Smyslov_Fan wrote on 05/03/12 at 23:32:51:


As every chess player should have a hobby - why not science which starts with mathematics?

I thought Zwischenzugzwang was joking. I certainly found it amusing. But since others are taking his formula seriously, maybe he wasn't joking.

I don't take my formula too serious, cf. the problems I mentioned four posts ago.

What you are talking about is the possibility that players know the correct mating procedure not "completely", but only to a certain percentage. Whereas your brother might reach 100%, you only reach 99%  :
Stefan Buecker wrote on 05/03/12 at 09:02:12:

Brown and Epishin are maybe 75% ...

My assumption of knowledge either being 0% or 100% also runs into another problem [Attention! Now some mathematics!]:

The "Logistic Function", which provides the aforementioned S-curve follows the formula y = 1 / (1 + e^z). If we take z = mx + b and do some calculation, we get ln ((1 - y) / y) = mx + b (with "ln" being the natural logarithm). With this formula it's easy to calculate a linear regression if the values for the x's and y's are known.
The problem with my assumption of y either equal 0 or equal 1 is that (1 - y) / y = ∞ for y = 0 and that ln ((1 - y) / y) = -∞ for y = 1, as the Logistic Function neither reaches 0 nor 1 (it just approximates those values). If I try to replace 0% by, say, 1%, and 100% by 99%, I can perfom my calculation, and get values for m and b. The same goes for 0.01% and 99.99%, and so on. Now the problem is that my results for the m's and b's don't converge the closer I get to 0% and 100% respectively. Just the shape of the resulting curve changes: Whereas it's quite smooth with a moderate gradient with, say, 10% and 90% resp., it becomes like a "sharp" step with 0.01% and 99.99%. 

So I still have to think about that ...

Stefan Buecker wrote on 05/03/12 at 09:02:12:

Taking these data into account would change virtually nothing in my results, as they predict the likelihood of a 2350 player to know KBN vs. K to be almost 100% (cf. x₁)

Here I'm quoting myself:
Quote:

I've got four answers yet, adding myself I have five data points. I set "knowledge" to either 0 or 1, and I count my own knowledge as "0".

From the data so far, I calculated the following linear trend:

y = 0,0015x - 2,53

where "x" is the rating (in DWZ) and "y"  is the likelihood of the particular player to know how to play KBN vs. K.
Two points of interest are: Where is the likelihood of knowledge equal 0 (x₀), and where is it equal 1 (x₁)?
From my data, I get:
-  x₀ = 1685 DWZ,
-  x₁ = 2352 DWZ.

Of course, there are heaps of flaws in this calculation: 5 data points are definitely not enough, a model with "knowledge is either 0 or 1" is too restrictive, and especially the assumption of a linear trend is highly implausible, not least because it gives impossible values for the ranges below 1685 (likelihood of knowledge < 0%) and above 2352 (likelihood of knowledge > 100%). A S-curve model is much more appropriate, but I have to find out how to calculate that (can somebody tell me how to do that in Excel?)

But despite all these flaws, I find the results given above (x₀, x₁) not too implausible. What do you think?

Best regards,

Zwischenzugzwang

TN wrote on 05/01/12 at 00:02:08:

A quite similar question: Did any of you ever have a position with K + N vs. K + P (rook's pawn) on the board, where the Knight's side can mate, because the Pawn's King is stuck between his own pawn and the edge of the board?
Nunn in his "Understanding Chess Endgames" (p. 49) gives as example the game Nogueiras-Gongora Reyes, Las Tunas Cuba ch 2001, where Black resigned in the following position:


It would follow 1.Nc1 a2 2.Nb3#.

This endgame is definitely not "exotic" material-wise, but rather "procedure-wise".

a great theme with the parity of colours, I remember learning it in a Paris chess club along with Marcello Mastroianni. OTB I had twice the much commoner theme of a bare King defending on the h-file against a h-pawn, Knight and distant King (the stronger side wins if he doesn't push on the seventh too soon). On the first time I waited, waited and finally the pawn was pushed one square too far : I had the draw but felt somewhat ashamed. On the second time, I resigned at once without allowing my opponent to blunder.