The poster to the left hangs on the wall of my office. Can you figure out the pattern to the sequence? Now can you estimate the size of the nth entry?
John Horton Conway died yesterday, a victim of Covid-19. His unique mathematical style combined brilliance and playfulness in equal measure. I came across his “soldier problem” at an impressionable age, and was astonished by the beauty of the solution. You’ve got an infinite sheet of graph paper, with one horizontal grid line marked as the boundary between friendly and enemy territory. You can place as many soldiers as you like in friendly territory, at most one to a square. Now you can start jumping your soldiers over each other — a soldier jumps horizontally or vertically, over an adjacent soldier (who is then removed from the board) into an empty square. The goal is to advance at least one soldier to the fifth row of enemy territory. Conway’s proof that it can’t be done struck me then as utterly beautiful, utterly unexpected, and a compelling reason to learn more about this mathematics business.
He invented the Game of Life. He invented the system of surreal numbers, a vast generalization of the usual real numbers, designed for the purpose of assigning values to positions in games but adopted by mathematicians for purposes well beyond its original design. (Don Knuth’s book on the subject is a classic, and easy reading). His Monstrous Moonshine conjecture is the reason I own a t-shirt that says:
which I am prepared to argue is the most remarkable equation in all of mathematics. He was the first person to prove that every natural number is the sum of 37 fifth-powers. More than a century after mathematicians first gave a complete classification of two-dimensional surfaces, Conway (together with George K. Francis) found a much better proof. He worked in geometry, analysis, algebra, number theory and physics. And reportedly, he could solve a Rubik’s Cube behind his back, after inspecting it for a few seconds.
Gone now, along with John Prine — another icon of my youth — and too many others. I got the word about Conway just as I was about to go to bed, and am typing this in a state of exhaustion. If I were more awake, it would be more coherent, but it will have to do.
Sorry to hear that…probably my first OWN use of a real computer (I practiced on a Texas Instruments calculator that read magnetic cards in high school) at the U or R using that account that all students get was to program it to print out Life-forms …oodles of printout pages wasted on that one!
Never knew about “Conway’s Soldiers”, thanks for making me look that up. I also looked up Monstrous Moonshine, but that one just hurt my head.
This is very sad news. He was both the first living mathematician whose name I remember reading and the first author of a book I’m currently perusing, The Atlas of Finite Groups. I’ll forever regret not talking with him at least briefly when I had the opportunity.
As for the Soldier Game (referred to as Conway Checkers), I learned about it a few years ago at a less impressionable age, on Numberphile. https://www.youtube.com/watch?v=FtNWzlfEQgY
(Numberphile offers layman’s explanation by real mathematicians of mathematical ideas outside the typical curriculum. It, and its sister channel Computerphile, are recommended to anybody who likes that sort of thing.)
Sorry to hear about this big loss for all of us. Like Jens Fiederer, I was introduced to Game of Life early in my programming career, and was fascinated how a few simple rules could lead to such complex systems.
Seconding the Numberphile recommendation. It’s possibly my favorite channel on Youtube.
Very sorry to hear. I was familiar with some of his work, mostly through Martin Gardner.
I had seen the sequence before, so was able to figure it out.
As for the length of the nth entry, is
16/9 ( 2 ^ (n-1))
kind of close?
It’s the worst. All of this is the worst. It feels like we live in a society where all the decent, kind people are working hard and caring for so many and are getting killed off while the grossest sociopaths out there (and not even the smart kind, the sloppy narcissistic kind with the attention spans of toddlers) are ascendant, the world over.
I’m so sorry to hear that. I too learned about Conway from Martin Gardner’s “Mathematical Games” column. I was fascinated by Conway’s Game of Life and the possibility that the universe might be a cellular automaton. And I started a Sprouts club in high school.
He will be missed.
@Jeff S. Ah, yes, Martin Gardener’s column at the back of Scientific American was a gateway drug to real mathematics for many of us.
@Drew Would it be a comfort to note that it has ever be thus? And that somehow we still have managed to make some progress despite of it? See “Why the Worst Get on Top” in Hayek’s Road to Serfdom.
I have seen nice tributes to Conway in other places. Here is one where someone implemented Game of Life in Factorio:
https://www.reddit.com/r/factorio/comments/g4jz95/in_memorium_of_john_conway_an_implementation_of/
Sub Specie: “Martin Gardener’s column at the back of Scientific American was a gateway drug to real mathematics for many of us.”
Every mathematician over 50 blames Gardner for his choice of career.
Given seven copper tubes, 5″ long, 1/2″ diameter, cut flush at
each end. Arrange them on a table, such that each unit touches
every other.
Or, show it’s impossible –
Richard D.: Ah, Martin Gardner! I confess I never subscribed to Scientific American, not living in the US at the time. But I buy every single book by him, both the collections of his columns and the original ones.
That along with Samuel Loyd’s puzzle books, Raymond Smullyan, and Douglas Hofstadter’s classic Gödel, Escher, Bach was not quite enough to make me a mathematician.
But it was enough to make me a hacker. theoretical physicist, practicing lawyer and, at present, a senior software engineer in the research division of major tech company.