Chess will never be solved, here's why

#1667
Yes, the idea is to examine for legality by providing a proof game from the initial position to the given position if possible. Inspection of the accuracy of the proof game then determines if that give position can arrive in a reasonable game with reasonable moves.

Okay.
I'll add that if white's pawns are moving up the board -
then it appears the game could end with Kc4 and Bxd3 double check with mate.  

And if white's pawns are coming down the board - its already mate.
I'll assume for now you're referring to that position.  
If the supercomputers had to generate a 'proof game' for every checkmate position - 
that would be Terrible.
And many of those would probably be Helpmates.

Much better:
Generate checkmate and stalemate positions and hopeless draws and other positions by first excluding all possibilities that couldn't be legal anyway.  Like adjacent Kings.
Then exclude some more by instantly finding them illegal.  In nanoseconds.
In other words - don't worry about how it got there.  

Someday the supercomputers might find a trillion positions that are legal looking on the board but somehow 'could never have got there'.
How many types of such positions could there possibly be?

#1670
White's pawns go up, black's pawns go down as is the usual convention for chess diagrams.
Indeed Kc4 Bxd3#.
The questions are:
1) Is the position legal?
2) What proof game leads to the given position from the initial position?
It is possible to determine that manually for 1 positions, but not for 10,000.

I'm looking at the pawns at a5 and a6. 
And how they might have got there.
Black is missing two pawns that he could have lost by them being captured - or by promoting them. 
But white is missing 3 pawns that he could have lost by them being captured or promoted.

#1672
So black has captured a pawn or a promoted piece ...bxa5.

I don't know if it would help if there's a proof that there has to have been at least one promotion here.

black's a5 pawn has to have taken at a6 or a5. 
It has to have come from b7 or c7.  
No promotions would mean a white b-pawn has to have taken on the
a-file to be taken by a black pawn taking on a6 or a5. 
No white d-pawn could get there because that would take 3 captures and black has only lost two 'items'.  
Which would mean in turn (with no promotions) that white's b-pawn would have had to have taken a 3rd black pawn on the a-file.
Which would have in turn have had to take a 3rd white pawn on the a-file - which is Impossible.
Therefore - there had to be at least one promotion in this position.

I've used Natch. How long it takes varies a lot depending on the computer it's running on. On my laptop, it's not much good for proof games longer than about 30 moves. A supercomputer would do better. Also, you have to tell Natch the ply count of the proof game you want to find, which complicates what you're wanting to do with it.

Anyway, I made a 47.5 move proof game for your position, and I'll give the position to Natch with a ply count of 95. Maybe it will find my proof game if I let it run for a week.

@tygxc I've proven two posts ago that there has to be at least one promotion in the position you posted in #1664.
But that wouldn't seem to interfere with legality.

#1675
Good to know for sure that it is legal.
1 week per position.
I was looking for a way to speed up evaluation of the 10,000 samples to determine how many could occur in a reasonable game wilth reasonable moves.

If you can find a cheap quantum computer on eBay you may be in business.

I do. It looks at the position, which includes not only information on where the pieces are, but also which of the 4 castling options are legal and if there is a possible e.p. capture.  What these options are for castling depends on how you set up the board.

• If a king and rook arise from removing other pieces from a full board, they do have castling rights.
• Any king or rook placed on the board (or moved on the board, including back to his home square) won't have castling rights

Those are the empirically inferred, reasonable choices of a Stockfish programmer.

To see what is going on when setting up positions, click the share icon at the bottom, then the PGN tab to see the FEN. The second text field can be a "-" or a string of some subset of {K, Q, k, q}, indicating castling rights for white and black.

Hope that summary makes it crystal clear.

@Elroch
Looking at the FEN would only tell you if its allowing castling - 
not how it got that way.
So the FEN does not determine whether castling is allowed - only reflects whether it is or not.
Hope that summary makes it crystal clear. 
To you.  

I wouldn't know if its legal or not. 
But there has to have been at least one promotion there.
Promotion of white or black pawn?  I don't know.  Yet.
Maybe there's a way to show there had to be more than one promotion.
Is there a 'snag' if there was five promotions?  I don't know yet.
For each pawn captured - that would be one less promotion possible.
But perhaps there were no pawn captures - 
which would mean there were pieces captured on the a-file.  Not pawns.
So far - there's a max of four pawns being captured.  
Could not be five - as at least one promotion is now proven.

#1680
It is legal.
@n9531|1 made a 47.5 move proof game

@Elroch
The FEN code tells Stockfish whether castling is legal or not - 
but that doesn't determine how it gets that way in the first place.
Also - the FEN code is not going to determine that - if there was no FEN code used to set up the position.  However much FEN code might be then generated to 'tell' Stockfish whatever.
Hope both points are now crystal clear.  To you.
You didn't seem to know about the 'clear board' point.
That was found out later.
All three points stand.
Clear yet?

And - the 'hypothesis' has not been proven to be incorrect yet either.
'Clear board' could represent a different mode than the mode of starting from an opening position.  That could be called 'game mode'.  

So if anything - the hypothesis has now been proven correct.
Not incorrect.
So that's now Four points that continue to stand.
You can keep trying to argue that the accuracy of one point negates the accuracy of another.
But it doesn't follow.  Perhaps five years from now - you'll still be on the same tack.  Like five years ago.  Its okay though.

I wouldn't know if its legal or not.  But there has to have been at least one promotion there.
Promotion of white or black pawn?  I don't know.  Yet.
Maybe there's a way to show there had to be more than one promotion.
Is there a 'snag' if there was five promotions?  I don't know yet.
For each pawn captured - that would be one less promotion possible.
Update:  There had to be at least one promotion in tygxc's posted position.
Which means there could only have been a maximum of four captures of pawns and there had to have been at least one capture of a piece. 
And that capure had to have been on the a-file too.

@n9531  tygxc's post referred to my post #1675 ... its there in the quote.
As to your 'game' proving its legal - that's nice.
As to 'having to tell' the number of ply to the program ...
is that a maximum number?  Could you clarify further?

As to how all of this pertains to 'solving' chess - it seems to underline my point that in generating positions - the computer shouldn't worry about if the position could 'get there' legally or not.
Just not generate positions that couldn't be legal - and also instantly dismiss those that take a nanosecond to so dismiss.   
Precisely how much of either could depend on the particular programming used ... and yes there's choices about that !  

#1684
"As to how all of this pertains to 'solving' chess "
As said before weakly solving chess needs 1 nanosecond per position.
As said there are 10^37 positions without excess promotions.
There is a sample of 10,000 positions without excess promotions.
It is now of interest to see how many of these, if any at all, can result from a reasonable game with reasonable moves.