chessrecreational-mathematics
The question concerns chess. I call a move forced if, in a given position, is the unique move consistent with the rules of the game. I wonder what is the largest integer $n$ such that there exists a legal position in which:
both black and white have forced moves for $n$ consecutive times;
the position is never repeated (with the same color having to move).
Notice that without condition 2., the answer is $\infty$.
Set is a card game that is very mathematical.
Set is played with a deck with $81$ cards. Each card corresponds to a point in affine $4$-space over $\mathbb Z/3$, with $3$ possible colors, shadings, shapes, and counts. The players must identify Sets, sets of $3$ cards corresponding to collinear points. Sets are also triples of cards which add up to the $0$-vector. The three cards pictured form a Set.
A natural question which arises during play is how many cards you can deal out without producing a Set. There can be $9$ cards in a codimension-$1$ subspace which do not contain a Set, corresponding to a nondegenerate conic in affine $3$-space such as $z=x^2+y^2$. There can be at most $20$ cards not containing a Set, corresponding to a nondegenerate conic in the projective $3$-space containing $10$ points.
Lest anyone think it is "just a game", this has led to quite a bit of research, including applications to computational complexity of matrix multiplication.
Here is a summary of solution proposed in this arXiv paper. A pair of positions on the $n\times n$ chessboard is constructed for which (1) there is a sequence $\sigma$ of legal chess moves leading from $P$ to $P'$; (2) the length of $\sigma$ cannot be less than $\exp\Theta(n)$.
The idea is to construct a position which consists essentially of $m$ tracks each of which is in some state in every moment. The set of possible states of $i$th track is a cycle group of order equal to $(i+3)$rd prime, and the position is defined uniquely by states of all tracks. A "move" increases every state by $1$ or decreases every state by $1$. Transforming $(0,0,\ldots,0)$ to $(1,0,\ldots,0)$ is possible by Chinese remainders but requires at least $p_5\ldots p_{m}=\exp(p_m+o(p_m))$ "moves".
(source)
The chess positions representing tracks use the above pattern, which is a specific example corresponding to $m=3$. The dots denote pawns, and we assume that the kings are located somewhere else on the board. In the paper it is explained (in different terms) why we can think of dark cycles as tracks, the positions of white bishops on them as states, and "moves" as minimal sequences of legal chess moves whose initial and resulting positions coincide up to a color of pieces.
Best Answer
This question has been answered at chess.stackexchange.com. It seems that if you allow promoted pieces, the current record is $n=9$.