Haft Schrödinger Chess¶

Description¶

The nature of the major pieces is unknown at first, and is determined by the moves they make.

The goal is, for a given series of moves, to check whether there exists an initial choice of piece natures so that:

There is no more than 1 king, 1 queen, 2 rooks, 2 bishops, 2 knights in each color
Every move played is consistent with these piece natures
No move played by a given color left their king in check

We discard the following rules:

$s \in \mathcal{S} = [1, 64]$: square of the chess board
$c \in \mathcal{C} = {0, 1}$: color
$i \in \mathcal{I} = [1, 16]$: index of a piece for a given color. Belongs to $\mathcal{J} = [1, 8]$ for the major pieces depending on their initial position on the first row, and to $\mathcal{K} = [9, 16]$ for the pawns.
$n \in \mathcal{N} = {K, Q, R, B, N}$: nature of a piece
$t \in [1, T]$: time, ie. number of turns played

At every given time we know the color of the player that moves: let’s call it $c_t = t [2]$.

These variables can be calculated at any time without knowing the actual nature of the pieces.

$p_t(s, c, i) = 1$ if square $s$ of the board contains piece $i$ of color $c$ at time $t$
$a_t(s, c, i, n) = 1$ if square $s$ of the board would be threatened by piece $i$ of color $c$ at time $t$, assuming its nature was $n$
$f_T(c, i, n) = 1$ if until time $T$, piece $(c, i)$ has performed a move forbidden for nature $n$

$d_t(c, s)$ is the number of pieces from color $c$ attacking square $s$ (number of dangers)

$$ d_t(c, s) = \sum_i \sum_n a_{t}(s, c, i, n) \cdot x(c, i, n)$$
$k_t(c, s) = 1$ if the king from color $c$ is located on $s$

$$ k_t(c, s) = \sum_i p_t (s, c, i) \cdot x(c, i, K)$$

Variable domain

$$\forall (c, i, n), ~x(c, i, n) \in {0, 1}$$
Variable interpretation: exactly one nature per piece

$$\forall (c, i), ~\sum_n x(c, i, n) = 1$$
Right number of pieces of each nature

$$ \forall (c, n), ~\sum_i x(c, i, n) \leq x_{max} (n)$$
Feasible moves: no piece can perform forbidden moves

$$\forall (c, i, n), ~x(c, i, n) \leq 1 - f_T(c, i, n)$$
Check rules: no player can leave its king in check

$$ \forall t, ~ \forall s, ~ d_t(c_t, s) \leq 16 (1 - k_t(c_{t-1}, s))$$