Knight’s Infinite Odyssey #SoME2

Chess is a wonderful game that is a source for many mathematical riddles and puzzles. One example is Infinite Knight’s Odyssey (actually it doesn’t have any special name, or at least I haven’t found one, so this name you just read is made by me. If the video gets popular maybe the name will get popular too). I think this particular problem is very interesting because there are so many ways to solve it and each of them is fascinating in its own way. I had so much fun in solving it on my own and sharing my solutions with you ❤️ oh and btw, this is my submission for #SoME2 Big thanks to: · my girlfriend for creating drawing character that guided you through the entire video, for creating the knight piece that was on the screen for probably 80% of the time and for helping me with animating stuff. Love u ❤️ here are her other art things if you want to check them out: · Grant Sanderson from 3blue1brown channel and James Schloss for organizing SoME and motivating me to creating this video · you for watching it and making me a happy person ❤️ References and sources: · Knight’s Tour: ’s_tour · Knight’s Infinite Tour: · Generalization of Infinite Knight’s Odyssey and it’s proof: · BlackPenRedPen’s video about Bézout’s identity: Music: · Unfoldment - Chris Zabriskie · Dances and Dames - Kevin MacLeod · Skipping in the No Standing Zone - Peter Gresser · Impact Prelude - Kevin MacLeod · Countdown to Myocardial Infarction - Peter Gresser · No Rocking in the Jazzhands Zone - Peter Gresser · Orbiting A Distant Planet - Quantum Jazz · Brain Power - Mela · Jingle Jazz - Quantum Jazz · AcidJazz - Kevin MacLeod Check out other places that you can go to and learn some math: · My Instagram page, where you can find lot of math content (also chess related): · Discord math community where you can talk to other math people from instagram and other social media, including me: Chapters: 00:00 - Introduction 01:35 - 1. Knight’s Tour 07:45 - 2. Flaws of the first solution 08:49 - 3. The elegant way 11:02 - 4. Generalization 12:24 - 5. Showdown! 15:44 - 6. The home straight Fun fact: the solution for generalized problem posed in the video, which is that a b must be odd and gcd(a,b)=1, can be written with just one equation: gcd(a b,a-b)=1 Corrections: 17:18 Clarification to the last part of the last proof: k((a,b) (b,-a)) l(2,0) is actually not equal to (1,0), but to (1,k(b-a)), which makes the proof incomplete. Indeed there is a much simpler way to find a (1,0) move: since a b is odd, a and b have different parity, so let’s assume without loss of generality that a is odd and b is even. Knight can do one (a,b) move, then it can do (b/2) times (0,-2) move to get to (a,0) and finally go floor(a/2) times (-2,0) move to get to (1,0), and we are done 😊 sorry for that one! Thanks to Kk Wai for pointing that one out!