[arjie]

Very clever and fun. Two tangential observations: the bird between two trains problem I remember from childhood when we were studying for an Indian entrance exam. I thought it was in I E Irodov's problem anthology, but I cannot find it there so this must be a false memory. Looks like it's from ancient times, practically Mathematics mythology. Does anyone know the earliest books that have it? No luck with LLMs since it's such a common question today the answers I get from GPT-5.4 and Claude 4.6 Opus with search are unhelpful.

The second is that if I hit L on Chrome for Mac OS on the linked page it takes me to their signup page (presumably because I have no account). So that's a keyboard shortcut to take you to the browser-use app page. But why 'L'? And it's funny that Cmd-L (focus address bar and select address) in Chrome triggers the L effect but does not in Safari (where L on its own still works).

[mohn]

Interesting question, a lot of search engine results claim that John Von Neumann was presented with the problem and quickly solved it by summing the infinite series instead reframing it as a constant speed for an easily calculated duration. Plausible, but sounds apocryphal. Here's the oldest reference I've found and verified by reading scans[0] of the source book:

Initiation Mathématique (1906) by Charles-Ange Laisant (1841--1920), number 53. Le chien et les deux voyageurs.

The setup here has two pedestrians walking in the same direction with a dog running back and forth between them. One of them starts out some distance ahead of the other but, because the one behind walks faster, they eventually intersect. It briefly mentions a variation where they are walking toward one another, as in the typical trains & fly version of the problem. Best of luck finding older, I wouldn't be surprised if it's out there!

[0]: https://i.imgur.com/vCCFgAQ.png

[mohn]

Whoops, my comment was supposed to say "instead of reframing", but the crucial "of" got lost when I reworded things. Too late to edit now.

[arjie]

Very cool! Thank you for doing the research to get that far!

[lukasec]

Great find! plan to add these variants to our parameter sampling. First time I saw this problem was when my game theory prof told this story. It's definitely folklore (see The Legend of John von Neumann by Halmos)

[krackers]

>I cannot find it there so this must be a false memory

It's also infamously the subject of a Von Neumann joke

>Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time a fly that travels at a steady 15 m.p.h. starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover ? The slow way to find the answer is to calculate what distance the fly covers on the first, northbound, leg of the trip, then on the second, southbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann; "all I did was sum the infinite series