Mathematicians finally find the infinite card game.

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More details about Jane Street's AMP: https://jane-st.co/SUM-AMP25 The full puzzle is below. Support me on Patreon and join in whatever thing I do next. https://www.patreon.com/standupmaths Please help me thank Mithuna for enduring all of that. Go check out her excellent physics channel: https://www.youtube.com/@LookingGlassUniverse Here is the main paper: A Non-Terminating Game of Beggar-My-Neighbor https://arxiv.org/abs/2403.13855 Huge thanks to Brayden Casella and Richard Mann for helping out with the video. Richard Mann's site: https://www.richardpmann.com/beggar-my-neighbour-records.html Beggar My Neighbour 1999 article by Marc M. Paulhus: https://www.jstor.org/stable/2589054 (paywall) The game playing out on mathstodon thanks to @mscroggs: https://mathstodon.xyz/@beggarmyneighbour All the best maths rumours start on mathstodon. https://mathstodon.xyz/@christianp/111912232261820441 The book I was waving around is The Mathematics of Games (Recreations in mathematics) by John D Beasley https://openlibrary.org/books/OL1854963M/The_mathematics_of_games The pub we were in is the fabulous Queens Head in Piccadilly, London. Good beer, quiet upstairs room. https://www.queensheadpiccadilly.com/ Mega thanks to my Patreon supporters. They keep me in the game. https://www.patreon.com/standupmaths CORRECTIONS - None yet (other than Katie Steckles found a typo), let me know if you spot anything! - Well, technically there was a mistake in the email I sent patreons where one line says 2,260 games instead of the correct 2,560 (first spotted by Takashi Toyooka) but that’s not really a video correction now, is it. Filming by Alex Genn-Bash Editing by Gus Melton Written and performed by Matt Parker Produced by Nicole Jacobus Music by Howard Carter Design by Simon Wright and Adam Robinson Rug courtesy of Lucie requesting it lives in my office not at home. MATT PARKER: Stand-up Mathematician Website: http://standupmaths.com/ Here is the full Guess My Number puzzle: I’m thinking of a number between 1 and 15 (inclusive). You get four attempts to guess my number. After every guess, I’ll tell you either “higher”, “lower”, or “correct.” If you correctly guess my number on one of your four guesses, you win. But there is a catch! If I ever respond “higher” three times in a row or “lower” three times in a row, the game ends and you automatically lose. (This rule applies to all three parts below, so do not forget it!) a) Your first guess is 9, and I respond “higher”. What should your second guess be in order to guarantee that you will win? b) Say we are playing the same game except I’m now thinking of a number between 1 and 40, and instead of four guesses, you get six guesses. Your first guess is 20, and I respond “lower”. What should your second guess be? Are you guaranteed to win? c) Now we’re playing the same game except I’m thinking of a number between 1 and N (inclusive), and instead of six guesses, you get nine guesses. What is the largest value of N such that you have a strategy that is guaranteed to win?