# Forgotten chess

As we watch a game of chess in progress, we can imagine what was happening up to that moment. How the players thought, how the game is going, how long it has been and what has happened…

However, this is based on a totally false premise for this game, the existence of a long “memory”. In fact, chess is a game with little memory …

Can you imagine how this game came to be in this situation?

In probability theory “Out of memory” is a well-known property of some distributions. It refers to the probability distribution not depending on the “transfer time” in a specific event. In a simple way, we can understand the lack of memory in the probability as the “indifference” between the previous results with the new events. For example, if we have a 1% chance of suffer a car accident every time we leave the house. So, if we go out once a day for a year, (1-0,01)^{365} is the chance of not having a car accident this year, which is something in the range of 25,5%. But this distribution has no memory, that is, whether or not I have an accident that day, it does not increase or reduce my chance of having an accident the next day.

But the focus here is to talk about something less tragic than accidents. A simple example that has no memory is a coin toss game. The chance of throwing an honest coin 10 times and it falls to the crown every time is (0,5)^{10}, something close to 0,1%. So, knowing that I threw the coin 9 times and got heads, what is the chance of my coin falling face down on the 10th launch? In this case, the result of the other 9 launches is indifferent to the new result, so the chance of getting expensive on the tenth launch does not depend on the result of the other 9, that is, we have a 50% chance of this happening.

Another example to warm things up: the chance for someone to play once and win in the lottery is 1 to 50.063.860. So the chance of someone playing twice and winning the lottery both times is 1 to 2.506.390.078.099.600. But the chance of someone who played once and won, played a second time and won too, is 1 to 50.063.860. That is, the next result does not depend on luck in the first confirmed result. These are examples of out-of-memory probabilities.

Just to exemplify a situation with memory, suppose there is a box with 1.000 screws. The chance that a screw was produced defective is 1%. That is, we estimate that of these 1.000, there are about 10 defective. Let us analyze some simple cases to show the memory of this event:

A) If we checked 900 of these 1000 screws and none were defective, the chance of taking one of the remaining 100 to check and be defective, is close to 10%;

B) If we checked 950 of these 1000 screws, and none of them were defective, the chance of taking one of the remaining 50 to check and be defective, is close to 20%;

C) If we checked 980 of these 1000 screws, and none of them were defective, the chance of taking one of the remaining 20 to check and be defective, is close to 50%;

D) If we checked 990 of these 1000 screws, and none of them were defective, the chance of taking one of the remaining 10 to check and be defective, is close to 100%.

With an idea of what memory is in probability, we can now talk about the game of chess. This is a game with little memory, but what do I mean by that?

A) There are pieces completely forgotten in this game: Queen; Knights; Bishops;

B) There are pieces with little memory: Rooks; Kings (because of Castling movement);

C) There are pieces with a “reasonable” memory: Pawns (they are not perfect in memory because in many cases, it is not possible to say exactly, analyzing the board, what actions were taken to reach this context).

In this regard, when looking at a board, players may have faced each other in different ways, repeated playing patterns, returned all their pieces to their original positions and resumed their combat from there. There are no rules that prevent players from having peaceful fun within the game. Thus, we will now analyze some actions that have “a little memory”.

A) Eliminated pieces, because although the pawns can replace them, they still compromise the existence of a Pawn and its ability to reach the eighth square, that is, it implies the cleaning of the path blocked by another Pawn;

B) Castling, because moving the King or Rooks referring to that action commits you, so it is as if there is a status “Castling free” or “Castling used”;

C) The famous “movement of two squares” by the Pawn, which can only be used when it occupies the initial position;

D) “El passante”, a special pawn movement that can only be applied to the next movement to which an enemy pawn passes through it using the famous “two square movement”;

E) The pawns’ inability to move backwards, so that each pawn’s advance is permanent in part of the game’s memory;

F) Pawns, either because they are eliminated or used to replace other pieces.

So the game of chess has some memory, but not enough to just observe the current state of the game and safely recreate what happened between the players.

This means that we can think, at most, from a certain instance, and not want to trust what happened before the game reached that state. You can imagine, two great chess masters (X, Y) are playing with each other, their game started an hour ago. And now two other great chess masters (W, Z) are starting to play.

After playing for a while, the board of both pairs coincides with exactly the same formation, Castling and the right to use “El passante” are in the same status.

The pairs are interrupted and ask players X and W to switch places, and so continue the game at the other table. But as both boards are identical and the game doesn’t have that much memory, this should in no way affect your strategies, even if their status is different for an hour of play more or less. It doesn’t matter how your opponent did to get the pieces in this configuration, since in both cases the games followed different strategies. And at each table, we have one player who mentalizes the game taking place in N minutes, and the other one who mentalizes the game taking place in an hour and N minutes.

A simpler example would be if players X and Y move only the knights for 16 moves, until they return to the starting positions of the game. In this situation, players W and Z would start playing. Then they ask player W to switch places with player X. The game for player X is already on the 17th play while for player W is on the 1st play. The same is true for players Y and Z.