Tobias Labs

While I don’t really play much Chess, I have recently been pondering something interesting about the game which led me to this train of thought I will share with you now.

I was thinking about Chess games and how the potential game “paths” are countably high yet approaching infinite, that is, If we consider the start of the game to the end of the game to be a path in a multi-dimensional space, the paths possible are nearly infinite in number.

Of course, if we consider the total moves possible, we must conclude that the possible paths are truly infinite: If there are no rules concerning the maximum number of moves, the game could go on infinitely. Obviously, in real life the number of moves is finite and therefore countable, but there are no hard and fast rules about when a game ends, so unless we cut it off at some arbitrary point we must agree that the possible paths are infinite.

In fact, if we try to define the possible “steps” on the path, we can quantitatively define them in a very simplistic manner: 8*8=64 squares, 6 pieces (KQBRCP), two colors, plus the empty cell makes 13 possible states for each square. Recall that we can write the possible configurations as 13^34, which equals the impressive number: 8.71*10^40. To help get a grasp of that scale, it would take roughly 2.6*10^33 gigabytes of data to hold every possible Chess board configuration.

Now certainly, the obvious problem is that many of those configurations are not legal. One example is the case where the board is entirely covered with only black kings. It is clearly absurd, yet is still counted as a possible configuration using this method.

So how exactly do we go about calculating how many positions are available? We could list every possible configuration and check each one for validity according to the rules of Chess, yet this too would yield multiple positions which are impossible.

The only way to go about this is to start at the beginning of the game and progress down the path until the “end”. So I began to wonder how one would go about this task—mapping every possible Chess game path.

Mathematically, I can see that each piece could be seen as a dimension traveling only integer units away from the origin, giving us 32 dimensions for 32 pieces. This appears to simplify a bit when we consider that most of the pawns have a very limited travel, yet on closer inspection we note that the pawn could become a queen, which gives an additional problem: A piece could change type.

How does this pan out in the end? A beautiful graphical illustration: Imagine a nodal network, each node being a configuration of the Chess board. If we give an upper limit to the number of moves, we know that the number of nodes will be countable yet still absurdly high. We could then compile a list of all past Chess games played, and each of those paths between nodes will be given a higher “strength”. Many of these nodes would cross over, perhaps multiple times, since duplicate configurations would be mapped to the same node.

When we put this all together we will see a centrally controlled nodal network which has clearly visible “strong” paths, yet on closer inspection will show a very complex network system.

I am trying to think of how to put this idea into a computer to generate an actual graphical map of the terrain. I only have about 600 gigabytes of space available in my hard drive, I wonder how far I could get…

Update: My fantastic room mate sent me the following link to the Wikipedia entry on the Shannon Number, an estimated lower-bound of potential moves in Chess. An additional interesting thought is to see what the significance of a smaller board or less pieces would yield. Also, the discussion of the disk space required for possible configurations does nothing to speak of the disk space to record possible paths, which would necessarily duplicate themselves and take an even larger space.