question for mathfags , how many legal game positions can exist

>>7666772
That's a question for your lawyer.

>>7666772
Allot.

52!

https://en.wikipedia.org/wiki/Shannon_number

Not all legal positions can be reached from the standard starting position.

>>7666834
No shit.

>>7666789
kek

File: illustration.png (35KB, 722x674px) Image search: [Google]

35KB, 722x674px

>>7666834

Ergo your "legal" positions are not legal positions at all; rather, what you have in mind is a larger (but must be simpler to calculate) set something like pic related, versus legal chess configs.

Can /sci/ formally show the conjecture in pic related? I waved my hands.

>>7666996

Now that I think a bit, you could have two bishops of like color on like-colored squares via pawn promotion, and I haven't thought through my proposed "initial-state" flip though. A board absent kings is patently absurd though.

>>7666805
Uh, I think there are more than 52 positions possible.

>>7667003
With pawn promoting the number of combinations it's fucking nuts. In theory there could be 18 queens on the board.

Furthermore there would be almost no way to tell if a certain board position involving an usually high number of promotions could be reached through any formula.

>>7666996
What chess rules are we assuming here. Since putting your king in check is generally illegal if I remember right all you would need is a board with the two kings right next to each other. Could probably get some fucked up pawn positions too.

>>7667021

>What chess rules are we assuming here

The fucking rules of chess you mong, which are so simple that I will enumerate them (possibly missing one or two, but I trust that /sci/ can push up its glasses and correct any omissions:

1) -Kings, Queens, Bishops, Knights, Rooks and Pawns all move (and capture) in such-and-such a way. They only move and capture in their respective ways, except for the below:
2) - special extra-move: castling (look up details)
3) - special extra-move: en passant (look up details)
4) - special extra-move: promotion. A pawn reaching the other side of the board is /promoted/ to any one of a player's choice of queen, bishop, knight or rook. Only pawns can be promoted and only in this manner, and once so promoted they remain in play as whatever they were promoted to either until mate or capture. There's no limitation on a player's choice apart from what has been given above: if you can get four pawns to the other side, you can have four more rooks (or) four more queens (or) two more knights and two more queens, whatever you like, etc.

5) -You may not move any piece in such a way as to place your own king in check, because then in the very next move your opponent can capture your king, which defeats the object of the game.
6) -Rather, the game is over when it is impossible for you (or your opponent) to make any kind of move which gets the king out of check. Happily, when a board is in such a state, it's pretty easy for the humans involved to recognize this.

Please let me know if I missed anything.

I guess an upper bound would be 64 choose 16.

Thank you and goodnight.

This is just what my intuition tells me, but is the number of ways the pieces can be arranged 64!/(64 - n)! (where n is the number of pieces) if you regard each piece as a unique and special individual?

>>7667124
>You may not move any piece in such a way as to place your own king in check, because then in the very next move your opponent can capture your king, which defeats the object of the game.
All I wanted to know. It seemed so obvious that you could get a counterexample from this that I wasn't sure if you were including this one.

this has not been mentioned, but who has to move is also a part of a position and it can make a difference
f.e. consider the starting chess position; this is obviously in L, but the starting setup of pieces with black to move is not in L

>>7666996
note that there are positions that look quite normal, but can't be achieved in an actual game - when trying to trace back the previous moves, it turns out none were possible
for example consider position:
white Ka6, pawns a7, b6, c7, black Ka8, white to move
this looks like a pretty normal position, but if its in L, then what was black's last move?

>>7667124
if we want to be super autistic, we can include 50-move formula, i doubt it matters here though

>>7667207
Also you must consider that one configuration of pieces can represent several different positions. For example, for the same configuration, en passant may be possible, or the right to so capture may have already been forfeit, or it might be the other side to move, so 1 configuration = 3 positions. The same is true for castling... if the configuration of pieces suggests both sides could castle either side, legally between 0 and 4 of these moves are actually legal, and if both sides possibly have an en passant capture as well, and depending on side to move... one configuration of pieces can represent a LOT of positions!

>>7666772
Incidentally OP, the position in your pic related is NOT a legal chess position!

http://kirill-kryukov.com/chess/nulp/results.html
here are some partial results
computation of any sort is surely too hard, but can we use some stastistical methods to get an estimate?

>>7667292
Good eye. White King and Queen are backwards.

>>7667124
taking en passant into account or not would not make any difference, as the positions resulting from this move can be simulated without allowing en passant

>>7667007
Top kek

>>7667292
>>7667888
That does not mean this position cannot occur, as the G7 pawn seems to be missing

>>7667016
there is a sweet real game that ended with something like 7 knights

>>7666772
i defeated the 98th rank chess player in the world, by simply forming a 90 degree titled square with two bishops, and two castles with one of them aligned.

i sometimes keep moving my pawns much more forward to get to the king.

i also triangulate my pawns using the techmuller theory.

would white/blacks turn be considered part of the calculation?

ie x position black to move being a different position from white to move?

>>7668449
Picture of your setup?

>>7666996
Something like this? [math]\in C[/math] but [math]\not\in L[/math].

File: Screenshot from 2015-11-19 20-22-24.png (4KB, 228x231px) Image search: [Google]

4KB, 228x231px

>>7669032
Maybe I could actually attach the picture.

>>7666772
10^120 possible games, fewer legal positions.

>>7668437
White pawns on b2,c2,d2,e2,f2 (just visible),g2, pawns cannot go backwards, therefore these pawns have not moved.
therefore the white bishops can't have moved
therefore in this game there has never been a free square for either the King or Queen to move to so they can swap positions.

However...
If this is a chess960 game (a trivial one, since it's merely a mirror of the classical chess starting position), then yes, a proof game can be constructed...
after 1. Nf3 g5 2.Nxg5 h5 Black plays Bh6-Qf8-g7-f6-Ke8-f7-g7-h7-Qg7-f8-d8-Kg7-f8-e8-Bf8 while white plays Ng1-f3-g1-etc.

>>7667016
>In theory there could be 18 queens on the board
Could there, though? I don't see how you'd be able to get all of the pawns to the other side without killing some of the other guy's pawns.

>>7669163
It can be done if each side captures 4 of the opposing pieces with pawns.

>>7667124
There are also the draw conditions (stalemate, 50 move, threefold repetition, and when neither person can checkmate due to lack of pieces).

>>7666772
Exactly 1.0 fuckloads

It's actually a surprisingly low number. Only 2,501 positions can exist.

>>7666772
A lot

>>7669155
I stand corrected

>>7669169
What's the minimum number of moves it would take to have 18 queens?

legal chess positions, 10^40 - the number of different possible games, 10^120.

>>7668449
w0t

who cares u fucking nerds

>>7671552
do you have the calculations? I'm just curious

>>7671536
It takes 5 moves for a pawn to advance from the 2nd rank to the 8th x 16 pawns = 80 (half) moves or a 40 move game; but 8 pawns need to capture pieces that aren't on the 1st rank. So there must be at least 8 (half) moves by pieces so bare minimum 44 move game...
maybe not... there might be some optimisations there.
Then there's the problem of clearing the promotion squares (more piece moves), and 'accidental' checks requiring non-pawn moves.
I'd guesstimate 46 moves?
google search reveals a 48 move solution.

>>7671552
>Hardy (1999, p. 17) estimated the number of possible games of chess as 10^(10^(50))
You seem to be off by a few orders of magnitude.

idk like six

>put two good chess programs vs each other
>have them play 1000 games
>average the number of moves

>>7672242
So 8 moves per player to clear the back file, and 40 moves per player to advance the pawns. Half of clearing the back file is synonymous with setting up the piece to be taken by a pawn. I'd imagine there is more than one possible solution. Additionally, it most likely can only be done with bishop and knight sacs, since sacrificing a rook would require twice as many moves, Furthermore, there are 14 possible moves the two bishops can make to be sacked. Multiply this number by the two moves the knight can make, and that leaves 28 possible moves. Although, the knight and bishop sacs can occur at different points in the game, so that complicates the matter further. The math required to determine the total number of unique games is beyond me. I'm not 100% sure everything I've said is correct either.

>>7666789
Kek, underrated

>>7672641
being this stupid