Knight's Infinite Odyssey

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: instagram.com/kreskazcieniem
· 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: https://en.wikipedia.org/wiki/Knight%...
· Knight's Infinite Tour: https://mathoverflow.net/questions/29...
· Generalization of Infinite Knight's Odyssey and it's proof: https://math.stackexchange.com/questi...
· BlackPenRedPen's video about Bézout's identity:    • Bézout's identity: ax+by=gcd(a,b)  

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): instagram.com/mathinity
· Discord math community where you can talk to other math people from instagram and other social media, including me:   / discord  

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!