Hi, Antonio. (I presume we are on a first name basis now :) > Why not? :)

<First, "scatalogical": > This is probably the first time I don't agree with you. See http://www.merriam-webster.com/dict....

<Henceforth, I will assume that <every> English word is in your active vocabulary ;>) > Hmm... you should, although I cannot guarantee that I'll understand <every> nuance...

<Second, "lunch time": I have a friend with whom I have had lunch weekly for 23 years. In some periods, I regarded myself fortunate when not subjected to "scatalogical" conversations before I finish the soup. (His conversation has its compensations.) > This reminds me of my brothers...

<(I have Latin but not Greek in my education.)> My greek ranges between zero and nothing but my insane curiosity likes etymology.

<I spent time on both sides of the Atlantic, leaving me uncertain about my spelling sometimes.> I have a similar problem: most of what I read was written by Americans but I rarely speak to them, mainly people from the UK.

<I happened to check the incorrect spelling on the web and found it in several "reasonably good" sources. I shall disbelieve them in future.> I often resort to Merriam-Webster when in doubt because they include the word's etymology.

<Just to follow up: I checked Liddell and Scott's Greek lexicon and found among the dung "scatophageo" (in obvious transliteration, to save me the trouble of going to UTF-8).>

scato: tumblebug business, phagos: one that eats. Peculiar choice of word with your "Enjoy your lunch" below...

<The biggest guns of all agree with you. No surprise there.> Thank you!

I played a lot of dominoes with both grandfathers (my mother's father was a very strong player) but I suspect I was already very compulsive-obsessive-inquisitive-... (How many '-ive's should decorate my personality to let my children live well by selling my memories?)

<johnlspouge: ... < <agb2002> wrote: <Terry McCracken: This is basic technique, hardly insane. > I agree. >

Maybe, but then I am unclear on the basic technique. Perhaps you gentlemen could formulate my general rule for me.>

Please see my week summary on my forum.

<johnlspouge: < <CHESSTTCAMPS> wrote: [snip] John, thanks for the catch! > No problem, Phil. Let me know anytime you want me to catch another bullet with my chest ;>) >

The puzzle position looks very simple but it shows some middle game characteristics:

1. There are several reasonable candidate moves (Kd7, g6, h5).

2. For any of them White has several non forced answers.

3. The combinatorics explode after a few plys.

Therefore, if the direct use of tactical methods is problematic then one should consider devising a plan (fix the opponent's pawns, attack them with the king, exploit the far passed pawn, etc.).

That's why I started my post outlining that plan and without any mention to material or threats, as I routinely do.

I have mixed feelings, mainly on the question of "training for what?">

The solution of a puzzle requires more or less effort which results in some training of something (for example, my resilience after having posted a horrible blunder...).

<On one hand, I am studying chess partly because I am interested in analyzing and improving my general problem-solving techniques by monitoring their performance objectively, with the relatively closed universe of chess problems.>

So we have another interest in common...

<(With the exception of our esteemed <DarthStapler>, by the way, it never ceases to amaze me how many people believe they "get it", when they do not bother to write down detailed solutions in advance.>

As you can imagine, I only pay attention to "statement & proof" posts.

<It's much easier to be humble, once you have made an arse of yourself in print.)>

Much easier and, frankly, much more realistic.

<To monitor my processes, I require "the" solution for comparison, which the computers and kibitzers generally supply for the midgame positions.>

I only require a suitable solution, although I'm interested in the best solution, of course. Lasker became a world class player with a few second and third best moves.

<Closure for endgame puzzles is harder to achieve.>

I like endgame's peculiarities: strategic aspects, very reduced material, difficulties to use pattern matching methods to assist calculation, etc. A curiosity: http://en.wikipedia.org/wiki/Chess_....

<On the other hand, it is my nature to welcome fresh challenges.>

Yet another coincidence.

<I am still digesting the novel environment of the K+P Sunday puzzle and learning from it, quite obsessively (one of your "-ive"s :) >

I enjoy a lot to review the (end)games of the usual suspects: Lasker, Rubinstein, Capablanca, Smyslov and Fischer.

<The novelty of the experience weighs strongly in its favor, as usual. For "balance", I can tolerate a few more endgames ;>) >

Endgames remind me of Occam's razor, a very helpful logic resource.

<Keep well, Antonio.>

<Strange.>>

Yes, indeed.

When I told my godfather and the father of a friend of mine that I wanted to study a math course at university they utterly disapproved my intentions and, to my surprise, replied in practically the same way (they didn't know each other): "You study what? This is what you will do: take a piece of paper and make a listing of the richest women around."

I chose the hard way.

<I definitely would have eaten better with other life choices (although fortune-telling was not among them).>

Who knows... I used 2002-7 iPod sales data at the beginning of 2008 to work out two estimates of 2008 sales (49.814 million units with a gamma model and 53.708 with a logistic model, the curves looked nice on my data analysis leaflet). According to Apple reports, they sold 53.828 million units. But that's not exactly fortune-telling.

<You could say that I became a mathematician, precisely so I could starve ;>) >

click for larger view

John Nunn gives this in his puzzle book (puzzle 54) and says that if Black “could reach the diagram position with White to move, then he would win easily”. He goes on to say (after some lengthy discussion) that “The above reasoning is absolutely characteristic of this type of pawn ending and it is well worth becoming familiar with this technique.”

Anyone you know <agb2002>? :) >

I spoke to my brother-in-law's father a few weeks ago and he told me that he never knew of a chessplayer called Frank Parr. Perhaps, a bit distant relative.

It's my turn now to beg pardon but I think there are at least two essential differences between the selected positions from A Sokolov vs Korchnoi, 1987 and Seirawan vs Kasparov, 1983:

1. The position in the latter game is much more static (black pawns only have one move).

2. The opposition plays a much more significant role in this game.

write <all> your wishes down in a wish list and give it to your Daddy. Give me a short notice if he hadn't fulfill them all until the end of the year.

Answer: No, the only function with the property described is identically 0.

Proof. For any 0 < y < x < 1, set a = b = y / x. Then, algebra yields

y f(y) = x f(x) / 2.

For any 0 < y < 1, pick a fixed x in (y, 1). For any natural number n, pick a sequence y < x [1] < ... < x [n - 1] < x [n] = x. Then,

y f(y) = x f(x) / pow (2, n).

Take limits as n tends to infinity, to show y f(y) = 0, so f(y) = 0 for all 0 < y < 1.

Some kibitz :) >

<johnlspouge: I missed the condition that x is not restricted to (0,1), but the proof obviously applies to all real numbers. >

Hi, John. What I find really interesting about this problem is the solving process (more interesting than the problem itself!). This is how I went.

Firstly, I assumed analytic functions and, after equating the corresponding expansions, found that all the coefficients must be zero.

Then, I chose a = b = 1/2 and noticed that f took the same value for any given x times any power of two.

Finally, I got x*f(x) = 2*y*f(y) for any 0 .LT. y .LT. x (or x .LT. y .LT. 0) and a = b = y/x. Thus, for 0 .LT. y .LT. z .LT. x we would have

x*f(x) = 2*y*f(y)
x*f(x) = 2*z*f(z)
z*f(z) = 2*y*f(y)

hence

z*f(z) = 2*y*f(y) = x*f(x) = 2*z*f(z)

that is, f(z) = 0 for any z <> 0. Now, one defines f(0) capriciously and gets some relative of Dirac's delta (just to be perverse...).

I vaguely remember a Gauss' quote about removing the scaffolding once the building is complete. This makes sense when one publishes results because probably nobody cares about identical function values for x*2^n with arbitrary n but the proper construction of an adequate scaffolding is so important (no matter it's about maths, chess, engineering or buildings) that I suspect that this approach has caused far more harm than the infamous, obnoxious expression "tedious maths" repeated ad nauseam. Is chess tedious because one has to calculate?

By the way, did you reach any conclusions about the endgame A Sokolov vs Korchnoi, 1987 ?

Take care,
Antonio

<I suppose we all know mathematical quality when we see - another topic :) >
Although I like minimal solutions (the difficult simplicity) I actually prefer constructive solutions. It's not only my nature, but also my profession because it is usually much easier to derive/produce software from them.

<Interestingly, I also started with the scaffolding a = b = 1/2. >
Surprise, surprise... :)

<Yes, Gauss was a master at hiding the origins of his solutions (notably, his systematic usage of complex numbers, which is a rather useful scaffold to keep hidden). >
Probably, knowledge was the only asset of most old mathematicians, so it is understandable that they wanted to keep their methods secret.

In my case, the only secrets I keep are my customer's trade secrets. I don't mind to make public my methods. See for example a quick and dirty solution to a problem (perhaps inadvertently) posed by Phil Van Dusen: chessttcamps chessforum. I'll return to it when I find the time.

<In the interests of clarity, I prefer to expose my scaffolding, if I can. >
Good point: if one can. I often notice that my own explanations about how I found a solution are not completely convincing, not long after writing them.

<As I have matured, however, I have noticed that certain bright authors look absolutely brilliant, mostly because they write obscurely. >
If I wanted to look absolutely brilliant I would have perfectioned my dribbling skills.

Now, if I want to look absolutely brilliant I only need to get a wig, drop some petrol on it, set it on fire and cover my bald head with it (according to my six year old daughter, I have NOT matured).

Interestingly, I have observed that writing obscurely is a good social deterrent...

<I should have known better :) >
Again, you're not alone...

<johnlspouge: < <agb2002> wrote: By the way, did you reach any conclusions about the endgame A Sokolov vs Korchnoi, 1987 ? > No. I am weaker on endgames than midgames, because the practice here (which is my main connection to chess) emphasizes midgames. >
From your comments, I have the impression that you rely almost exclusively on pattern recognition (just a constructive criticism!).

<I am still constructing the scaffolding for my thought on endgames (with some success), but my learning process uses a top-down approach, rather than an immersion in calculation and details. >
According to Andrew Soltis (The Inner Game of Chess), Capablanca, one of the most accurate players in history, was a tireless calculator. You solved the functional equation by an immersion in calculation and details. Assuming that problem solving processes in maths and chess are essentially the same, aren't you contradicting yourself?

<I am therefore more interested in general principles than specific conclusions. >
I lost my faith in general principles when I realised that I was able to contrive a number of counterexamples for any of them. My main interest is to find a reasonable move algorithm.

In the AVRO 1938 book tournament, Botvinnik mentioned Capablanca's move algorithm but didn't give any details. I suspect that he didn't proceed with candidate moves systematically but with some kind of position scan to discover what was going on so that the move would logically/naturally follow. I might be utterly wrong, but this is the approach I'm currently experimenting with CG's puzzles.

The solution through the Cauchy functional equation imposes extra constraints, but suggests how someone generated the problem.> >

Hi, John. I was assuming f(x) = a*f(a*x) + b*f(b*x) for all a, b in ]0, 1[ and all x <> 0, as stated in Erdos.

To derive the Cauchy equation you take a = x/(x + y) and b = y/(x + y) so that a + b = 1 and f(x) = c, for any c. Therefore, you can find particular solutions to particular conditions on a and b.

I found this when I tried the series expansions: if you assume f(x) = SUM(i=0,inf; ci*(x - x0)^i) then for any a, b in ]0, 1[ such that a^(k+1) + b^(k+1) = 1 we have f(x) = ck*x^k. In the case of the Cauchy equation k = 0.

for every x,y > z > 0 and or y < z,x < 0.

Thus, z' f(z') = z f(z) whenever z and z' have the same sign. The constant c = z f(z) satisfies c = 2c, so f(z) = 0 for every z != 0. >

In other words, for any nonzero z, z' with the same sign it is possible to choose nonzero x, y with the same sign as z such that

|z|, |z'| < |x + y|

and

x f(x) + z f(z) = (x + y) f(x + y) = x f(x) + z' f(z')

which implies f(z) = c, c + c = c and finally f(z) = 0.

My motto is from "Galaxy Quest": "Never give up, never surrender." >
I prefer the classics, like Pink Floyd's "Hey You" :)

You are correct, of course. I just love being wrong - ask my wife ;>) >

My wife enjoys my mistakes. Love is so strange...

<By the way, as Cyrus the Virus said to Garland Green in "Con Air" ...>

Hola, <agb2002>.>

Hola John.

<Toga evaluates 25.Rxg6 as a forced mate; 25.Qd4, at a mere +10 P or more, along the variations people indicate.>

I considered 25.Qd4 because I often notice, to my deepest astonishment and even deeper anguish, that the first move I find is NOT the best move... But seriously, Lasker's suggestion (you know, find a better move, etc.) makes even more sense when solving puzzles than when playing a game with time controls.

<It's nice to see <The Bish> back in form today, after taking a sting from the "<bellringer>" yesterday. Until yesterday, I cannot recall <any> faulty analysis from him. Maybe he was just trying to get his kids out the door too :) >

Hi, <agb2002>. The sound you hear from my post is the crisp crack of self-flagellation.>