Puzzles and Problems

Are all positions reachable?

This first problem relates to most of the other problems, which are likely to be attacked by constructing board positions without necessarily constructing a way to reach them from the standard starting position.

The question: starting from the standard starting position, what class of board positions are unreachable. I can think of two:
diagram of a matching position All positions with two or fewer stones (of any mix of colors) are unreachable, because one of the players would have won before the last move was made. 
diagram of the described position All positions with no stones of one color are unreachable (because the game would have passed through "one stone of color remaining", which is a win. 
My initial conjecture was that all other positions are reachable, but that has been mooted by Joseph (JaKe) Kisenwether , who pointed out two additional classes of unreachable positions.
Diagram of the position Positions in which no piece has been captured and both white and black have all of their pieces united. The reason that this is unreachable is that whoever didn't move last must have united his pieces on the move before last and the game would have ended. 
Diagram of the position Positions in which both white and black have all of their remaining pieces connected AND every piece borders a piece of the other color. This is unreachable for a similar reason to the class above: even if the last move was a capture, the player who made the preceding move would have completed a winning configuration and the game would have ended. 

Jorge Gómez Arrausi  has pointed out a related class of unreachable positions
Diagram of the position Positions in which one player is connected , but  neither player could have created the connected position.  In positions such as the one at the left, it is straighforward to generate all possible white predecessor positions, which have one of the white stones at some other position on the board, and possibly has a black stone where the white stone is now.  In the board at the left, there are exactly 330 such hypothetical white predecessors.  Similarly, black must have created this position by capturing an isolated white stone, and since all black stones are adjavent to white, black could not have just created this position. 

The common thread in these classes of unreachable position seems to be the argument that the game would have already ended. Are there any unreachable conditions that dont depend on this argument? 

How Many Checks are possible?

Actually, several related questions. Starting from a position where neither player has passed up an opportunity to win, what is the maximum number of winning moves available?

Related sub-questions:

  1. What if the loser cooperates?
  2. What is the maximum number of threats to win that can be refuted? That is to say, the situation changes from "no immediate threat" to "n threats to win" and back to "no immediate threat".
Send in your solutions.

Shortest Game Constructions

What is the shortest possible game, assuming both players cooperate to end the game as quickly as possible. The best known constructions are 9 moves for the standard setup, and 11 moves for the scrambled setup. The 11 Move construction for scrambled eggs was discvered in May 1999 by enthusiast JorgeGómez Arrausi Jorge Gómez Arrausi, who improved on the previous best result of 13.

The Stalemate

Are positions possible where neither player can move? I've fairly convinced myself that it can't occur on a normal board, but with enough extra pieces (for example, a fully packed board!) it clearly becomes possible. What is the minimum number of pieces on an 8x8 board to create a stalemated position?
Diagram of the position This elegant solution uses just 24 pieces, the same number as the normal starting position.  This position was devised by Joseph DeVincentis, as solution to a puzzle for a rather strange game known as  Ackanomic.

His message describing how he derived this solution is quite informative 

This stalemate position has a predecessor, seen at the right, but of course, that position has more than the standard number of black pieces, so could not arise in a regulation game. 

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