Setting a chess puzzle

I was delighted this week, when I thought I managed to construct a delectable Chinese chess puzzle, pretty much by accident and chance. It contained a somewhat less commonly used theme as well as an accidentally set sacrifice. Happily, I chugged it into the engine for sidelines, and maddeningly, about 3 strange defences materialised out of thin air, defences I had never considered. In the end, I hereby resign to my sad puzzle-setting fate and present you the somewhat correct (but not a mate) puzzle.

Uhh... how do I write the plaintext for it again? Meh.

###AK#PR#
####A##H#
EP#######
##P#P##H#
P####P##H
#########
P#####PPR
####E####
######RP#
R####K###


Red to move and win, of course.


More on how my puzzle failed in a later blogpost, if I remember and can be bothered to make the effort. It's late and I'm braindead too. But in the meantime, enjoy the puzzle.


Just in case I remembered my own notation wrongly, the one presented is as follows:
P -- the most important, powerful, and prevalent piece (Darwin's theory!) on the board.
E -- elephant, minister, whatever. The mascot of Chinese Chess (Xiang Qi)
A -- The advisor, or "shi".
R -- Rook, or "ju", but probably not "che"
K -- The lousiest piece on the board -- never have I lost a game not because of it
H -- Knight (or was it N?). Either way, the "ma".
C -- Not featured in this puzzle, the sneaky cannon, or "pao".

A badly played Chinese Chess match

Clearly I haven't been playing Chinese Chess seriously in quite some time; I'm blundering pieces left, right, and center, for the most trivial of reasons. But the thing that takes the cake (it's a lie group!) is that I've not been sacrificing for initiative as much as I probably should. That will take some time to reset. And being cooped up in a place without access to the Internet (well, not much of it) for five days every week (all right, maybe six), I don't think I'll have time for quality chess any time soon. However, I did manage to find an average-ish player on a particular chess website today, and lo and behold! -- here is the chess match, with a slight bit of commentary on why I moved certain moves and my failings.

(As a side excuse, it was a 5 0 game.)
Black: Me            Red : ???

 1. C2.5 c8.5   2. H2+3 h8+7   3. R1.2 r9+1
 4. C8.7 h2+1   5. H8+9 r1.2

At this point I was waiting for the response R9.8 to play C2+4. No I don't remember the theory, but that kind of moves look theory-ish. Also, we probably had deviated from main theory lines by then, so...

 6. P9+1 r9.4
 7. A4+5 c2.3  

 At this point I rather liked my setup. Sure I wasn't exactly winning, but it looked like I might have an attack along the right side (in my POV).

 8. R2+4 p5+1

The start of my troubles.

 9. P5+1

Doh!

 9 ... a6+5

 ?? move. Now I'm in a little bit of trouble. Sigh. Since the formatting looks a bit weird, I guess I'll just take this space to comment on my next few moves. His pao, sitting on the middle, is strong and attacking my king. Even though my cannon is pretty much doing the same thing, I have insufficient coordination, and hence I thought it might be safer to just trade every thing off the board... and my cannon could easily be replaced anyway. Following that, I just threatened to kill his xiang as best as I could, and I would even have happily sacrificed my ma for it, but... he just didn't accept. Aw... so much for being an aggressive maniac.

10. P5+1 c5+5  11. E3+5 c3.5  12. R2+2 r4+7
13. E5-3 r2+4  14. C7.5 c5+5  15. E3+5 r2.5
16. R2.3 h7+5

Yet another occasion that I played a dubious move. Now my xiang dies. And before his. Better was probably R5+3 R3+1 R5=7 R3+2+ A5-6, and now I'm threatening a stupid one move mate, and his shuai must move out, whereupon I can check him and put his king in an extremely dubious place vulnerable to harassment by the side, which is rather sneaky and difficult to defend against. My usual defence against this kind of things is just hoping my opponent doesn't see. Either that or trade off the 'ju's off the board and run away in fear.


17. R3+3 a5-6  18. H3-4 h5+ 19. P3+1 h7+5


Just moving my ma along with the flow. I wanted to go there anyway. Yet the bing move was somewhat necessary to prevent a faster check (which could be fatal since his shuai cannot move).

20. R9.8 h5+7  21. H9+8 h7+9

And I thought I was being obvious. Not obvious enough, apparently :(.

22. H4+3 h9+7

Oops, unstoppable mate out of nowhere in particular. And from this moment on I was just being an irritating troll warlord. But he resigned :(. Aww.

23. K5.4 r5.6  24. A5+4 r4+1
25. K4+1 r6+3  26. K4.5 r4.5  27. K5.6 r6+1

0-1

In conclusion, I was still down by a bing at the end. This reflects a deterioration of my skills. *sigh*

Till next time then.

Vocabulary (from a different person)

Thought I might as well post something after yet another long period of inactivity. Definitions come from google. Words come from... hmm?
Extolling: Praise enthusiastically
Ember: A small piece of burning or glowing coal or wood in a dying fire
Penumbra: The partially shaded outer region of the shadow cast by an opaque object
Inimitable: So good or unusual as to be impossible to copy; unique
Forevermore: Forever (o.O)
Foreboding: Fearful apprehension; a feeling that something bad will happen
Threescore: Sixty
Perchance: By some chance, perhaps
Afore: Before
Hereinbefore: Before this point in this document
Repose: Temporary rest from activity, excitement, or exertion, esp. sleep or the rest given by sleep
Thence: As a consequence
Pallid: Pale, typically because of poor health
Dissonance: A tension or clash resulting from the combination of two disharmonious or unsuitable elements
Intoning: Say or recite with little rise and fall of the pitch of the voice
Forlorn: Pitifully sad and abandoned or lonely
Corvus: A small Southern constellation (the Crow or Raven), south of Virgo
Heretofore: Before now
Maven: An expert or connoisseur
Dirge: A mournful song, piece of music or poem
Wherefore: As a result of which
Entreated: Ask earnestly or anxiously for
Nefarious: Wicked or criminal
Delectable: Extremely beautiful
Emancipate: Set free
Spurned: Reject with disdain
Mirth: Amusement

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.