Nimbers (Part 1) - Tipping the Balance

[@1ToTheInfinity]

Wow, great video, I haven’t seen a lot of videos covering nimbers so it’s great to see this explained so well as someone who doesn’t know much about game theory, also great visuals too!

[@mostly_mental]

Thanks! I'm glad you liked it.

[@Djake3tooth]

The other (nim) cube-roots of 2 are ω⊗2 and ω⊗3, because exponentiation distributes from the right on multiplication, just as we'd expect . And you gave the hint that 1, 2 and 3 are all cube-roots of 1 :)

[@mostly_mental]

That's exactly right. Well done!

[@lexinwonderland5741]

please keep up the good work! i love your videos!! they deserve to be on 3 blue 1 brown's contest series

[@vedantsingh5786]

can you provide the proof that the sprague grundy value of a position in the game of twins is the nim sum of the sizes of piles

[@mostly_mental]

Let's (for example) say we have a game of nim with piles 1, 3, and 4. Those have nim sum 6, which means there's a move which reaches a position with each value from 0 to 5. Now look at the corresponding game of twins, with red coins in spots 1, 3, and 4. Take the move corresponding to each of those nim moves, and you'll reach positions where the nim sums of the red coins are each value from 0 to 5. Also note there's no way to reach a position with nim sum 6, since the corresponding move in nim would also have Sprague-Grundy value 6, which isn't allowed. So we can always reach positions with each smaller nim value, but not the same value. And the end position (with all blue coins) has nim sum 0. That's exactly the relation we used to define our Sprague-Grundy values. So the Sprague-Grundy value of the twins game is the same as nim sum as the piles.

[@vedantsingh5786]

@@mostly_mental Thanks a lot! Can you provide me with some resources that can help me understand how to formally prove stuff like these (game theory, number theory stuff)? Love your content btw

[@mostly_mental]

@@vedantsingh5786 Glad you like it. If you're looking for a proper textbook, I first learned about the nimbers from "A Course in Game Theory" by Thomas Fergesen. I also highly recommend "Winning Ways for Your Mathematical Plays" by Berlekamp, Conway, and Guy. There are also plenty of good free resources online (I usually just search for "Combinatorial Game Theory PDF"). I don't know much number theory, so I can't help you with resources there.

[@bernatesteve1418]

Can anybody tell me the rules of the game rayles? (6:04)

[@mostly_mental]

On your turn, you draw a loop through one or two points. You can't reuse a point that's already part of a loop or cross over an existing loop. As always, last player to make a legal move wins. And in the other game, green hackenbush, you can cut any edge, removing that edge and all edges that are now disconnected from the ground.

[@jerryrowen1]

You talk too fast and your video is out of sync. Get it together please!😢 And;your voice is irritating !!!!!!!😢

[@mostly_mental]

Thanks for the feedback. There's a nice setting that lets you adjust the video playback speed, if you want to slow it down a bit. I've just rewatched this and it seems synchronized to me. Do you have a timestamp where it gets off? I'm afraid there's not much I can do about my voice. That's just how I speak.