Mathematical questions about Chess

So, there are a lot of interesting mathematical questions in chess, so I wanted a forum for asking and helping answer some of them.

Heres one I was just thinking about:

What is the maximum number of possible moves in a single legal position?

As a trivial upper bound, there must be less than 321 possible moves(assuming all of your pieces have their maximum mobility, even though this cannot happen)

Does that upper bound take pawn promotions (i.e. having multiple queens) into account?

Realistically, though, I'msure the number is far below 321. My guess is around 100.

Another fun question: How many moves is the longest possible game? This is obviously assuming draws are automatically claimed after 3 fold repetition and 50-move rule. 

Yeah, the upper bound assumes that you have 9 queens, and assumes that each piece has its maximum possible number of moves(EG each queen has 27 possible moves) even though you definitely can't arrange the pieces in a way that makes this happen.

As for the question about maximum possible moves, I was discussing that on another thread, and we came up with the answer 5948 moves or 5898 if KvK is drawn instantly due to lack of sufficient material

Another interesting one: In some positions, more than 50 moves can be necessary between irreversable moves to make progress, so the 50-move draw rule is really stupid in this case. There is some limit though on the number of necessary moves between irreversable moves though, so what is the smallest number that you could change the draw rule to that avoids these cases? 

I've heard something along the lines of-- there are more possible configurations of pieces on a chess board than there are molecules in the universe.  This seems too large to me-- does anyone know what the actual comparison is?  

thanks-- wow

jpr, it depends on how you define "configurations"

if you mean the number of ways that you could possibly put chess pieces on 64 labelled squares, thats 13^64, which is approximately 10^71. the number of atoms in the universe is estimated to be around 10^84, which is much larger than 13^64.

This though would count a lot of "configurations" that make no sense, and couldn't ever happen in a real game(such as 64 black bishops on a board)

even counting only the configurations that have legal numbers of pieces could give you an overcount of legal positions since other factors come into play when deciding if a position is legal or not.

The scarrily big number that comes into play with chess is when you think about the total number of possible chess games.

That is a hard number to calculate, but an(incredably unsharp) upper bound for it is 321^5948 or approx 10^14909, which is an astonomically large number compared to the number of atoms in the universe 10^84

This upper bound is a rather bad estimate though...

 

The total number of ways that a chess game can be played is far above the number of atoms in the universe. The number of board configurations, however, is pretty far below I think.

Number of rectangles on a chessboard is fun.  Placing 8 queens on an empty board such that none threaten each other is a fun one.

You may have seen these before though.

You can also generalize it.  When placing n queens on a nxn board how many possible configurations are possible?

I'll have to think about the number of rectangles.

Yeah, I heard of the 8 queens years ago, even though I've never thought about how many posible configurations there are(especially generalized)

There's a lot of good results for generalized knight's tours, any square board with sides a multiple of 4 can have a knights tour, any hexagonal boards(with hexagonal spaces) with sides bigger than 5 I think, you can have a knights tour, and triangular boards(with hexagonal spaces) with sides bigger than 6 I think, you can have a knights tour.

I'm not 100% sure about the lower bounds of 5 and 6, but I do know all the large boards have knight's tours.  

I'm sure the mathematicians who developed chess thought about this.

Thank you Benzodiaepine for your completely non-trolling comment.

-The professor 

No problem. But you forgot a Z there, professor.

@SeanEnglish--  thanks for the more detailed content.  Very helpful clarificiation!

@AlexDyer-- thanks very much 

Let me ask you something Sean....do you believe that the grand total # of possible (yet legal mind you) moves is infinite ?

Lola, I'm sorry I don't quite understand your question.

what kind of moves are you talking about?

the # of moves in any one position is finite

the # of distinct moves on a chess board(aka the number of ways you can set up a chessboard[legal or illegal position, doesn't matter here], then make a move[again legal or illegal, doesn't matter]) is finite

the # of distinct moves in a game(aka on move X piece A moved to square B) is infinite[irregardless of legality of the moves/games] since theoretically games can be infinitely long.

If you meant something other than these three, please clarify:) 

the 50 move rule and repeated position rule still need one of the players to claim the draw.

If you do put those restrictions on it though, chess becomes finite with the longest possible game being 5948 moves long(at least as best as I was able to calculate)  assuming that KvK isn't instantly called a draw by insufficient material.

I'm sorry, not moves. The grand total # of legal possible positions.

Finite or infinite ?

Finite.