The Conway Claim Conundrum

Let’s say you’re me and you’re reading the Wikipedia article for the card game Beggar My Neighbour. You scroll down to the math section (which every Wikipedia article should have, honestly) and find that it’s an open problem whether every game of Beggar My Neighbour ends. “Hm, the game seems pretty simple, weird that no one’s solved this yet,” you think, as you continue reading. You then find that “John Conway once listed this among his anti-Hilbert Problems, open questions whose pursuit should emphatically not drive the future of mathematical research.” “Lol, good one Conway, he doesn’t miss,” you think as you prepare to turn your eyes towards the next sentence when suddenly a thought comes in your head. “Conway has a list of anti-Hibert problems? That sounds pretty interesting, I wonder if he has stuff like the Collatz conjecture on there.”

Naturally, you look up (using Ecosia, if you use Google consider switching) these problems, hoping to find a list. The page loads, and almost nothing looks like a nice pdf of problems. Richard P Mann’s site repeats the same claim as the Wikipedia article. It does contain a list of the Beggar My Neighbour hands which take the longest to end, which you find pretty interesting, so you bookmark it for later. Marc M Paulhus’s Beggar My Neighbour paper has a quote from Conway in it, confirming that Conway claimed to have a list of anti-Hilbert problems, but no new information on said problems can be gleaned from it. What’s more, no source is provided for this quote, so the trail immediately runs cold. Finally, a Wikipedia discussion turns up in which user SartorialBenefit has the exact same question you do with no good answers!

“This is a mess, let’s check where that original Wikipedia article sourced this claim from,” you think. The sentence that initially inspired you to go on this wild goose chase has one reference from the article “Unsolved Problems in Combinatorial Games” along with a little quotation next to the reference. The file linked to in the Internet Archive doesn’t have the article’s pages available publically because of course it doesn’t, but after a bit of sleuthing you find an archive on the journal on Google Books that once again, claims that Beggar My Neighbour is one of Conway’s anti-Hilbert Problems, and that this list was made about 40 years ago. Since the article was written in 2002, this would date the creation of the list at around 1962, when Conway was 25. It then references the Marc Paulhaus article from before.

Based on this research, I (and by I, I mean you) believe that Conway did actually create list of anti-Hilbert problems, but that this list was never published / shared publically. Richard Guy, who coauthored “Unsolved Problems in Combinatorial Games” was a notable coauthor of Conway and it is unlikely that Guy would hear about this supposed list and include it in his article without confirming its existence with Conway. However, it seems that the contents of the list have been lost to history, with the exception of its inclusion of Beggar My Neighbour. However, it doesn’t seems that this loss is a big deal to mathematics as a whole, given that this list was explicitly not supposed to drive further research. Still, it would be interesting to see what non-serious problems Conway decided to put on the list. If there are any new discoveries related to the Conway Claim Conundrum, please let me (and by me, I mean me) know!

Lucas Jacobs
27 August 2022

(Edit 5 December 2023: I misspelled “Hilbert” in the first paragraph and wrote “seems” instead of “seem” in the last paragraph, but I am not correcting these errors since I think it’s bad practice to edit the past. However I did edit the link to Richard P. Mann’s site from http://www.richardpmann.com/misc.html to https://www.richardpmann.com/beggar-my-neighbour-records.html since I think it is good practice to remove dead links.)