13 January 2009

Undefended Pawns in Chess960 Start Positions

Returning to A Followup, an Error, and an Insight, the 'insight' was a possible relationship in SP024 between Black's difficulties (a 76% success rate for White) and the characteristics of the start position ('the e-Pawn is undefended, while the g-Pawn is protected only by the King' where 'both Pawns can be attacked in two moves by the Queen and the d-Knight'). This started me thinking about another question: 'Is there a general relationship between the characteristics of chess960 start positions (SPs) and the resulting tactics?' Considering the weakness of f2/f7 revealed by the Fools Mate and the Scholars Mate in traditional chess (SP518), the obvious answer is 'Yes'. How to verify this?

I decided to investigate undefended Pawns across the 960 different SPs. I went back to my Database of Chess960 Start Positions and developed a method of counting defended Pawns for each SP. For example, in any start position a Rook always defends exactly one Pawn, the Pawn in front of it. A Bishop always defends two Pawns, except when it starts in a corner and defends one Pawn. Combining these basic observations allows a count of how many times each Pawn is defended in a specific SP.

Taking SP518 (RNBQKBNR), the eight Pawns are defended in a pattern that looks like 1-1-1-4-4-1-1-1; the a/b/c and f/g/h Pawns are all defended exactly once, while the d/e Pawns are each defended four times. This symmetric result for SP518 is partly because the King and Queen protect the same number of Pawns from a particular start square.

It turns out that there is only one other SP that has the same start profile as SP518, its 'twin' SP534 (RNBKQBNR), where the King and Queen swap places on the central squares. I've introduced the term 'twin' to denote two SPs which are the mirror image of each other; the pieces in one twin (running from the a- to the h-file) are in the same sequence as the other twin (running from the h- to the a-file). It follows that the profile of defended Pawns for twins also presents a mirror image.

One property of twins is that they present exactly the same opportunities for the initial development of the pieces. Only when castling becomes possible do they start to take on individual characteristics.

Another classification of the various SPs is by the total number of times that Pawns are protected by pieces. The Pawns in both SP518 and SP534 are protected a total of 14 times by pieces (1+1+1+4+4+1+1+1 = 14). As shown in the following table, there are 442 other SPs where the Pawns are protected 14 times:

12 -     8
13 - 148
14 - 444
15 - 316
16 -   44

SP295 (QNBRKRNB) and its twin SP370 (BNRKRBNQ) are two examples of the eight SPs where the Pawns are protected 12 times. SP404 (RBBQNNKR) and its twin SP750 (RKNNQBBR) are two examples of the 44 SPs where the Pawns are protected 16 times.

My database tells me that there are 360 SPs where all eight Pawns are defended at the outset. That leaves 600 positions where at least one Pawn is undefended. The 600 SPs are summarized in the following table, which shows the number of SPs having one, two, and three undefended Pawns.

1 - 452
2 - 144
3 -     4

Three undefended Pawns(!); how is this possible? Both SP059 (NNRQBKRB) and SP075 (NNRKBQRB) have the eight Pawns defended according to the same profile -- 0-0-3-3-2-2-3-0 -- and their respective twins are SP897 (BRKBQRNN) and SP881 (BRQBKRNN).

Start Position 059
Three undefended Pawns (a/b/h)

Do any of these database discoveries help to play better chess960? I doubt it, but they might help to differentiate the 960 SPs more rapidly. At first glance they all tend to look very similar.

1 comment:

Tom Chivers said...

I've been thinking about the 960 start positions too.

In particular, I've started questioning whether comparing % scores of any of the 959 positions with chess makes any sense at all. After all, the 959 positions are basically played out with zero theory: so it would make more sense to compare them with chess results around the time of Greco, not chess now.

This leads to a further thought. Do the %'s change with rudimentary theory applied? For instance, in the position where 1.Ne3 c5 2.d4! is extremely good for white, does 1.Ne3 e6 score much more moderately? Perhaps one way to test this would be using 'Monte Carlo' analysis?