topologically equivalent to a sphere.
Conway was a free-range mathematician, working in a wide variety of areas, ranging from the classification of finite simple groups to his popular Game of Life. Of the 26 sporadic groups, three are named after Conway.
Here are some more posts where I’ve written about John Conway or cited something he wrote.
- Never applied for a job
- Opposite of Parkinson’s law
- Ternary Golay code
- Almost prime numbers and almost integers
- Subprime fib sequence
 Because this list of dates is symmetric in month and day, the rule works on both sides of the Atlantic.
10 thoughts on “John Conway and mental exercise rituals”
For many years, I’ve had a ritual based on a memory system my Dad taught me when I was a tyke. I go over the 26-letter NATO alphabet, a memory-peg word, a Hebrew word, and two words that encode ASCII for the capital and uncial letters. It begins “Alpha, tea (1), Aryeh (lion), Jail (65), Back (97),” goes through “November, tar (14), Nakhash (snake), Cuff (78), Tights (110),” etc. It puts me into flow-state, or sends me to sleep quickly. Variations of the ritual exploit the accidental (?) association between the 52 glyphs in the Latin alphabet, 52 cards in a standard card pack, the 52 weeks of the year, and the 156 (3 x 52 = 2 x 78) faces of a tarot pack, obverse and reverse. I have a combinatorial supply of visual memory tricks!. I move on to an algorithm for day-of-the-week from year that is totally different from Conway’s because it’s based on my memory pegs. It’s easier, too, requiring only arithmetic mod 4 and 7. My method was published by Bruno Furst in many books and pamphlets through the 1970’s. Finally, I have variants of the system that work on hex and Radix-36, allowing me to memorize SHA hashes and other obfuscated data. I don’t practice any of that enough to be fast, but I know people who do and are quite amazing!
From time to time I have short a piece of cord lying around and practice knots as a mental exercise ritual. It’s a great way to learn new knots and to burn familiar knots into muscle memory.
I mentally practice the piano before falling asleep. I find that tricky parts or reciting learned passages creates similar stumplers when I only imagine playing as if I really do. I also get better playing for real by only playing in my mind. The disadvantage is that you do not have the sound to fact check – ie if you practice a section incorrectly on the real piano it sounds wrong too – not so in my mind.
Doesn’t anybody count sheep anymore?
The doomsday algorithm is my go-to method when I need to figure out the day of the week for a date, and I have no electronic tools at my disposal. Thanks for this article.
In addition to the dates that you cite, the following dates also fall on Doomsday:
The last day of February
July 4 (Independence Day in the U.S.)
October 31 (Halloween, or the last day in October, if you prefer)
The day after Christmas (often, but not always “Boxing Day”)
You can remember the May 9/September 5 pair, together with the July 11/November 7 pair, with the sentence “I work from 9 to 5 at the 7-Eleven”.
Finally, I agree – determining Doomsday is quite complicated, but once you know Doomsday for one year, it’s easy to determine it for the years close to it. Doomsday always advances one day per year (365 mod 7 = 1), except in leap years, when it advances two days (366 mod 7 = 2).
So, because Doomsday is Monday in 2022, it will be Tuesday in 2023, and Thursday in 2024.
I memorize Bible verses. I started in 2019, I’ve already memorized all of Galatians and I’ve done the first 5 chapters of Ephesians. I use Anki to create flashcards and when I have some spare moments, I review the deck.
Normally my recall is around 80-90% per day. When my daughter was born last year, there were days when my recall dropped to 50%.
Seth Roberts also did arithmetic problems every morning. He measured his performance (how fast he could solve them) and tracked many other things. He found that eating half a stick of butter every day could shave off 30 milliseconds: https://quantifiedself.com/blog/seth-roberts-on-arithmetic-and/
Like Conway, I enjoy the game of go. Studying the endgame in go led Conway to develop surreal numbers. He also first explored Game of Life patterns on his go board before someone offered to program it for him.
I have found lately that doing joseki (opening patterns in go) in my head puts me to sleep very quickly! I’m sure the same thing would work with chess openings.
I find it interesting that almost all commenters – myself included – posted sheep-counting routines rather than morning exercise routines.
Mine: 1) Collatz. Take a natural number and do Collatz iterations on it until you reach a power if two or fall asleep; 2) Recite the Rosary