# I Who would win a perfect game of chess?

#### phyzguy

OK, then how would you try finding this optimal strategy?
I know how to prove that there are infinite prime numbers.
I mean without some trial and error I don't see how can you find an optimal strategy?
I didn't say you can find it, only that we know it exists. The fact that there are a huge number of moves and finding the optimal strategy is so difficult is why we don't know the answer to the OP's question. You can say, "I think it will be a draw," but there is no reason for the rest of us to believe you.

#### PeroK

Homework Helper
Gold Member
2018 Award
I didn't claim to have a solution, it's just my strong belief which I see is unjustified.
So why in checkers for example the advantage in starting isn't sufficient to enforce a win?
What mathematical argument was given for this?
Checkers is a simpler game and has been crunched by a computer.

There's no reason that the solution for chess is the same. It might turn out that white can win.

#### jbriggs444

If, for example, you can show that it's always possible for white (or black) to reach a position with a forced mate, you only have to evaluate the possible moves leading to that position.
That sounds like the essence of alpha-beta pruning. If you are trying to decide between a forced win for white and a forced draw for black then black can prune away all moves other than to a known forced draw. White can prune away all moves other than to a known forced win.

#### dRic2

Gold Member
Although it might be intellectually/mathematically pleasing to think about it, what an horrible game would be a "perfect" game ?

I'm not criticizing the OP for the question (very legit), but I noticed that this kind of discussions seem to reduce chess to just "maths" (logic). A non-player (an outsider) might get the wrong idea! Chess is about art and phycology and culture and a lot of other stuff!

Now hate me for this useless comment

#### MathematicalPhysicist

Gold Member
Now when I think of it, white will always win, it's not the same as in checkers.

Try to play against the computer in level 10 with the hints with white in chess.com. I know it's not a proof, but it seems plausible that in this game the one who starts will win, if he played perfectly.
It took 109 moves, if I were to repeat these game will the moves of the black changed?

I have exams on Condensed Matter Physics and Particles theory 2 so no more time chatting here.
I stand corrected.

#### MathematicalPhysicist

Gold Member
Although it might be intellectually/mathematically pleasing to think about it, what an horrible game would be a "perfect" game ?

I'm not criticizing the OP for the question (very legit), but I noticed that this kind of discussions seem to reduce chess to just "maths" (logic). A non-player (an outsider) might get the wrong idea! Chess is about art and phycology and culture and a lot of other stuff!

Now hate me for this useless comment
You are in the wrong website.
:-D

#### bahamagreen

Last time I did programming was mainframe Fortran IV...

Given 64 squares and 32 pieces
Let all states of the board (including improper ones) be represented by a base 31 number of up to 64 places.
Assign the values 0-31 to the pieces, even white, odd black, as "digits" of this number.
Example of the initial board state at the beginning of a game might be something like:

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Treating these states as either numbers or strings, exclude states that represent:
- no pieces
- one piece
- two pieces where at least one of them is not a king
- more than one king of either color
- bishops on incorrect color square
- pawns on their back row
- too many of same color piece (due to pawn promotion... this would calculate how many duplicates of a piece are possible and allowed based on comparing the possible number of pawns promoted vs pawns not yet promoted and available to reach the back row)
- addition impossibilities in a game

Distinguish states that represent proper moves.
Set the initial starting game state as move number zero

For each set of all proper final move states* (see below) where the set's move number is one less than the present move number but not negative
For each piece
Test all states that re-position the piece to every square and eliminate states from the list that:
- re-position an incorrect color piece for that move sequence in the game
- re-position a piece to the square it is already occupying
- move a piece through an occupied square, inclusive (expect knights, and proper captures, castles)
- move a piece improperly per game rules of piece movement (including second same color castle)
(above piece movement tests done algebraically)
Mark or associate all proper final move states* for this move number with an indication of the move number and the moved piece number
Increment the move number
Repeat
This provides a disordered list of all proper piece position move states (including capture, castle, and promotion) for all move numbers of all proper games

Relate these proper move states into strings of all proper games
Using the proper final move states with their indication of the move number and the moved piece number

Set move number to one

For all proper states of the move number
Test all proper states where move number is one greater
If greater move number state is a proper move from the previous move state number, mark or associate the two
(include test for third of same move, move number greater than 50 if desired)
Increment move number
Repeat

Use the proper move associations to produce all possible proper games state number lists
Filter this list for however you define a "perfect game"

#### PAllen

Chess is unsolvable with traditional computers

Claude Shannon noted that a true chess solution would require storing 10^120 moves. This gets into age of the universe type computational times and impossible storage requirements with any conceivable computer technology other than maybe a huge quantum computer
No, that number would have to be analyzed but not stored. The result of analysis so far could use one of the compact tablebase representations. I once worked out this would only require a number of bits similar to atoms in the moon to play any position perfectly.

#### PAllen

Chess is kind of weakly solved when 7 pieces (including the two kings) remain on the board (discarding the castling moves and en passant). I don't know if they're working on 8 pieces, etc. An idea to have another indication that chess may be a draw (or a win/loss) with perfect play is to set symmetrical starting positions with the few pieces on the board and see the outcome of perfect moves.

I also guess that it leads to a draw, but it's just a pure guess.
Some chess variants are easier to deal with than chess (giveaway or suicide chess), whilst others are more complicated (crazyhouse).

Another very interesting question is whether all starting position of chess 960 (Fischer random chess) lead to the same outcome than regular chess, with perfect play.
I would say it is strongly solved for 7 or fewer pieces. The winner as well as best play for both sides, for any such position, can be generated ( even though the sequence itself is not stored).

#### phyzguy

Now when I think of it, white will always win, it's not the same as in checkers.

Try to play against the computer in level 10 with the hints with white in chess.com. I know it's not a proof, but it seems plausible that in this game the one who starts will win, if he played perfectly.
It took 109 moves, if I were to repeat these game will the moves of the black changed?

I have exams on Condensed Matter Physics and Particles theory 2 so no more time chatting here.
I stand corrected.
So you've replaced your guess that it would be a draw with the guess that white would win. Well, it's one of those two. If we knew which, we could answer the OPs question.

#### PAllen

Chess is kind of weakly solved when 7 pieces (including the two kings) remain on the board (discarding the castling moves and en passant). I don't know if they're working on 8 pieces, etc. An idea to have another indication that chess may be a draw (or a win/loss) with perfect play is to set symmetrical starting positions with the few pieces on the board and see the outcome of perfect moves.

I also guess that it leads to a draw, but it's just a pure guess.
Some chess variants are easier to deal with than chess (giveaway or suicide chess), whilst others are more complicated (crazyhouse).

Another very interesting question is whether all starting position of chess 960 (Fischer random chess) lead to the same outcome than regular chess, with perfect play.
Some of the starting positions in chess 960 are considered extremely favorable to one side, based on computer plus human analysis. Nothing is proven, but I would guess some are forced wins for one side.

#### jbriggs444

So you've replaced your guess that it would be a draw with the guess that white would win. Well, it's one of those two.
Is it proven that black cannot win against best play?

#### Buzz Bloom

Gold Member
The 50-move rule limits the maximum number of moves to just under 6000. There's a thread about it here:
Hi PeroK:

The 50 move rule was changed for a while but restored in 1992. It may well be modified again sometime in the future.

https://en.wikipedia.org/wiki/Fifty-move_rule
All of the basic checkmates can be accomplished in well under 50 moves. However, in the 20th century it was discovered that certain endgame positions are winnable but require more than 50 moves (without a capture or a pawn move). The rule was therefore changed to allow certain exceptions in which 100 moves were allowed with particular material combinations. However, winnable positions that required even more moves were later discovered, and in 1992, FIDE abolished all such exceptions and reinstated the strict 50-move rule.​

Regards,
Buzz

#### PAllen

I think it is important to note that Alphazero is a major breakthrough in machine learning, but says almost nothing about solving chess or perfect play in chess. The breakthrough is learning to play chess with no strategy input or example expert games, and reaching a level well beyond human experts. Perfect play, or even best computer was never a goal of the research. A few random observations on the theme that alphazero is very strong, but still far from an oracle of perfection:

- Alphazero mainly played stockfish 8. Stockfish 10 wins against stockfish 8 by at least as much as Alphazero did. Alphazero has not played against stockfish 10 because winning computer chess tournaments or even research in chess is not a priority at all for the deep mind team.

- In the Carlsen-Caruana world chess championship match, there was a game where stockfish 10 running on a supercomputer found a forced mate in 43 moves. Alphazero analyzing these games was unable to find this line.

- There are Alphazero losses indicating that its highly selective search plus deep position understanding (self developed), has weaknesses of the expected kind. There is a game where Stockfish 8 crushed Alphazero in 22 moves in a position with extremely deep tactics. This was exceedingly rare, and only uncovered when a thousand games were played, but it is sufficient to prevent making assumptions about solving chess based on Alphazero.

My opinion is that chess is likely drawn with best play, but this is purely an opinion.

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#### PAllen

To answer the titular question of this thread: a perfect chess player

#### PAllen

Previous link titled: Solving chess - Wikipedia. More generally, Game complexity - Wikipedia, Solved game - Wikipedia

Solutions can be
• ultraweak - the outcome of the game if both players play perfectly. In some cases, one can do the solution non-constructively, like with strategy-stealing.
• weak - an algorithm that gives one player the best outcome, no matter what the other player does, when starting from the initial position.
• strong - like the above, but starting from any position.
Tic-tac-toe is strongly solved, and it is easy to solve it with brute force. Connect Four is more difficult, but it has been solved in its classic configuration, 7 wide and 6 high, and other small sizes. Likewise, Go has been solved for 7*7 and smaller sizes, though Go is typically played on a 19*19 board.

The most difficult board-game solution so far is for checkers, and is required an enormous amount of computing power. Computers Solve Checkers&mdash;It's a Draw - Scientific American, Chinook - World Man-Machine Checkers Champion, Checkers Is Solved | Science The abstract:
Note that checkers is weakly solved.

#### PeroK

Homework Helper
Gold Member
2018 Award
Hi PeroK:

The 50 move rule was changed for a while but restored in 1992. It may well be modified again sometime in the future.

https://en.wikipedia.org/wiki/Fifty-move_rule
All of the basic checkmates can be accomplished in well under 50 moves. However, in the 20th century it was discovered that certain endgame positions are winnable but require more than 50 moves (without a capture or a pawn move). The rule was therefore changed to allow certain exceptions in which 100 moves were allowed with particular material combinations. However, winnable positions that required even more moves were later discovered, and in 1992, FIDE abolished all such exceptions and reinstated the strict 50-move rule.​

Regards,
Buzz
I know.

#### mfb

Mentor
Chess is unsolvable with traditional computers

Claude Shannon noted that a true chess solution would require storing 10^120 moves. This gets into age of the universe type computational times and impossible storage requirements with any conceivable computer technology other than maybe a huge quantum computer
No, that number would have to be analyzed but not stored. The result of analysis so far could use one of the compact tablebase representations. I once worked out this would only require a number of bits similar to atoms in the moon to play any position perfectly.
You don't have to analyze 10120 games. As an example: If both players move their knights out and back in again before starting a "normal" game this is a separate option in the game tree complexity of 10120, but you go through the same state as other games. No need to analyze them separately once you reach a common state again. You only need to know the value of each position and you have to consider each move from this position, you don't need to analyze all games (=all possible combinations of moves in the whole game).

We had this topic on page 1 already.

#### fluidistic

Gold Member
I would say it is strongly solved for 7 or fewer pieces. The winner as well as best play for both sides, for any such position, can be generated ( even though the sequence itself is not stored).
I'm wondering why you say strongly solved when en passant and castling (which are allowed in some positions with 7 pieces or less) are ignored from the current tablebases. Could you please explain what you have in mind?

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