OMG I necro'd the blog...

It's been such a long time since I posted, and I have had a break from the school of school. For these two years anyway. As such, I have mostly stopped writing, and right now it feels a little unnatural to be typing over here, being inactive for the past 5 months or so. As such, my writing style would probably have changed, but it matters not, for it is the same person bringing you time-wasting paragraphs of contentless rants!

Nevertheless, I am sad to say that my brain has degenerated in fields of... everything ranging from sudoku to maths to even... *drumroll* chess! And as such I have actually not thought of material to post here. However, I hope that I will be able to think of more content soon, and resume the monotony of chess, chess, chess, maths and chess posts. In the meantime, a random link: http://www.kongregate.com/games/Tukkun/anti-idle-the-game?acomplete=anti+idle. Wonder where this goes...

The culmination of laborious scribing

Here is THE list of quotes from a totally unknown source. It's *chh!* OP! *ss...*



"I told my doctor i broke my leg in two places. He told me to quit going to those places."

"Who is John Galt?"

"It takes only one drink to get me drunk. The trouble is, I can't remember if it's the thirteenth or the fourteenth."

"My doctor gave me two weeks to live. I hope they're in August." -Ronnie Shakes

"The greatest pleasure in life is doing what people say you cannot do." - Walter Bagehot

"The secret to creativity is knowing how to hide your sources." -Einstein

"Experience is a good teacher, but she sends in terrific bills." - Minna Thomas Antrim

"Go, and never darken my towels again." - Groucho Max

"Sometimes I think the surest sign that intelligent life exists elsewhere in the universe is that none of it has tried to contact."

"A boy can learn a lot from a dog: obedience, loyalty, and the importance of turning around three times before lying down." -Rob

"Students achieving Oneness will move on to Twoness."

"It's fun to charter an accountant, and sail the wide accountancy..."

"No opera plot can be sensible, for people do not sing when they are feeling sensible."

"If you believe everything thing you read, better not read."

"It is impossible to enjoy idling thoroughly unless one has plenty of work to do."

"Under capitalism, man exploits man. Under communism, it's just the opposite."

"Those who believe in telekenetics, please raise my hand."

"Never be afraid to laugh at yourself, after all, you could be missing out on the joke of the century."

"Assuming either the Left Wing or the Right Wing took control of the country, it would probably fly around in circles."

"Alcohol may be man's worst enemy, but the bible says love your enemy." -- Frank Sinatra

"When I was kidnapped, my parents snapped into action. They rented out my room." -- Woody Allen

"Kleptomaniac: A person who helps himself because he can't help himself."

"A good lawyer knows the law. A great lawyer knows the judge."

"Ninety percent of the politicians give the other ten percent a bad reputation."

"Just because nobody complains doesn't mean that all parachutes are perfect."

"Hard work never killed anybody, but why take a chance?" -- Charlie McCarthy

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" -- Jerry Bona

" 'I am returning this otherwise good typing paper to you because someone has printed gibberish all over it and put your name at the top. ' -- An English professor, Ohio University, also probably the fate of most of my essays"

"No one can have a higher opinion of him than I have, and I think he's a dirty little beast."

"The trouble with having a open mind, of course, is that people will insist on coming along and trying to put things into it."

"Isn't it interesting that the same people who laugh at science fiction listen to weather forecasts and economists?"

"Mistakes are part of the dues one pays for a full life."

"After all is said and done, a lot more will be said than done."

"Does this rag smell like chlorofoam to you?"

"Never try to tell everything you know. It may take too short a time."

"Brain: an apparatus with which we think we think"

"Income tax returns are the most imaginative fiction being written today." -- Herman Wouk

"We've heard that a million monkeys at a million keyboards could produce the complete works of Shakespeare; now, thanks to the Internet, we know that is not true." -- Robert Wilensky

"The direct use of force is such a poor solution to any problem, generally employed only by small children and large nations."


Cheers!

P.S. "dammit... foiled"

Paradox regarding infinities?

Mm… today I read about Cantor’s diagonalisation argument, arguing that there exists infinities which are bigger than others. At first, this sounds obvious, but when considering that for an infinity (let’s call it X), X+1=X, this suddenly seems a lot less obvious.

It would seem of utmost importance to give a general idea of the definition of equal sizes of 2 sets. For finite sets, this is already obvious by, in short, counting. The size of {1,2,3} is 3, which is less than the size of {-1,-2,-3,-4}, but for infinities, this is again less obvious (bah… infinities just have to be this hateful).

However, we note that it is possible to compare sizes of finite sets in a different way. Let’s just say that we wish to compare the sizes of the two abovementioned sets. We attempt to achieve a bijection between members of the 2 sets, say 1->-1, 2->-2, 3->-3, and lo and behold, we are stuck, for there is no other element in {1,2,3} left to biject to -4. So how do we know that it is impossible to choose a different mapping from the first set to the next that produces a bijection? In this case, it is simple, and trivial by Pigeonhole Principle (finite sets ftw!). However, as we shall see the in the case infinite sets, this is no longer as straightforward nor convenient (again, infinities are irritating; bzz…).

Firstly, let me brief explain why X+1=X. Suppose X is the cardinality (size) of the set of natural numbers, because other infinite sets are probably just analogous, but infinitely (pun!) more irritating to write an explanation for. So, a bijection is given as follows: let the new element representing the +1 be 0 (yes, 0 isn’t a natural number to me, now stop that argument already). A bijection (from set of natural numbers to the set also with 0) would be 1->0, 2->1, 3->2, …, n->n-1,… where n is, well…, any natural number.

Taking this one step further, now let us “prove” that X+X=X*2=X. Again, suppose that X is the cardinality of the set of natural numbers. Let the new X elements be a1, a2, a3,…. Now, a bijection (from the set of size X to the set of size 2X) is given by 1->1, 2->a1, 3->2, 4->a2, and so on, with n->(n+1)/2 if n is odd, and n->a(n/2) if n is even, for all natural numbers. (Ok, I admit set notation looks cooler, but I’m typing.)

As an attempted Mathematician, let me enthusiastically take this just one step further (the hour’s getting late!). The next thing in the series is to prove that X^2=X. This is cooler.

All right. Time out. I just thought of something to whine about infinities in general. The main problem with infinities is that unlike finities (is there such a word? O.o), inductive approaches do not work. For example:

1=1+0+0+0+0… = 0+1+0+0+0…=…=0+0+0+0+…=0 (?).

Clearly, there is a step in the middle where the induction just fails. Hence, infinities tend to refute everyday logic of things like… … … … … … … … … an irritating thing.

Back to proving X^2=X: now let us consider the set of positive rational numbers, given in (sometimes improper) fractions. A rational number is one expressible in terms of a/b, where a and b are integers. Hence, a positive rational number can be expressed as a/b, where a and b are rational numbers. Yay! Now we shall arrange the rational numbers in a square with a top left hand corner and the other corners at infinity. Being a nitpickish person, I shall arrange it as such:

1/1       1/2       1/3       1/4       …
2/1       ----       2/3       ----       …

3/1       3/2       ----       3/4       …

4/1       ----       4/3       ----       …

.           .           .           .

.           .           .           .

.           .           .           .

  Now, imagine a grasshopper from Bremen, Germany starting from the top left hand corner of the square. It hops in this order:

1          2          9          10        …

4          3          8          11        …

5          6          7          12        …

16        15        14        13        …

.           .           .           .

.           .           .           .

.           .           .           .

  Now we formulate our bijection in the order of the leaps our dear grasshopper makes:

1->1/1, 2->1/2, 3->2/1*, 4->3/1, 5->3/2, and so on. Now, every rational number has a corresponding natural and vice versa. Therefore, by our definition, X=X^2. Wonderful.

*since 2/2 has been nitpickily replaced by a dash for being the same as 1/1

  Readers are like, totally encouraged to attempt to prove that X=^XX (see tetration), and owing to the awesomeness of Conway, a->n->3, a->n->4 (see Conway’s chain notation). They, by the unjustified basis of scientific induction, should hold true as well.

Honestly, the original purpose of this post was to raise a contradiction I convinced myself of, but that has since been resolved. Nevertheless, it would probably be interesting to figure out where the mistake lies. Let the hunt for the fatal flaw begin.

Cantor’s diagonalisation argument (omg finally on to the point of the post) can roughly be stated as follows:

The power set of a set S refers to the set of all subsets of S (or so I believe). So, illustrating with a finite example, the power set of {1,2} contains the null set {}, {1}, {2}, and finally {1,2}, the set itself.

The argument concludes that the power set of any infinite set is larger than the set itself.

For the sake of convenience and easy reading, we shall assume that the infinite set involved is the set of natural numbers.

Firstly, a bijection between the power set and a real number (haven’t really thought much about this yet, might be wrong) between 0(inclusive) and 1(exclusive) can be established as follows:

Let such an arbitrary real number be denoted as 0.abcdefghijklmnopqrstuvwxyzα…, written in binary. Now let a subset of the natural numbers S map, in the bijection to the following real number: 0.abc… where a=0 if 1 is not an element of S and 1 otherwise, b=0 if 2 is not an element of S and 1 otherwise, and so on. Therefore, both the set of real numbers from 0 to 1 (yes, I might regret saying this) and the power set of the set of natural numbers are of the same cardinality.

Now let us prove that the cardinalities of the power set of the set of natural integers exceeds that of the set of natural numbers. We approach this via a proof by contradiction.

Suppose otherwise; i.e. there exists a bijection between the set of natural numbers and the power set of the set of natural numbers. Then there exists a bijection between the set of real numbers between 0 and 1 and the set of natural numbers. Let the bijection (from natural numbers to the power set) be as follows:

1->0.…

2->0.…

3->0.…

and so on, where (naturally) represents either 0 or 1.

  However, we observe that there is a real number between 0 and 1 that is definitely not equal to one of the real numbers already bijected to. A possible real number can be expressed as: 0.b1b2b3b4b5b6b7…, where b1 is NOT() (binary operators!), b2 is NOT(), b3 is NOT() and so on. Hence, the bijection is not a bijection, i.e. there exists a contradiction, meaning our original assumption of a bijection between the set of natural numbers and its power set is impossible.

Intuitively, a corollary is that 2^X>X, since the cardinality of the power set is larger than the cardinality of a set itself.

However, if we consider the powers of 2, we notice that there are an infinite number of perfect powers of 2, and for convenience, we shall call the infinity Y. Now, 2^Y and Y are not equal, meaning that this set is smaller than 2^Y, by the above argument. However, if we map n->2^n, we notice that the number of powers of 2 and the number of integers are in fact equal since the bijection holds. Therefore, 2^Y indeed equals the number of natural numbers. However, if we consider the first k powers of 2, there are 2^k natural numbers smaller than or equal to the largest power of 2 mentioned. By extension, the number of natural numbers is roughly equal to 2^Y (arguably with a factor of 2). But that gives 2^Y>=X/2=X=Y. Clearly, something is wrong somewhere. Can you find the fatal error?

My solution is given in (hopefully) the same colour as the background below. Highlight to review.

Whoever said that the power set of an arbitrary infinity X had cardinality 2^X? The combinatorial argument for this does not appear to hold for infinities as it is inductively based. See what I mean about infinities being irritating and counterintuitive?

YAY LAME JOKE

Q: Why don't students brush their teeth before big exams?
A: Because they want to cheat during exams via yellowtooth.

Random Cchess match

Truly random.

1. C2.5 c8.5 2. H2+3 h8+7 3. R1+1 r9.8
4. R1.6 a4+5 5. R6+7 h2+3 6. R6.7 c2+2
7. P7+1 r8+8 8. A6+5 a5+4 9. H8+7 a6+5
10. H7+6 c2.8 11. R9.8 c8-3 12. C8+7 a5-4
13. H6+4 h7-5 14. R7+1 h5-3 15. C5+4 a+-5
16. H4+5 e7+5 0-1

RJT chess -- General Ideas 2 (Attack series)

Here is a nice picture I made using screenshot and Qianhong (Go try it.), showing a position (well... yea) with red to move.



Red appears to be winning. However, if for example, R2=5, K5=6, and there is (for now) no mate! And in this game, that's a scary thought. It's tiring to think of a mate...
Instead, there exists a (hopefully I'm correct...) miraculous move in this situation! Can you find it? (Yes you can...)

C1-1!

This move prevents the shi from being any useful in the defence of the king. Amazing move!... right?

Let's assume black totally does not see the danger of this, and plays a random move!

... P5+1

Suddenly...
R2=5+
...K5=6 (K5=4 is just suicidal)
R5-1+
...K6-1 (forced)
C1=4, planning R5+1 the next move. Good game; well played!

Edit: Swapping the first and second moves totally appears to work better. Awww...

RJT chess -- General Ideas (Attack series)

Notice that I did not call it strategies, as the ideas I shall mention are really quite vague, but nevertheless somewhat (surprisingly) applicable to a real match of RJT chess.

This post focuses on the attacking madness bit of RJT chess, and (hopefully) another post will eventually be made (might not be by me) on the defensive bits of RJT chess and its strategies.

Big strategies
--------------
1) Getting in sufficient firepower:

Firstly, it is practically impossible to kill with just a single piece, no matter how menacing it may look, under general conditions, as any piece may interpose and hence stop its rampage. *sad face*

Hence, an important attacking idea is to ensure that at least a few pieces (depending on how good your tactical skills are as well as how blind you think your opponent is) can get in to a threatening distance of your opponent's king. (usually as long as it restricts the movement of some of your opponent's defenders; for example, a cannon (pao), as denoted conveniently by C, may pin 2 pieces to your opponent's king. Similarly, A single well placed piece may disrupt the movement of advisors (shi, denoted by A) in the palace.)

Hence an important concern is how to get sufficient firepower next to your opponent's king.

2) Make your opponent commit:

With an action as simple as stuffing a chariot (R) in front of your opponent's king, you typically would have made him commit as to which side he is keeping open and which side is more closed to attacks. Hence, it is more convenient to marshall your forces as you know where they would want to go.

3) Remember your small (but not unimportant at all) BINGS (P):

When your attack looks like it's stalling, bring in the P if there are still open lines and 'ma's (H) if there are no open lines. Since you have 5*P, that can really tie down a good portion of your enemy's pieces especially if he does not defend strategically.

4) Crowd a place when all else fails:

If your normal attacks are not going through, it is possible (and has worked before with great success) to crowd out your last checks with pieces (power of the bing!) and hence win the game by claiming that you are left with no checks at all.

Small strategies
----------------
1) Attempt to get your C directly into the line of sight of your opponent's K with absolutely nothing in between. Since nothing may be stuffed in between, a final strategic single check may prove to be absolutely fatal. Plus this really messes up your opponent's defensive formation. Also, it sounds like a good idea to move your C rather far back to create a zone where your opponent absolutely cannot place his pieces. Happiness ensues.

2) Liberate your horses! They are typically the pieces which finally gets in the important check, as it is rather simple to block off checks from all other pieces with just... well 6 defenders (2 on each side) of the king.

That's all the attacking ideas I can think of for now, and I will (may) return to give more details on each of the strategies at a later date when I finally manage to conjure up semi-professional-sounding ways to explain the how and why of each strategy. That or when I get bored again.