Let ##G = (V, E)##. A node in the graph represents a distinct ephemeral position. Formally, an ephemeral position is a tuple ##P = (B, C, E, H)## where ##B## is an 8x8 matrix representing the board layout, with each element ##B_{ij} \in \{ \text{pieces} \} \cup \{ \text{empty} \}##. ##C \in \{ \text{true, false} \}^4## represents castling rights (White Kingside, White Queenside, Black Kingside, Black Queenside). ##E \in \{ (i, j) | 1 \le i, j \le 8 \} \cup \{ \text{null} \}## represents the en passant square, if any. ##H \in \mathbb{Z}_{\ge 0}## is the half-move clock (used for the fifty-move rule). Let ##V## be the set of all possible ephemeral positions, and let ##M = |V|## denote its cardinality. The edges of the graph, represented by the set ##E##, are implicitly defined by the legal moves in chess. Define the adjacency matrix ##A## of size ##M \times M##, where each element ##A_{uv}## is defined as ##A_{uv} = \begin{cases} 1, & \text{if there exists a legal move from position } u \in V \text{ to position } v \in V \text{ (i.e., } (u,v) \in E \text{)} \\ 0, & \text{otherwise} \end{cases}##. Here, "legal move" is defined according to the official rules of chess, including piece movement, captures, checks, checkmates, and special moves like castling and en passant. The total number of distinct sequences of moves up to a maximum length ##n_{\text{max}}## can be expressed as $$ N = \sum_{k=1}^{n_{\text{max}}} \sum_{u=1}^M \sum_{v=1}^M (A^k)_{uv} $$. Here, ##(A^k)_{uv}## denotes the element in the ##u##-th row and ##v##-th column of the matrix ##A## raised to the power of ##k##, which represents the number of walks of length ##k## from node ##u## to node ##v## in the directed graph ##G##. Apparently identical board layouts can occupy different nodes due to differing histories (e.g., castling or en passant rights), leading to different possible future moves from the same board arrangement. The threefold repetition rule introduces further complexity by allowing for early termination (draws) upon certain repeated states, effectively adding structure to the graph. Although ##A## is extremely sparse (most positions connect to few others), the magnitude of ##M## makes the explicit computation of ##(A^k)_{uv}## for large ##k## computationally infeasible. This problem belongs to the class of #P-complete counting problems, for which no known polynomial-time algorithm exists, even on quantum computers. While a universal quantum computer might offer advantages for certain algorithms, no known quantum shortcut exists for computing this full enumeration of walks. Any universal computer, whether quantum or classical, would face astronomical memory or time requirements to handle all entries of ##A^k##. So the precise calculation of ##N## remains effectively unattainable.
As a side note, this reminds me conceptually of Wolfram's "
Ruliad," representing the entangled limit of all possible computational processes.
