Test Week

Hello again! This week definitely wasn't as busy as the last week, but it was still pretty fun. As for research, I've been debugging some of my programs (I'm not very good at programming) and reading literature on other variations of nim. One interesting concept I was looking at was the idea of a Grundy Value.

A Grundy Value is a number that a certain position of an impartial game can be written as. Grundy Values are usually looked at only in impartial games, meaning that it's only looked at in games like nim (since most boardgames aren't technically impartial).

Using binary, the Grundy Values allow us to associate each nim game to a number. This binary representation also allows us to determine 'safe values' in nim. These safe values would be points that a certain player would be guaranteed a win at. Looking at safe values can also check whether a board game is impartial or not. Having a certain set of safe values for a game could mean that the game is not impartial.

My problem is applying all these concepts to nim with a handicap. Once the handicap is added, the game is no longer impartial. Either some of the terms I was learning about don't apply at all anymore or I need to modify them to still apply with the handicap. I'm still pretty lost, but hopefully I can translate some of the theorems for impartial games to nim with a handicap.

In terms of my math class at ASU, things have been going well. I've caught up with the material for the most part. I've really enjoyed the lectures so far too! As for the test...

I took the first midterm for the differential geometry class at ASU. I thought I knew the material, but I turned out to be wrong. There were five problems on the test (and an extra credit problem). I solved three of them. Then I got stuck on the fourth. After wasting a lot of time on the problem, I realized I had another problem to do.  I moved on to the fifth, but I didn't know how to do that one either. The test was hard for me. It's a lot different in college apparently.

I'll study harder for the next one. The test was still fun despite me being bad at it. I don't have anything else to say...

Thanks for reading~

I have a test coming soon

So this week at ASU was also pretty interesting. In terms of my research project, I'm still writing programs that can calculate win probabilities of certain game scenarios in nim. I'm trying different types of recursive approaches in hopes of getting a better understanding of what actually is happening in the game. One approach I'm using is looking at the win probability of one player in terms of the other and vice versa. The other approach tries to eliminate the second player from the game entirely in the recursive function. Now, there really isn't a point in writing two programs that do the exact same thing, but I was hoping that looking at the problem from two different angles would help give me insight mathematically (since computer programs just spit out numbers and that's no fun). I'm still programming, but I'm gonna try to bring you guys some cool math for next week.

Also at ASU, my differential geometry class has been going well. I really like the class (and the lecturer) and have almost caught up. We had a quiz on definitions and basic problems today that went well. I forgot to name the book we're using so i'll do that now. It's called Differential Geometry on Curves and Surfaces, written by Do Carmo.

What's also cool is that we have our first midterm on Tuesday, meaning I gotta study ( a lot). This is going to be my first non-Ms. Bailey college level math test so I'm pretty excited. I also need to catch up on homework in the class, since I joined late. I might post a bit about some of the cool things I learn in class or about interesting problems I studied  during my time at ASU.

Lastly, ASU has really great math seminars. Once every two weeks, ASU hosts ASU math circle, where the math department gives a lecture on a cool new topic of research in mathematics in a manner that is understandable to high schoolers. Also, each friday, ASU math department also holds a geometry seminar. While these might be a bit harder than the math circle talks, these are really cool in that the professors at ASU present on topics that they are studying or hope to study soon. Going to these talks is a nice way to see how research at ASU math is going.

Funny thing is, this week, Noam Elkies came to give a talk at ASU. For those of you who don't know who he is, he's a pretty big mathematician. He's the youngest tenured professor at Harvard and just a straight up genius. I had been looking forward to his talk for weeks now, but I totally forgot about it this week. So yeah... that was sad

OK thats all
bye~

Starting my SRP

So this week was strange... I spent a lot of time working on my senior research project at ASU. I did a lot of cool things.

First, I wrote a couple of programs for finding win percentages of different variations of nim. The variation of nim that I considered first was similar to fibonacci nim. From the last post, we know that in nim, there is a set number of stones in the middle, with each player taking turns to remove a certain amount from the pile. The goal of each player is to take the last stone. However in fibonacci nim, after each turn, the other player is allowed to take up to a fixed multiple of the previous players move.  
For example, consider a game with a 100 stones in the middle. If a player took 5 stones in his previous turn, fibonacci nim would allow the next player to take up to 2*5 = 10 stones from the middle on his turn. This would mean that the players would not only have to consider the number of stones in the middle, but also the number of stones removed in the previous turn.

As for handicapping this game, the method I considered was changing the "multiplying factor" of each player. In fibonacci nim, we can say that this factor is 2 for both players, since each player is allowed to take up to twice as many stones as the previous player. However, one method of handicapping would be to limit the multiplying factor of the better player. This would limit the possible moves he could make, reducing his chances of winning. This would mean that the skilled player, who plays perfectly, may not be able to always defeat the unskilled player, who plays randomly.

The second cool thing I did at ASU was attend a talk (there are weekly geometry seminars on Friday) about subriemannian geometry. Now I wish I could say I understood the talk, but I most definitely didn't. I understood the definition of a subriemannian manifold and that was pretty much it. The talk seemed cool though. There were a lot of pretty pictures and stuff. Other people seemed to like it.

The last thing I did was sign up for Math 494 at ASU. So over the next quarter, I'm going to be attending a differential geometry class, taking the tests and doing homework with the students there. The class actually started four weeks ago, so I've got a bit of studying to do. But the class seems fun, so I'm gonna try to learn the stuff.

Ok that's all. Thanks for reading~