HACKENBUSH: a window to a new world of math 

NOTES & DISCLAIMERS: This video uses some slightly different terminology than is established in Winning Ways. This is a conscious decision made in service of a smoother educational experience, with the understanding that adjusting to the book's terminology will be easy after absorbing the video. These terminological changes are as follows: 1. Winning Ways mostly uses "game" to mean "position in a game". However, it does frequently use "position" instead, and also sometimes uses "game" in the more common sense (what else would you call Hackenbush, generally, if not a game?) Solidifying the distinction between these two terms, and referring to positions exclusively as "positions", seemed like a good choice for this video. 2. Winning Ways calls the two players in a combinatorial game "Left" and "Right", rather than "Blue" and "Red", respectively. The book introduces the vertical bar notation pretty early on, so it makes sense to name the players based on whose options are on which side of the bar. It's also a lot more necessary when dealing with games that don't involve blue and red things. However, since this video is focused on Hackenbush, using "Blue" and "Red" saves time and a lot of mental overhead. 3. Winning Ways refers to Hackenbush positions as being comprised of "edges", rather than "branches", and refers to "erasing" them rather than "chopping" them down. My decision to use the latter terminology here was just a stylistic choice, motivated by the fact that in an animated environment, we're no longer limited to pencil-and-paper modes of thinking. 4. Winning Ways uses "number" only to mean "surreal number". Things like ﹡ and ↑ are not called "numbers", but either "games" or "game values". It seems like existing on a number line, being directly comparable to numbers, and having defined behaviour for interacting with numbers via addition, subtraction, multiplication, etc. is not enough to be considered a number. Nevertheless, I think using the word "number" for these things is a lot less confusing, and helps a lot more to drive home the point of the video. The very few times that I need to refer to surreal numbers specifically, for example in defining what a switch is, it's only a small sacrifice to add the word "surreal". ﹡ and ↑ aren't numbers in the "traditional" sense, but neither is √ω. Tradition is what we make it. 5. Winning Ways prescribes the pronunciation "slash" for the vertical bar, rather than "bar". It only mentions this once, as the book, being written with text, never needs to actually pronounce the symbol. Saying "slash" every time a vertical bar appears on screen would have been too jarring for me, and it would have taken a few more words to introduce that pronunciation, so I went with what seemed more natural. 6. Winning Ways does not use "epsilon" to refer to the reciprocal of ω. However, this is a somewhat established usage of "epsilon" - the Wikipedia article on surreal numbers uses it, for example - and I talk about tiny numbers enough that it's useful not to have to say "one over omega" all the time. In addition, the introduction of dotted lines representing "fragile" branches is nonstandard. Unfortunately, switches are not possible in standard Hackenbush, so to get around this, Winning Ways introduces a separate game called "Childish Hackenbush", with the rule that you're not allowed to leave any branches floating in the air. I thought it was simpler and more elegant to add to the existing game, rather than changing it. This is also one reason why I chose not to make the ground a dotted line, like it is in the book. Smoothness of animation and the ability to transform the ground into a number line were other reasons. Finally, the book uses shorthand like 1﹡ for 1 + ﹡, and 1↑ for 1 + ↑. They explain this as being analogous to how we write 1½ for 1 + ½. This notation is convenient given the ubiquity of such values in the book, but I didn't feel the need to introduce it for this video. 1 + ﹡ is, thankfully, still a valid, and much more immediately intuitive, way to write 1﹡. BACKSTORY: Although this video was submitted to the first Summer of Math Exposition contest, it was not originally created for that purpose. It began as a short Twitter thread on June 8th, where I pompously declared that, although I had never made a video with anything except Screenflick and iMovie, I would someday make "the definitive mathtube video on Hackenbush". Following that, I started the first draft of the script on July 10th, finished it on the 13th, and began making the animations on the 14th. It just so happened that the contest was announced a couple days later, and it gave me the perfect deadline to strive for. From July 10th to August 22nd, I spent a good portion of my waking hours every day working on this video. Making the animations was about half of that time, totaling over 100 hours. The rest was writing the script, recording the voiceover, adjusting timings, producing the soundtrack, and applying layers of general polish. It was a massive chore, and the only reason I'd ever do something like it again is if there were another topic that excited me just as much and that didn't already have a dedicated video on it. Which seems unlikely, but who knows :)

"A branch is a branch, you can't say it's a half" got a solid laugh out of me, wonderful

Came back to this mathematical masterpiece for a second viewing only to discover that Owen has passed away. While I didn’t know him, this news is still a punch to the gut. I love maths and this video is so inspiring, shedding light on some greatly underrated work in the field. I don’t know what Owen was going through, but it makes me think back to my own struggles with mental health and suicidal ideation, some of which continues to this day. I’m glad Owen was able to leave a positive impact on so many people, including with this video. I like to think that Owen was able to inspire and create future mathematicians with this video, as he has definitely inspired me to keep going further. I hope his family is doing okay and handling the loss well. Owen, you are an inspiration to many, through your music and this beautiful video, it truly is one of my favorites.

The learning curve of this video: - Water is wet - 1+1=2 - Compute the meaning of life - Draw the thermograph of Oof

This video has really touched me, one of if not the best math video I've seen. Rest in peace Owen, you've left behind a true masterpiece.

Amazing video! The perfect mixture of approachability and depth. I know they're a lot of effort to make, but I'll definitely be watching if you make more :)

Rest in peace, Owen Maitzen. ♥

Rest in piece, Owen. I never knew you, but this video put real delight into my life for whatever that's worth. Your appreciation for beauty and whimsy in equal measure was a bright light to me, and you helped me understand the love I hold deeply for math. Thank you, and godspeed

"Throw away Cantor and submit yourself to Conway" sounds like a calling of a religious cult, but those are usually less crazy than this

YES. THIS IS THE ESSENCE OF MATHEMATICS. I WANT TO BUY THIS BOOK NOW.

The “Down the Rabbit Hole” section reminds me of the descent into insanity at the end of Tantacrul’s video where he reviews Sibelius. Or his review about Dorico.

Owen I hope you lived a life you enjoyed and if you were alive I would have hoped from the bottom of my heart that you create a life for yourself that you love. I saw this video 24 hours ago and ever since I have had the urge to quit my corporate job and work in academia. I have not been able to shake this feeling as I had anticipated. You have at the very least changed one solitary life; mine.

This video brought back some happy memories! When I was in Grade 9, 1978-79, I was in a math club lead by Richard Guy. (It was for participants in a city-wide math contest, which I had won.) Attendance dwindled to just him and myself, and then he began to introduce me to what he was working on with John H. Conway: they were writing Winning Ways. So I learned about Hackenbush and other games, the { | } notation, defining integers as games, * and ↑ and so on. He also mentioned the "Bűvös kocka" of Ernő Rubik. I bought a copy of the 2-volume first edition of Winning Ways when it was published. Prof. Guy also taught a calculus course I was in, in 1982.

I've watched this video three times and will almost certainly watch it several more. I only just noticed "on plus on - holy hell" this watch through.

I love the Mario 64 music slowly creep in when he mentions the “1/2 A press” of the red branch.

I think Grant needs to have some kind of "Honorable Mention" list for things that were inspired by his SoME1 challenge, but did not get included for whatever reason including missing the deadline, but was made known (if not actually released or draft-released) in time for his wrap-up and postings about it.

man, coming back to this for a fourth watch or something, i did not know owen had passed away. thank you from the bottom of my heart for this little gem i adore and all the other things i never knew you for. rest in peace.

I just saw his obituary. Rest in peace.

It’s such a sad shock to hear of Owen’s death. Owen, you and I never knew each other, but you touched me, in your own small way. I just so happened to come across this video, and was blown away by the math, yes, which is amazing, but also the MUSIC! I’m also a composer myself, and after finding your music channel I listened for quite a while to your improvisations, which I found totally enrapturing and inspiring. You seem like you were similar to me in a lot of ways, and definitely the kind of person I would’ve liked to have known. A lot of people miss you. Rest In Peace.

Brilliant reference at 6:23 -- not only are you referring to the "0.5 A-presses" joke, but the notes playing at this part of the video ACTUALLY play the Title Select Theme from Mario 64. Bravo, holy cow.

pure delight at the reveal that "over" stunningly is less all the positive surreals, as crazy as they are, but then manages to be *greater* than the already introduced up series... what a wonderful video, with wonderful pacing and storytelling. thank you for making it.

i have to admit that 53:54 actulally got me laughing at a maths video just because of how absurd and mind bending it had gotten at that point

This is a really good video!! It's filling the Vihart-shaped hole in my heart -- perhaps not stylistically, but in the sense of exploration and mathematical play. I've always been somewhat confused by the surreal numbers, but in their original context of game theory they seem not just useful, but completely natural -- of course they're defined in terms of what they're less than and greater than, because in games that's often all you can tell. It gives a lot of motivation to their definition, which in turn really helps me understand their behavior. Also, the music is really great!! It reminds me a lot of Untitled Goose Game, not just stylistically, but in how it seamlessly ebbs and flows to perfectly match the video. Really good job with that, it's so good

Quality standards for maths content just got surreal. From sound to animation, the presentation is as polished and sharp as one would expect from but the best maths RU-vidrs out there; the flow of topics and pace of explanations are just right; the subject itself is immensely enthralling and your exposition more than makes it justice. This essay leaves any mathematician in an excited position for more, I will probably rewatch this ω times until someone (hopefully you) explores the colon principle in a follow-up.

You already know this, but this is beautiful. Thank you so much. This video makes me want to be better at pursuing the things that fascinate me and presenting them to the world. And to be a better pianist. Even if you never produce anything like this again, this will remain one of my favourite channels.

Maybe it's just because It's 2am and I've had a long day, but finding out Owen is dead filled me with so much sadness and anger that I broke down crying. I just finished watching this whole video straight for the third time, I was in adoration of your work, thinking that it may well be my favourite video on the Internet and now I find out you killed yourself? This is so unfair, my friend. You were even in Earth Strike like me. I can't even express how devastated I am. Rest in Peace you beautiful soul.

This is exactly the intersection of math, philosophy, art, game theory, and dumb pannenkoek references that I needed. Even if you never make another video, know that this one was absolutely enough. More is welcome though ;^)

When I got to 44:00 and heard "You can dive headfirst into the world of on, off, dud, over, under, ..., upon, hot, ono, ...", I couldn't help but think... "oh *no* indeed" -- But yeah, this is incredible! I feel like every 5 minutes I've only been playing in the kiddie pool of numbers compared to the universe coming up.

"hours of chill piano improv to relax slash study to" lol the deadpan humor never stops and-hold on, that was all original music?! You pulled a sneaky on us; this video was actually about adding songs to my study playlist with Vihart's Pi Day improv! I like how it's a similar vibe, calmly philosophizing about math memes... Speaking of, 2:03 isn't the Loss meme, but now we can rigorously define it :P

This video is incredibly well-executed from start to finish. And for a first(?) video essay, too! Really stupendous work. If you do choose to make more of these, I'm sure you'll excel. And if you don't, well, you can at least rest assured you've made an all-time classic. P.S. If you haven't seen them, I suspect anyone who enjoyed this video would enjoy "30 Weird Chess Algorithms" and "There Are 48 Regular Polyhedra." Same vibe, same vein of hyper-niche but obscenely funny math lectures. Edit: Just learned of Owen's passing. It is a tragedy on many levels. Take care of yourself, whoever you are that's reading this.

Thank you for this video! When I was in highschool we were tasked to write a short (10 to 15 pages) "paper" to prepare us for university. I chose surrreal numbers as my topic and spent almost a year with Conways work through On Numbers and Games and lectures of him I found here on youtube. I wasn't in a good state of mind at the time and didn't put in as much effort as I wanted. But still during this time I gained a deep respect for Conway and he became somewhat of a mentor figure for me and someone I dreamed of meeting in person one day, knowing full well it was quite unrealistic. So when I heard he died from covid last year it felt like a personal loss, as I lost this little dream of meeting him one day. All of this just to say thank you, once again, for this video, as it is a wonderful homage on the work of these men and some of the most beautiful mathematics I ever encountered.

guy

i love the pannenkoek reference with "a branch is a branch, you can't say it's only a half"

Amazing video! Thank you for sharing. RIP sir.

Absolutely stunning video! Definitely confused with any best RU-vid videos one could imagine

This video finally explained to me what Conway was saying in one of his lectures when he said that using his surreal notation you can create numbers that are neither positive nor negative but also not a zero.

I am positively blown away by this video. Never have I felt so close to the brilliant mind of Conway...not even when he has explained it! Bravo^omega!

i took a combinatorial games course last year where we explored our way through the surreals and this is an absolutely incredible video, captures a lot of the coolest stuff in the basics of combinatorial game theory. genuinely one of the best math videos ive seen in a long while

This video is amazing. The wonder, the confusion, the discovery, the humor, it's all perfect. Your passion for maths is contagious, hopefully thousands of people will see this.

By far the best SoME video IMO.

Ok people are finally seeing this, excellent. One of the truly essential videos on this site imo.

At 6:22 the piano starts playing super mario 64 music as you make the 1/2 a press joke, you are amazing

Possibly the best video on RU-vid

Thank you for the legacy left with this presentation masterpiece, RIP. I'm a mathematician at heart and would have never learned about the work behind Winning Ways if not for this video. I will surely buy the book now and can only hope one day I can pass along the curiosity to my family. Glad to see views racking up recognition, crazy that Owen is gone just 4 months after this release.

0:32 I mean illustrating Guy with some stock photo Guy was hilarious , and yeah 54:28 is the actual Guy. But awesome video. Even if this will be the only video like it, I am still subscribing : )

This is a very valuable video, thank you very much

40:21 THIS, the ?, was the first time I saw a non-well-founded set in the "wild" when I first watched this video, and this make me curious about those objects, which are actually "forbidden" in ZF set theory (due to Regularity axiom). Respect and RIP.

So sad to scroll down to the comments and learn about his death. This video is a great legacy.

This… is probably my favorite video I’ve seen on RU-vid. I’ve been trying for a couple months to find someone who wanted to play around with surreal numbers with me. You just showed me how much deeper this world goes, with an absolutely beautiful piece of work. Thank you for making this. I loved your music as well, I will go check out your other channel now. Thank you thank you 🙏 ✨❤️

This video deserves the world.

Rest in peace Owen. This was a truly brilliant work, and we're all sad to have lost you.

This is the video I'm going to share when I try to explain what mathematics really are. Thanks for making this video, for surreal

This completely blows my mind. And when they sneaked thermodynamics into it I almost fainted. I mean, what the actual...

Great vid. If this is your only math vid, you're certainly going out on a high note

We have a tradition in my family of watching documentaries together, i watched this twice and was so enamour i convinced my family to see it with me, thank you for a great memory and a new wish Owen, and rest well knowing you showed at least 200 thousand people the beauty of math and little games to overthink.

Definitely one of the best recreational math videos I’ve seen on this platform. Incredible stuff!

I finally watched this, after it being on my list for over half a year. It was an incredible experience, and it reaffirmed my love for math once again (which has been dwindling lately from college burnout). I share your deep appreciation of Conway, and his death in 2020 really saddened me. However, I hadn't found the time to read any of his books, other than cursory glances. I should start fixing that. Thank you for making this. Edit: I just found out that the person who made this video died three months ago. Rest in peace, Owen. I didn't know you, but I hope you knew you were making the world a better place. You won't be forgotten.

This video is a masterpiece in every way.

Wonderful, wonderful, wonderful video! I can't imagine a better introduction to all these concepts. Thank you very much, Owen, and rest in peace.

The more i watch the layers of greatness this video builds and finally in the end it turns out its just an unpaid advertisement for some obscure book, that better believe that I am putting it on my wishlist. Wow, really enjoyed!

This is super inspiring to see a RU-vidr make a very high quality math video of their chosen topic without being an established mathtuber. Great work and I want to do the same.

I solved the addition table thing for nimbers in my head using Cayley tables from abstract algebra. I never thought pure math could be this useful but here we are

The best vídeo ever made about it!!! And its your first vídeo! This is surreal.

This might be my favourite math video ever. I've watched it 3 times now and I've just decided to order the book. I'm extremely thankful to you for doing this, you legend. I don't know you personally but I really hope you're having a good day when you read this❤

okay i have a question. so, when figuring out the bracket notation for 1/4, we discard the 1 in {0 | 1/2, 1} because red would never want to choose 1, making it extraneous. why don't we apply the same principle when figuring out the notation for *2? going for * instead of 0 (i.e. chopping the top branch) would be the wrong choice for both players, so why don't we discard the * in {0, * | 0, *} and simplify to just {0 | 0}? my guess is that discarding extraneous numbers is less a hard and fast rule and more something we can decide to do when it's useful..? considering that in any *n position it's always best to just chop the bottom branch, if we discarded all the other possible continuations we would end up representing every *n as just {0 | 0}, and that just wouldn't make for a very good system. that makes sense to me, but i'm wondering if there's a better explanation i'm missing?

This was incredible all the way through, from the animations, to the explanations, to the background music that somehow always fit the topic. Awesome!!

Imma be honest. This might be my new favorite math video on RU-vid. I love how playful everything is.

A fascinating topic, packed into a sleek and approachable presentation that's easily on par with the likes of 3blue1brown and Mathologer. Thank you!

Incredible video! This deserves at least 10^7 views.

this is oh so insanely good, probably the best advanced math educational video ever produced. Please do more

My sincerest thank you! These is likely the most beautiful video I've ever seen.

Bravo! One of the best math/algebra videos I've ever watched!

This video was simply mind-blowing, and by far my favorite video from the Summer of Math Exposition contest! Not to dis any of the other entries, but I can't fathom how this wasn't selected as the overall winner. Congrats on a stupefying feat of math exposition - you captured precisely what it is that makes mathematics so wonderfully strange and beautiful. I am forwarding it to everyone I know!

To Owen Maitzen: thank you for the video. Many years ago I was reading about Conway s ideas that connect games with sets of ( original) numbers. The idea to start with beautiful games and to connect this to some new set of numbers!!!!!!! It is so beautiful ❤️❤️❤️❤️❤️❤️❤️❤️❤️❤️

8:57 epic bongcloud opening

16:43 That transition from stars to dots is SICK holy shit

Thank you, Owen, for this video, which is not only about Hackenbush and combinatorial games and a book review but a love and passion for math.

NIM: player must remove at least one or any number of objects provided they all come from the same stack. PRIM: players must be co-prime to stack size DIM: players must evenly divide stack size

In the last few days I have Discovered the world of Conway’s Surreal Numbers. Seeing how much is appears in so many things keeps adding to its majesty. I am still taken a back by the aesthetic beauty of the Tree Diagram and Surreal Forms!

this is a work of art

Ohmygoodness someone recommended this video to me and I'm so happy I watched it. I haven't laughed so much about math in so long, whilst also learning so much. This is so awesome and I would love it if you made more videos like this. I am immediately subscribing and I'll recommend this video to anyone I think would like it. Thank you so much.

You have literally launched me into a stratosphere of math. Thank you so very much!

wow, almost deleted this video from watch later, it would have been a mistake, awesome video

This is one of my favorite videos I've ever seen

I keep coming back to this video, I just love it so much. It's incredibly effective at helping the viewer understand whats going on and it preserves the playful tone of the original books. Fantastic work!

Thank you for such a brilliant tribute to Dr. Conway!

Fantastic. Jist.......fantastic!

"on plus on - holy hell" I almost didn't catch this and now I'm dying of laughter edit: I was lucky enough to see a Conway lecture about 15 years ago, when I was at a math camp in middle school. I wish I could remember everything he showed us but I know we learned about omega and the Doomsday algorithm, and it's one of the coolest afternoons I've ever had. Def gonna go buy this book now

this is an amazing video and a fantastic intro to game theory, combinatorics AND surreal numbers, all in one! thank you so much!

Quicksand is the perfect font for this style, you really nailed your visual style 100%! This is pure art, mate. Congratulations!

I don't know how I didn't see this earlier, got to be one of the craziest math videos I've seen!

I've watched this video about six times. It delights me.

it's unbelievable that this is your first video, it's just so good, if you make more of them I'll certainly watch

I predict this to be a winner of the 3B1B SOME1 competition.

god this video is so good. i rewatched it, like, five times already and i will continue to.

What an incredible video. I'm just lost for words. Every math teacher should make sure their students see this video. How could you not fall in love with numbers after seeing this.

Hi Owen: Thank you so much for making this video. I started reading Winning Ways myself after watching your video, and I'm absolutely fascinated with everything about it. Thank you so much for introducing me to CGT and surreal numbers.

words cannot describe the feeling you gave me with this video , Owen. I am a silly little year one CompSci student and Ive never been this excited to learn something. Thank you for this beautiful piece of art that is this video. I want to get this book as a christmas present.

This is a beautiful summary of one of my favorite series of math books, really absolutely gorgeous. Great visuals, wonderful narration --- thanks for sharing this piece of art. That this currently only has 26k views just shows how bad a gauge the view count is as a measure of quality, and it makes me wonder what other gems I might have missed because of RU-vid's algorithm.

Man I gotta make a comment on this video I first heard about combinatorial game theory on Discord, with a message that mentioned the confusing terminology introduced by the original mathematicians that's kinda just stuck Then today (which, for the record, is 2023-12-22, or 2023 December 22nd in a more readable format that I totally didn't just make up), I was watching RU-vid, and somehow I ended up in the niche of long math-y video essays, and so eventually I ended up here. I watched the entire thing, and I found it funny, informative, and just plain wild at times - particularly the comment about the "wordbuilder's almanac", which I took note of and found very funny indeed because of my interest in conlanging (which may or may not be diminishing in favour of Scrabble nowadays? I honestly couldn't say.). Anyway, after watching it, I read the comments and discovered of Owen's passing. This struck me in a way that would be impossible to replicate - in that the thrills that I experienced watching the video were turned somber. But my point is that this was an amazing video - I loved it, and I'm absolutely certain that many others have done and will do as well. If you're the sort of person that reads comments before watching videos, and you're reading this, then please watch this video. I beg you. It is possible that you will be delighted and horrified in many different ways that I'm not exactly sure how to put into words (because I'm no good with words, so please be forgiving), but I promise that you *will* enjoy it. :) Thanks for watching my Ted talk ig - now go watch this video! ;)