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Archive - Nov 2008

Date

Useless information for AIME

Posted November 28th, 2008 by Mgccl

AIME
Math

For AIME, there are always questions ask m+n or mn where m and n and relatively prime integers. It's easy to infer the only way to m or n = 0 is the other one is 1. The answer of AIME are always integers from 0 to 999.
Suppose m and n are distributed evenly as long as m+n<1000 and mn<1000. How is the distribution of m+n and mn? Because if m and n can't be found easily, an educated guess can increase the chance.

Case 1: For the m+n case, the amount of (m,n) pairs for x are $\varphi (x)$
Case 2: For the mn case, the amount of (m,n) pairs for x are $2^{p(x)}$
Where $\varphi (x)$ is the Euler's totient function, p(x) is the number of unique prime factors for x, I can't find any named function with that definition.

So for m+n one, it's better to chose a larger number, even a prime number.
For mn case, it's better to chose numbers with a lot of unique prime factors.

Case 3: It happens sometimes when m and n doesn't have any other requirement except m+n<1000. Then we have a linear relationship. The amount of times m+n = x is x-1.

Case 4: Sometimes there are Squarefree requirement for n, then it is still approximately linear.

A nice plot. Green = Case 4, Brown = Case 3, Purple = Case 2, Blue = Case 1 * 50
Then one just have to have a random number generator, and then convert it into a pick in the plot.

Taking a break from study--Cave Story

Posted November 28th, 2008 by Mgccl

Cave story is a casual game.
It must have been played by millions by now.
Eye candies:

This post contains SPOILERS. the entire post has SPOILERS! Skip it if you want to try yourself. Or if you are like me who doesn't care about spoilers, read on.

I have played all three endings and feel my soul have been upgraded.

The story is extremely simple, but the provoke ideas requires lot of thinking.

The fate of irony isn't really hard to grasp. During the game, there are a few moment worth remembering. The death of Toroko and King. King who tried to defend his village ultimately failed. Quote had to kill Toroko even that who he tried to save. Irony of fate.

Are there anything important about memory should be learned from this work?

Memory is history in personal level. When someone else knows the history, they can exploit it.

Doctor convinced some Mimigas to work so they can fend of something in the future that could be the recurrence of the past, but they are actually leading themselves to their doom.

The main protagonist Quote is a robot who lost it's memory and got entangled to stop the Doctor, and never regained his memory back.

After Curly Brace recovered from her memory loss, Cthulthu said he liked her without the memory better. Strange. Does this means anything?

Why are Curly and Quote, two robots seems so human, they need air, they can eat and they also have emotions. Are their emotions created by having memories?

There are times where people scream when their hard disk fail and all their data are lost. They often say all those important memories are gone with it.
So humans have that many important memories they want to remember, but their brain is not capable to store that much. Is it true that some memories are made to forget?

Humans are just shells, like Curly and Quote. Memory is what really defines a person.

If one can change his own memory, would he? As some memory can make him the person he wants to be, but also means he ceased to exist to himself.

Why is this game called cave story when the island is actually floating in air?
Because Quote started in a cave, and end up in caves until he finally found out it is not a cave during middle of the game. That's my opinion...
The ending is quite a happy one.

Small summation problem

Posted November 20th, 2008 by Mgccl

Math

$\sum _{i=1}^{n-2}\sum _{j=1}^{n-1-i} (n-i-j) = 2\sum _{i=a}^{b}\sum _{j=c}^{d} (n-i-j)$
n is even.
Find a,b,c,d, where a,b,c,d are all linear combination of n/2, i and 1.

I stumbled though this problem because I was trying to solve a even harder one
$2\sum _{i=a}^{b}\sum _{j=c}^{d} (n-i-j)f(i,j) = \sum _{i=1}^{n-2}\sum _{j=1}^{ n-1-i} (n-i-j)|f(i,j)|$ where f(i,j) = -f(j,i)

I don't know the difficulty of this question, but I don't have any technique for solving problem like this. So I did exactly what the question asked. Decompose the variables into linear combinations of n/2, i and 1 and brute force all the solutions.
$2\sum _{i=k_1\frac{n}{2}+c_1}^{k_2\frac{n}{2}+c_2}\sum _{j=k_3\frac{n}{2}+m_1i+ c_3}^{k_4\frac{n}{2}+m_2i+c_4} (n-i-j) = \sum _{i=1}^{n-2}\sum _{j=1}^{ n-1-i} (n-i-j)$

There are a lot of solutions. I put an restriction on all the coefficients. But I found a particular nice one:
$2\sum _{i=1}^{\frac{n}{2}}\sum _{j=1}^{\frac{n}{2}-i} (n-i-j) = \sum _{i=1}^{n-2}\sum _{j=1}^{ n-1-i} (n-i-j)$

And... I don't like my current MimeTex system. The output equations looks ugly.
I need to find other servers. Currently MathBin is nice and fast, but it is created to be a pastebin. Ahh. I want a delicate server. Or leech from CodeCogs LaTeX Equation editor. Wordpress one is good. but blocked in China.

Mathematica 7 released

Posted November 19th, 2008 by Mgccl

As a Mathematica user(MUHAHAHA, that's what's so good about having a parent work at a research institution. MUHAHAHAHA!!!), the release of Mathematica 7 is an exciting news.
Mathematica has becoming Photoshop for math. In the future I will edit photos by typing equations instead of mouse gestures. Love this thing.
Can't wait to get my hands on it...
While it is good. I can't help to notice Mathematica 7 is way bloated and a released too soon. Consider Mathematica 6 was released only a year ago. Does anyone really need speech output system for Mathematica? lol.

I can get my own copy if I gold USAMTS. The last problem for this round is quite hard consider I use the naive and obvious sum up all area and divide method. Still searching for nice identities.

Fuck intel. No more.

Posted November 16th, 2008 by Mgccl

Intel
Rant

Since my Intel has already been sent, and just today I realized what I forgot to do.
I forgot to create a cover page for my research paper.
Big screw up for me.
But, Screw Intel.
This is how Intel get started

"hey. we want to make a contest judge people's research paper. What do you think? So we can judge people by how they do in their research. In their real scientific talent."
"How about instead of just research paper, we want them to make the paper looking good with cover pages + and a 20 page form with more information than college application + 5 essays about themselves which we would not even care to read + some recommendations. So we can judge people in every way instead of what the contest is really about, finding a person who does science and does it good."
"Brilliant!"

Fuck it.