Nim is the only impartial game

I was joking with Eric Kim, a classmate of mine, this week that the game he was studying could be reduced to a certain position of nim. His work on Richmann games studies another very interesting unsolved game theory problem. A Richmann game is a combinatorial strategy game. Given a finite graph and a token placed on a starting vertex (point) of the graph, the game begins. Each player also starts with a fixed amount of money. Then on each turn, both players secretly write on a card how much of their money they are willing to give in order to move the token closer to their destination vertex. The person who bids more is then allowed to move the token one step closer to their destination vertex. However, the player who made the move must pay their opponent how much ever they had on their card. The game ends once one player gets the token to their destination point.

Lots of interesting theorems come out of this game. For example, there is a ratio, such that if the amount of money one player has divided by the amount of money the other player has is exceeding that value, victory will be guaranteed for one player. The game also allows for Richmann values to be designated to positions and Richmann costs be designated to moves.

Just within the setup of the game, I noted some similarities to nim. In both games, we are able to assign value to certain positions. Both games also follow the strict alternating turns pattern.

As I was noting the similarities, things started to add up for me. I saw (very intuitively and roughly and not very mathematically at all) how the games seemed similar. So I stared at my notebook for some time. However, something bothered me. A simple rule of Richmann games seemed out of place. The rule was that addressing the occurence of a tie. If two players both are willing to wager the same amount of money, the rules of Richmann games say to flip a coin, with the winner of the coin flip allowed to move the token towards their destination. Given that the probability that two players wager the same value is quite literally zero (since they can wager any non-negative real number, of which there are infinite), this shouldn't actually affect the game. This still bothered me, so I started to look into this a bit more (yayy for new problems). Thats where my week has been in terms of nim.

With Differential Geometry, I think I've caught up, but I wont really know till the midterm next week. Hopefully I'll be able to answer all the problems this time.

Thats all for now.

Thanks for reading~

DIfferential Geometry is hard for me

So Spring Break is finally over. My week away from academics actually proved to be quite relaxing. It allowed me to catch up with friends I hadn't talked to in a while, read books that I wanted to, sleep in (which was honestly the best part), and just relax after four grueling (but super fun) weeks of my senior research project.

Spring Break was also why I did not have a blog post up last week, since I was obviously taking time off. As for this week, a couple of things happened. The MathCounts team I had been coaching did amazingly at the state competition. We had the overall winner and a 3rd place contestant, both who will be moving on to an all expense paid trip to National MathCounts. I'm really excited to see them do well at the next level and even next year, when I'm not there.

And back to my senior research project again. Differential Geometry has been incredibly difficult for me. Not only does it require a solid understanding for both Linear Algebra and Multivariable Calculus, but it also requires rigorous proofs. A lot of differential geometry proofs have been simple to state, even in layman terms, but have incredibly technical proofs. My textbook, de Carmo's Differential Geometry on Curves and Surfaces, has started to get very hard for me, so I've looked at a few other books to learn from.

I might try to read Needham's Visual Differential Geometry and Beltrami's Hyperbolic Plane in order to catch up with what I've been struggling with.

The class has been very analysis-like, so I also thought about reading more complex and real analysis. A few textbooks I've wanted to read are Baby Rudin, Visual Complex Analysis, and Pugh's Real Mathematical Analysis.

I'll start reading some of these books and maybe introduce them and my opinions on them next week. Differential Geometry, despite being so very incredibly horribly terribly difficult has been super rewarding for me. Seeing a college level class six months before I head out has me aware of how difficult college will be.

My senior research project has been a mix of both nim research and differential geometry (more differential geometry recently), but I really have learned a lot from it. I've learned what undergrad research is like and what taking undergraduate classes is like. Getting the whole experience as a hopeful math major has been eye opening.

Thats all for now~
Bye

Mostly Differential Geometry

I'm still having problems programming, so the majority of the blog is going to be about differential geometry at ASU. We went over our midterm this week and continued with lectures. Though the classes have all been really interesting, the schedule of a college student seems pretty routine. Go to class, learn the material, reread the book to make sure you actually get the concepts from lecture, solve practice problems, and repeat.

This week was a lot harder for me than the previous weeks in terms of keeping up with the material. I've had multivariable calculus at school (yay BASIS/Ms. Bailey) and have learnt a bit more afterwards, but the class requires a fluency in it that I don't have. I have the same problem with linear algebra in the class. Luckily, after a good amount of reading and rereading, I've been able to get through the material.

We've been looking at diffeomorphisms this week and how to create maps between two surfaces. A big part of the course studies how surfaces can be smoothly transformed. This week, our lecturer, Dr. Kotschwar, was out of the state for a conference, so we had Dr. Julien Paupert fill us in. This week, we focused on  orientation of surfaces.

Orientation looks at how many 'sides' a surface has. With something like a hollow sphere, we could distinguish the sphere's inside and outside. But with surfaces like mobius strips (https://en.wikipedia.org/wiki/M%C3%B6bius_strip), we cannot distinguish 'sides' in the same way.

Also wikipedia isn't as bad as people think it is, especially in math.

For the most part, a lot of the theorems that we have proved have been intuitively 'obvious', but rather difficult to prove. Soon, we're going to get to the really cool stuff.

You guys are probably also wondering whats going on with my game theory research. I've still been programming/spending most of my time studying differential geometry, so I'll have cool results hopefully by the next post. I don't have much else to say about my project. Next time, I might try to keep my post more organized. I might try to keep myself limited to three things I've learnt at differential geometry and focus on my game theory project. I'll also share some of my (very ugly) recursive formulas I have.

I'll also post pictures (for real this time) since everyone else seems to be doing that. OK, thats all I got. Thanks for reading again.