∫iv A perfect game of Chess

In principle finding a perfect chess game seems to be a reasonably straightforward formulated problem in discrete mathematics.

To find a method and to give a simple but precise definition of the ultimate game involves finding the/an ultimate best move that follows after each best one and such could be in this way established by brute force. This in turn would mean in effect physically playing out (almost) all possible games.

The obvious question is does a perfect game exist? And if then, what is the result? Which side wins? It is essentially a problem in combinatorial applied mathematics.

We are looking at chess from a purely mathematical viewpoint as philosophy with no consideration at all of actually playing recreational chess or participating in competitions.

Brief observations

One would actually be able to formulate a simple description of an ultimate a perfect game, thus of an ultimate best move that follows after each best. It is a game with no mistakes. It would however be very hard to predict the final result the two sides seem in almost perfect balance and it would probably stay that way throughout.

We may need to have a question in a spirit similar to in “the hitch hiker's guide to the galaxy” and even the famous very ambitious profound questions of the late Hawking. I suspect that in our situation too the hardest part is for you to know exactly what the question is before attempting the answer.

It could be that it is the question which is unclear. In other words the real difficulties might in fact be in making a correct and thorough analysis of and exactly formulating the problem in this way before attempting any actual calculations. Furthermore in attempting concrete calculations you really do have this dilemma.

As in answering which came first, the egg or the hen?

A perfect game and best moves

To begin with exactly what do I mean by “best move”? Surely it does look like a kind of rhetorical question. How do you compare two games? Save for the obvious? It is so that if you can compare each  two and you could determine which is the best of the two (or whether they are equivalent) it is in theory possible to do and in fact it is so because there is a finite number of games however many there may be. It is how a human plays too only he is extremely restricted.

According to the method described the whole process is repeated and once again, and then again. In this way for each position the best subsequent one must be found. Then the one after that because a best move always precedes a best move and the others are eliminated one by one. Eventually only the best ones in the whole set of all games remains and no game is better. It is then a universal best, the/a perfect game.

Some practicalities

I believe that to demonstrate, to find and formulate, reason and prove that a suitable algorithm exists must be straightforward. However for say an average length game we have apparently unlimited calculations and going well past all types of notation the calculations would still take essentially forever to complete.

You would need some kind of near infinitely fast computer and just as unthinkably immense are memory requirements anyway it all would be of little practical use due to the unbelievable number of all imaginable possible positions.  I fear that even the speculated quantum supercomputers would be totally hopeless.

What is more positions can be duplicated or equivalent lines may split up and diverge. This complicates matters considerably. Indeed note that different paths may often be followed while the same positions can be arrived at (visited) for instance when openings change and transpose back into each the other with returns and repetitions. Especially in endgames with very few men on the board such situations abound.

Different routes may be followed but with the same result and destination. In fact there is most probably more than just a single perfect game and we expect many more but of course the argument is not of much practical consequence.

Algorithms and existence of solutions

Is there then a perfect game of chess, or the same, a best move in any conceivable position? This would obviously settle the question: Who eventually wins? White, or black, or a draw, stalemate? A chess game cannot end in another way because the rules would prohibit an infinite game.

It means that from the starting position you must play every legal game right through to the end in order to know what the first move has to be. Can we then design and prove the validity of an algorithm? Is there an algorithm? I believe there is. There must be. In effect it could be described in a relatively simple computer program.

Unfortunately now you end up with incredibly unbelievable unknown complexity in a recursion setting. Even for by far the most any given single position it has to be practically impossible to find and to demonstrate such a concrete universal best move.

And finally ...

The game would develop as follows: What would the best first move be? And black's best reply? And then again back to white for his next? For the very first move I would first have to consider each direct possibility and to know which one is the best. For that once again for each, one step deeper and it continues and proliferates more and all the more rapidly.

Therefore in conclusion I claim that for this apparently simple board game it is in practice impossible to know who wins, white or black otherwise whether it is a draw. Any answer can only be purely intuition and in the end really just a wild guess.