19 =/= 17

I won't act like I'm not ashamed to be the only person on this blog to have posted not on the 17th in the last 5 posts... but nonetheless: this class. Fun stuff. It is both fortunateand unfortunate that I have already been introduced to most of the proofs we have done so far, fortunate because it gives me a better understanding of what's going on in the class and unfortunate because it feels in a sense 'unfair' when I contribute to the class. Which is something I dislike, because I often end up feeling that way in math classes that I take.
I don't doubt that this class will become my most challenging high school math class thus far (maybe ever?) and for that I am excited.
A note: Cory, that proof was sweet. I was all but convinced that you could populate the board with a wide array of numbers but, well, you can't argue with mathematical proof.
<(^.^<) (>^.^)>
...aaand I'm out

Thoughts and Definitions.

I really like this class because of the proofs that we are doing. Beautiful proofs. Delicious proofs. Rigorous proofs. The fact that we are working with so low-level concepts and are rediriving a lot of what is considered trivial makes me jump with joy because now I can actually trust it. Also, the course basically explains why stuff works, something that I find is completely neglected in pre-college math from my experience. Yay!

Of course, I also have several things that I am dissatisfied with, namely the usage of things we have not defined. What is a set? I badly need a definition. Also, we haven't defined the symbolic notation we use to represent integers in terms of digits. I have tried devising my own definition but it is really tricky and have failed so far.

I'm also looking forward to analysis as from what I've heard it is really interesting and it seems as a very fundamental part of all research mathematics.

Half-YSP, Half-Karafiol...

...is basically how I'm feeling the class to be like. YSP in the sense that we're doing YSP-esque things, but Karafiol in the sense that we do things your way (we prove things, no lecturing). I really like number theory, because it makes me feel like I really know what I'm doing in mathematics, so this section is really rewarding to me. I think analysis will also be really interesting as I really don't know that much about it, and it just sounds perplexing. At this point, I'm wondering a couple things. The first of those is what in the heck is a chaos fractal (I guess I'll learn from the lecture)? The second is are we going to be doing much set theory? Not hardcore (or not so hardcore) Galois Theory like some of us did over the summer, but just basic stuff like monoids, groups, rings, etc.? Back to the subject of how class is going now, I think I'm having a lot of fun. I like thinking critically about these proofs, and I want to refine my technique for proving things. For me, the toughest steps are the first and the last, so getting those down is a goal for me. I think there's a lot of good to come from KAM as we go and I hope everyone is excited as I am.

mostly what nausicaa said: when other people prove stuff, i get it, but they use techniques that i never would have thought of. everyone says that writing proofs just takes a lot of practice, so i'm hoping that it will get easier. on friday during english i had an epiphany and i figured out how to prove one of the problems, for the first time in my life, and i was SO excited.
i love love love kam, because it's so great to be in a class where everyone really cares about the subject, but i feel like i have to struggle to understand the things that other people are saying. i find kam--actually, more like the brains of the other people in kam--very intimidating, but in this case the intimidation is enjoyable.

also i adored one two three infinity and i hope we discuss it at some point.

Fun With 0.3

(centered so the 1's work out. Meh.)

We sort of skipped over problem 0.3 and I thought given our use of the well-ordering principle it would be fun to go over this one. 0.3 goes like so:

On an infinite chess board each square is labeled with a positive integer and each such label is the average of the four labels above, below, left, & right of it. One square is labeled 17. What is the sum of the square two above and two to the right of it?

Way 1: Well, a board of all 17s works and the problem DOES say what "IS the sum" not "what ARE the sumS." So 34.

Way 2: OK, Way 1 is evil, let's be fair. We want to know if any other combination of numbers is possible (other than a board filled with all 17s). Consider any board where the conditions are met. Now, by the well-ordering principle, pick the smallest # out, let's call it x.

To make everything simpler, let's subtract (x - 1) from all the squares. This still leads to a valid board as all the averages are shifted down (x - 1) too. (e.g. the average of (4 - 2), (5 - 2), (6 - 2), and (5 - 2) is two less than the average of 4,5,6, and 5) Now our smallest number is 1.

So 1 is somewhere on the board:

1

But what numbers can surround 1? Well, no numbers below one (the board is all positive; 1 is the smallest #). So it must be numbers >= 1. But if we have a number LARGER than 1 surrounding 1, we must have a number SMALLER than 1 by it. Oh wait, we just said we couldn't have a number smaller than 1. So 1 must be surrounded by all 1s:

1
111
1

But those 1s must be surrounded by 1s...

1
111
11111
111
1

... etc. So our board becomes all 1's, or, translating back to our original board, all x's (the smallest # on the board). But since there's a 17 all the numbers must be 17.

Hope that was educational, see you all tomorrow.

English Language Is Simply Insufficient

...To describe my frustration, adoration, and utter consternation when tackling proof this year. I've had this expectation over the years in other math classes that everything would come to me with great ease. There had existed this "buffer zone" of information on the subject I had obtained from other sources mathematically that could carry me through the majority of a class so that really, I would only be learning a fraction of new material over two semesters. My previous mathematical knowledge here in KAM ran out somewhere around day two, when we surpassed the general knowledge of proof and number theory summers at YSP afforded me.

But I am not nearly as "paralyzed by fear" as my opening statements lead you to believe: it's hard, sure, but never have I experienced such a community of the mathematically enthused that don't seem to mind that I can't get everything right. On a more personal level, it's showing me that it's totally OK to be wrong about math. And dang, it's just a fun class.

blog trouble

so I got the invite and followed it here, but I haven't actually been asked whether or not to join so I'm unsure as to whether I'm actually a member of this blog or not. I also got a message a few times that says there was an error with the invitation. anybody else having trouble?

-LT