Chess, Nim and Symmetry

I have a certain interest in symmetry in chess and plan to explore the subject in some future entries. However, first I need to clarify a few concepts and I don't really know where to start. So instead there will be some initial random thoughts.

In chess, symmetry normally refers to a position which is symmetrical along the horizontal mid-axis (mirrored along the 4th and 5th rank). This probably is because it's the kind of symmetry which occurs most frequently and therefore is of most practical interest.

However, it is also possible to reach positions which are symmetrical along the long diagonals.

This pawn endgame for a period was a popular wild variant on ICC:

It is completely solvable and reminds of the ancient game of Nim which has long been mathematically solved (at least the most popular versions).

I have heard rumors that this game was removed from the server because it attracted gambling. If that's correct it's strange because it's a game that in principle can be mastered completely and doesn't involve any element of chance.

That of course doesn't stop it being a devilishly difficult game to play perfectly. I assume most of those playing it for big sums of money must have had access to tables with computer generated solutions.

For a beginning understanding it may be useful to analyze this position which obviously is closely related to the initial position:
My guess is that White to move loses. But although I analyzed this position (or a similar one with kings on b7 and g2) 25 years ago I must admit that I don't remember anything about it - except that it isn't a draw.

Preliminary conclusion: Diagonal symmetry occurs relatively rarely in practical chess and tends to lead to races and occasional zugzwang and not to drawish positions.

In future entries I will concentrate on symmetry along the horizontal mid-line.


PS: Today I added a chess gadget to my blog. You can see it at the bottom of the main window. I have no idea how good, hard or varied the exercizes are. It seems you get a new diagram every time you reload the page [F5].

Another Shortest Proof Game

At Chess Publishing Forum, IM John Cox has suggested a new 'shortest proof game' task: What's the shortest game ending with '0-0, mate' or '0-0-0, mate'.

So far the best tries are:
1.d4 e6 2.Qd3 Ke7 3.Bg5+ Kd6 4.Na3! Kd5 5.Qf5+ Kxd4 6.0-0-0, mate


and:

1.f4 f6 2.Nh3 Kf7 3.e3 Kg6 4.f5+ Kxf5 5.Bc4 g6 6.d3 e5 7.0-0, mate.

Both achievements are quite good but I wouldn't be surprised if they can be improved upon.

One would think that short castling must take at least four moves: one knight move, one pawn move to open for the bishop, one bishop move and castling but that isn't necessarily so because Black can capture pieces too.
The same of course goes for long castling.

For those of you still toying with symmetrical mates from my blog entry of April 5th, here are some short games I found:

Knight: 1.e4 e5 2.Nc3 Nc6 3.Nge2 Nge7 4.g3 g6 5.Nd5 Nd4 (Dia)







6.Nf6 mate






Rook: 1.Nf3 Nf6 2.Ng5 Ng4 3.Nxh7 Nxh2 4.Nxf8 Nxf1 5.Ne6 Ne3 (Dia)







6.Rxh8 mate







Bishop: 1.b3 b6 2.Bb2 Bb7 3.f4 f5 4.e3 e6 5.Be2 Be7 6.Bxg7 Bxg2 (Dia)





7.Bh5 mate









Pawn: 1.g4 g5 2.f4 f5 3.gxf5 gxf4 4.Nf3 Nf6 5.Ne5 Ne4 6.f6 f3 (Dia)






7.f7 mate








King: 1.f3 f6 2.Kf2 Kf7 3.Kg3 Kg6 4.Kh3 Kh6 5.e3 e6 6.Bd3 Bd6 7.Bg6 Bg3 8. hxg3 hxg6 (Dia)






9.Kg4 mate.








Any better?

Addendum July 12th
There is a quite readable article on Shortest Proof Games at Chessville. It appears to be the first in a series.

Extreme Symmetry

I will be very busy until the 15th or so but I will try to fill the silence with some mini entries. In my entry of February 22nd I offered two reconstruction tasks:

1) Black's 5th move is to promote a pawn to a bishop with mate.
2) Black's 5th move is to promote a pawn to a knight with mate.

Here are the solutions:
1) 1.c3 d5 2.Qb3 d4 3.Kd1 dxc3 4.Kc2 cxd2 5.Qc3 d1B mate (Dia)

2) 1.d4 e5 2.Kd2 exd4 3.b3 d3 4.Kc3 dxe2 5.Kb2 exd1N mate (Dia)

Today's task can be seen as a prelude to a theme to which I hope to return shortly: Symmetrical positions and the advantage of moving first.

Can you construct a symmetrical game in which White's 4th move is mate?

The usual conditions apply:

All moves must be legal but obviously they don't have to make sense by conventional chess standards.

Unfortunately there are two and a half solutions. There are two different mate positions and one of them can be reached via two slightly different routes.

Good luck!