Are the two related or proportional? I happen to be good at both. What do you think?

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Chess and the Cartesian plane

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  macer75
  #1 5 days ago

  It has recently occurred to me that, given the fact that we currently represent moves in chess using algebraic notation, which is essentially (in part) a system of coordinates, it is possible to represent the chess board using a Cartesian coordinate system. Specifically, files a-h correspond to 1-8 on the x-axis, and ranks 1-8 the same numbers on the y-axis. Under such a system, we can observe that the movements of the chess pieces model various mathematical functions:

  1. Rook moves model the function x=n or y=n, where n is a natural number between 1 and 8, inclusive.

  2. Bishop moves model y=x+n or y=-x+n, where n can be negative, 0 or positive natural numbers (there are additional restrictions, but I have not as yet given enough thought to the issue to describe them with complete accuracy).

  3. When a knight is situated on the square represented by the coordinates (p,q), all of the squares that it can reach on the next move (on an empty board) lie on a circle with (p,q) as its origin and a radius of √5. In other words, they are all points on the function (x-p)^2 + (y-q)^2 = 5.

  Given these observations, it may be helpful to think about what representing chess on a Cartesian plane can contribute to the study of chess. As I am not an expert in chess theory, this is not a question on which I have much to say as of now. Nevertheless, I can think of a situation where the Cartesian plane can have practical applications, namely in blindfold chess:

  Suppose that you have a bishop on b1, and you intend to move it in the direction of the long diagonal (in other words not to a2). However, on his last move your opponent moved his knight to d6, and you do not want to put the bishop on a square where it can be taken by the knight. What are the squares that you must avoid? Using Cartesian coordinates, it is not difficult to find the answer. All you need to do is solve the system of equations

       y=x-1

       (x-4)^2 + (y-6)^2 = 5

  and you will find the two results (x=5, y=4 and x=6, y=5) that correspond to the coordinates of the squares to avoid.

  Now, I know that there are many people here who are well-versed in chess theory. And to those people (but not only to those people - beginners like myself are also welcome to respond) I ask: what are some of the ways in which you think the use of Cartesian coordinates can benefit the study of chess? Tell us your thoughts in the comments below!

Im talking about simple arithmetic like +-×÷ without using calculators and pen n paper. I believe if one cannot do simple arithmetic mental calculation like 85x45= 3825, it is difficult to visualize board after 3-5 plies. My proposal is practice mental arithmetic and chess visualization improves.

With  chess  you  have  concrete pawns and  pieces on a  chess board that aid visualization patterns in the brain.  If you could in a concrete way explain this practice of mental  arithmetic.  And relate  it to chess  visualization patterns.

I'm a big skeptic of the idea you can get better at chess. studying something else.

OTOH

I've long seen kids get much better at the game with no study at all.

this is not what I mean.  Kids can and do get better at the game (without study)... they grow more emotionally mature, more logical, more methodical and more patient and these are huge advantages.

but if we take a kid have him learn math and take a kid and have him do school,etc.   I would expect that identical kids would do equal at chess... even though the one kid was actively studying math.

meaning math skills is not the same as chess visualization.

I read a book in French which deals with this matter called "Psychologie Des Grands Calculateurs et  Joueurs D'Echecs" (roughly translated: Psychology of Great Calculators and Chess players ) written by the French psychologist Alfred Binet. He questions some famous calculators like Inaudi and Diamandi and some famous chess players of that time like Goetz, Zukertort, Tarrasch, Blackburne etc... He found that while something is necessary to reach a decent level of chess (intelligence?) both calculators and chess players use a huge amount of past memories to be able to calculate faster. He also investigates how some players can play blindfold chess without any loss in level and how they visualise the board. He conjectures that they also use memorized matches to achieve this feat. For example Morphy could easily recall matches he played 18 months ago .Dr. Tarrasch could play a couple of simultaneous blindfold matches because he could remember not only the moves but the general ideas and reasoning behind a position. That's why they try to play different openings for each match etc. etc.

It's an interesting book but I don't know if it is available in English.

Morisien wrote:

It's an interesting book but I don't know if it is available in English.

Who would have believed this many years later.  Julia Child and you faced with the same problem.  Julia with French cookbooks  only in French, but not in English.  You with Alfred Binet's book in French, but not in English. lol

I used to amaze my physics and chemistry students with my mental arithmetic but I've never been good at chess visualization.  Members of the high school chess team I coached and I tried playing games without a board.  I'd cross paths with one walking through the halls between classes and he'd yell out "e4." I'd reply "e5" - mainly because it led to less unsymmetrical positions.  At bullet pace, we'd get to about move's 7-10 and start to lose the game.
We also tried using smaller boards to see if scaling up from a 5 x 5 board (no P's, Q's, or R's) to 6x6 helped.  We could handle 5x5 with only 5 pieces on each side, but as we scaled it up and added pieces and pawns we lost it.  Maybe we should have started on a 5 x 8 board with P's.

The cartesian coordinate thing seems going a little overboard in needed effort - my formal math education is through Multivariable Calculus - but there may be 8x8 matrices or determinants that fit a little easier. But it's an idea worth exploring.  It would be interesting to know how the engines set up the board mathematically.

I don't think these assumptions/computations will apply in the real chess world....when I was active playing chess more than 30 years ago, every time I play it seems I already have played the game. This is because I  memorized matches and could recall great games played by great players.  I agree that math skills is not the same as chess visualization.