Let us define a pawn structure as any legal configuration of pawns on the chess board. Calculate the total amount of discrete pawn structures that can arise from legal positions for White.
The placement of other pieces on the board is irrelevant, as is the amount of moves it takes to arrive into this position.
So this:
http://en.lichess.org/analysis/rnbqkbnr/ppp2ppp/4p3/3p4/4PP2/3P4/PPP3PP/RNBQKBNR_w_KQkq_-And this:
http://en.lichess.org/analysis/4k3/8/8/8/4PP2/3P4/PPP3PP/4K3_w_KQkq_-Are counted as equal
Since a pawn that arrives to the 8th rank is no longer a pawn, it is not considered a part of the pawn structure. However its absence does create a distinct pawn structure (although this would not be counted any differently than if it were taken by another piece.)
Another thing to consider is that the a pawn being on any other file also creates a distinct pawn structure, but it can only travel as far as the f file in this case though.
I've yet to try and calculate this myself, but it seems like a fairly difficult challenge.
Also I neglected to mention at if the b pawn moves to the c file and the d pawn is gone, then you won't be able to tell just by looking at the structure which pawn it was. Consider this a distinct position from if the d pawn moved to the c file and the b pawn is gone.
maybe I give it a try..
what about to find first a minimum and maximum Border.. so you can say there have to be between x and y positions of that kind.
It's easier and you are more related to the problem.. and the next step is easier?
#1 Question: does this exercise assume the standard chess starting position, or any possible starting position?
Let's assume that the chess board is not 8*8 (effectively 7*8 since the 8th row does not count), let's assume it is m*n and perhaps we can find a recursive formula or some other interesting things.
Let M(m,n) be the number of pawn configurations. So our job is to calculate M(7,8).
#1 Please define clearly what constitutes a pawn configuration (PC). In #2 you said that even if the pawns are located in the same places the very fact that they come from different columns make it a different pawn configuration.
May I ask:
1.Assume that there are only two pawns, c3 and c4 in the pawn configuration. If c3 comes from b2 while c4 comes from c2, does that constitute a different configuration from the situation where we have the same pawns but the b2 pawn is now on c4 while the b3 pawn is now on c3? (i.e., do we need to label these pawns and a different labelling constitutes a different PC?)
2.Does a PC depend on what happened before the PC appears? For example if we have a situation where we only have one pawn from c column on e7. Does it make a difference if the f pawn get removed before the g pawn or the other way around?
#4 Yes assume standard stating position, and all positions that can be achieved in a legal amount of moves (I don't think the 50 move rule should be a problem but I maybe wrong)
#5 since whites pawns start in the second rank it can be generalized further to a field of 6 by 8
#6 Alright I'll clarify that by generalizing it a bit. Assume that every pawn is labeled with its starting file. Any unique placement of pawns across the board is then considered a unique PC. The rule being that you have to be able to obtain that position legally (i.e. the a pawn can never reach the g or h file).
Hello!
I decided to sign in since I could not resist a mathematical challenge...... ;)
However, I'm sorry but your question is very hard to understand.
Would you like to know the possible PC's by move 50 and only 50 or is that rule absent in your challenge?
Would you like the challengers to take into account of En Passant?
What about when the challenger does not wish to capture his opponent's pieces, may the challengers claim the position dead?
I urge you to establish more definite constraints or else this challenge may not be mathematical, but instead philosophical! xDD
In any case, I'm on my holiday and will publish my result in the days to come as I understand it now.
Please pardon my English.
Friendly Greetings,
FinnishMathematician
#8 the majority of the positions will probably be completely dead, but since they can be achieved somehow they should be counted! For example:
http://en.lichess.org/analysis/4k3/P7/P7/P7/P7/P7/P5PP/4K3_w_Q_-I believe it can be reached, but I think it would require for black to promote a few pawns in order to move those pawns
en passant should definitely be taken into account but I don't know if that'd really matter because any position obtainable through it is probably obtainable without it. Don't know about 50 yet because I'm having a hard time seeing if there are really any positions that complex.
