on the other hand - if (as many people believe) chess is a neutral game where perfect play will always end in a tie, then solving chess is much simpler given that you could play from the beginning for a game ending in either three fold repetition, stalemate, or a draw.
Math as a Mathematical Formula using Matrix/Algebra?
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johnny263 wrote:
j_king: i'm not sure that we're disagreeing. i looked over some of your old posts and it looks like you agree that solving chess by a using a position tree is theoretically possible but currently impossible given our current means (storage space and calculating time, etc.). so thanks for the wikipedia links but judging by what you have been saying, i do not disagree with any of the facts presented in them. on the other hand - i think it is interesting to consider the possibility of solving chess by means other than using a position tree. i'm NOT talking about an equation that would calculate the best possible move by considering all the possible positions for move 1, and for move 2, and for move 3, etc. and then telling you the best move on move 1 (which is essentially a position tree approach). i AM talking about an completely new way to approach the "best move" criteria. you asked me for an example and i'll try. imagine if it was discovered that there was no way to stop fool's mate. you would have "solved" chess in creating an unbeatable line for white and you would not have had to calculate all possible chess positions. now, obviously that's not the best example since fool's mate is easily stopped, but it's the best i can come up with. i'm not claiming to have solved chess here, i do, however, think that it is solveable.
Well, to be formal, another poster did point out the difference between an algorithm and a function.
The reason why that is important is because from a blank board, White has approximately 20 moves available. The Fool's mate example requires a very specific sequence of moves in order to be possible. In order to "solve" chess, this equation would have to produce a Fool's mate 100% of the time by playing Black. That is simply not possible if White doesn't follow the specific sequence.
A function simply cannot exist that can solve chess.
The only avenue I can see which one may pursue besides the modern algorithms we've been developing for over fifty years is a state-machine approach. Such a machine would probably be enormously complex; relying on sets of algorithms for determining the next move of play based on its current state. However; even if such a machine were possible, it would only be able to weakly solve chess as by the conditions of solvability I linked to in my previous posts.
"Solving" chess is a goal pursued since some time in the mid-30's. Some of the greatest minds in computer science and mathematics have thrown their weight at it. However, it's complexity will be (thankfully) out of the bounds of a strong solution for the foreseeable future. If it were solved already, it would be an awfully boring game -- which we do know without a doubt that it is not. :)
I love it - every math genius in the room poked there head up at this question, and me also - however, I have the answer...
NO!
Now instead of investing hours in a formula, learn how the pieces move. Your opponent will lose on time trying to apply the equations!
Haha well from a mathematical point of view there's two main thought that come to my mind the first is;
As mentioned by Jonny236 chess is generally accepted amoungst the mathematic community as being a nuetral game - however any concievable 'golden strategy' would be so much more complex than the equivelant strategies for Naughts and Crosses and Checkers that it would be impractical to apply this set of rules to a human Vs human game of chess.
Personally I believe that the reason there is yet to be a strategy of this form is that chess is a dynamic game - the true relative value of each piece and each square changes based on every move that is played aswell as every move that is NOT played. It is constantly evolving and as such any set of rules that is applied to a game with have to take account of this fact.
The second idea that springs to mind is also only really applicable to a computer but how about an algorythmic probabalistic analysis of games that then defines the optimal move based on a data set collected of previous games - assuming that you can get a large enough uncorrupt set of data this seems to be the strongest method for writing a super chess programme.
A similar programme would be something like 'sender base' the online threat management database created by Ironport of Cisco. The way in which this works is hundreds of thousands of networks are installed with an 'Ironport' which monitors and detects all digital attacks against said network - it then records the information of ALL successful and unsuccessful digital attacks against the network and stores this information on the 'sender base' database. This data is then compiled and used to probabalistically analyse the probability of an 'email' or 'ping' etc being malicious.
Similar if you could inbed a algo based programme into a popular online chess site you would be able to collect a massive data set of moves, sequences of moves and entire games that could then be combined to create an unbiased analysis from a purely probability driven pov. Obviously this would take the role of a modern day IBM project LOL.
This is just an idea but I'd be really interested in hearing what the other 'maths and chess' crowd's thoughts are on such a thing?
Ta for listening to my rabble!
Sam
perhaps in future years there will be mathematical equations which will solve for the best move on the chessboard, but it is a long way away. I believe that computers use a similar method, but they only analyze a massive number of possible moves and a massive number of possibilities which could result from each move. The thought of trying to solve chess problems using an equation is certainly an interesting concept. Perhaps there will one day be an equation or identity that is derived for the game of chess in the future. Perhaps you will merely just have to plug in the coordinates of each piece on the board into the formula and you will be able to solve it using simple linear algebra to determine the most logical move. But this presents us with another question; is the most logical positional move always the best move?
Someone has done it. His name is Bernard Parham and he created Matrix Chess.
Feb 13, 2010
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#48
really. I figured it would use matrix algebra but it has been done. Wow! I will google him. Thank you.
darius wrote:
Here's a thought--can you make a mathematical representation of chess? Since the moves are within a given field that seems like a matrix, and the patterns are predetermined with some rules, can a formula be created that recreates the game in mathematical terms that have some functional use. If so, we can use our math to beat computers. Not being a good mathematician (I can add and subtract but division, hm...) it's beyond me, but I wonder if someone in the field could figure something out. All computers do is calculate moves and then use an algorithm to judge a position. Mathematic formulae may be superior. Again, beyond me.
Yes it's possible how do you think computers work :P?
They search all possible moves and assign values to them then play the one with highest value.
Durial wrote:
perhaps in future years there will be mathematical equations which will solve for the best move on the chessboard, but it is a long way away. I believe that computers use a similar method, but they only analyze a massive number of possible moves and a massive number of possibilities which could result from each move. The thought of trying to solve chess problems using an equation is certainly an interesting concept. Perhaps there will one day be an equation or identity that is derived for the game of chess in the future. Perhaps you will merely just have to plug in the coordinates of each piece on the board into the formula and you will be able to solve it using simple linear algebra to determine the most logical move. But this presents us with another question; is the most logical positional move always the best move?
I think that Linear algebra would be the best approach.. But as you so wisely stated.. the most logical position isnt always the best.. When trying to use rational equations to solve irrational human thought.. I find error. I make sacrifices and elude the other play to allow them to precieve a "winning persona" to low and behold sneak in for a win. . . A data set of all possible moves would be nearly infinite depending on the desired result... a check-mate based attack.. or the position of a single piece... Although this thought has crossed my mind only recently. I may pursue said linear expressions..
-JustJulian-
First we should probably understand what linear maps are.
A linear map is an operation, let's say L, that takes some object x to L(x) in accordance with the following properties:
L(0) = 0
c*L(x) = L(c*x)
L(x+y) = L(x) + L(y)
These objects x and y which L acts on are usually vectors, that is, points in n-dimensional space. One way to understand linear maps is that, given a shape S in n-dimensional space, the shape L(S) [i.e. every point in S transformed by the map L] looks the same as S, just stretched/shrunk in some directions and rotated in others.
Every linear map can be expressed as a series of stretches in one direction (e.g. turning a square into a rectange, or a circle into an ellipsoid), followed by a rotation.
So how does this apply to chess? Well, we aren't doing anything linear in chess, about the only thing that's linear at all is counting up the material (since the material can be expressed as the vector [pawns, rooks, bishops, knights, queens]).
Suppose there were a linear map C taking every chess position to a position that could occur on the next move. Not even the best next move; any legal move will do. By the formulas for linear maps, we have
2*C(starting position) = C(2*starting position)
...but wait, how do we multiply chess positions by 2? Before we're even started, this is a mess. There is an expression for turning chess positions into other chess positions, but it's most definitely not linear or anything nice and simple like that. The simplest you can get it is probably the standard "if it's a bishop, it has to go diagonally, also you have to see if that would put you in check, etc..."
(Can someone who knows this stuff better elaborate/correct me a little? We can make this the 'teach everyone abstract algebra' topic!)
you are correct for this post starter. I believe we can do this. I also you can also do this. Mathematicians are nothing but humans who try to represent every thing with symbols or to say they see a language which handles large amount of data in to symbols. I am not talking about applied science of mathematics which is inferior to original logical mathematics. I believe if any one can try they can build the equation of chess in an year or so. even if you dont have any basics in mathematics.
j_king:
i'm not sure that we're disagreeing. i looked over some of your old posts and it looks like you agree that solving chess by a using a position tree is theoretically possible but currently impossible given our current means (storage space and calculating time, etc.). so thanks for the wikipedia links but judging by what you have been saying, i do not disagree with any of the facts presented in them.
on the other hand - i think it is interesting to consider the possibility of solving chess by means other than using a position tree. i'm NOT talking about an equation that would calculate the best possible move by considering all the possible positions for move 1, and for move 2, and for move 3, etc. and then telling you the best move on move 1 (which is essentially a position tree approach). i AM talking about an completely new way to approach the "best move" criteria.
you asked me for an example and i'll try. imagine if it was discovered that there was no way to stop fool's mate. you would have "solved" chess in creating an unbeatable line for white and you would not have had to calculate all possible chess positions. now, obviously that's not the best example since fool's mate is easily stopped, but it's the best i can come up with. i'm not claiming to have solved chess here, i do, however, think that it is solveable.