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There are $A_{m,n}$ pieces, rotated giving $B_k$ with values $B_{k,s}$ pokiness. Introduce new variables $V_{m,n}$ to indicate right wall of $(m,n)$ cell. It will either be true or false. $\top=\langle$ and $\bot = \rangle$, Similarly $V_{m,n}$ to indicate bottom wall of $(m,n)$ cell. We will have $M$ and $N-1$ dimensions for these to avoid extra clauses for the end of the walls. Similar logic for bottom walls $H$ $X_{k,m,n}$ and $Y_{k,s}$ continue to indicate one-hot mapping of their location and orientation. (Remember B takes 3 values, and H takes two values) \begin{align*} X_{k,m,n} \land Y_{k, s} &\implies (B_{k,RIGHT \oplus s} = V_{m,n}) \\ X_{k,m,n+1} \land Y_{k, s} &\implies (B_{k,LEFT \oplus s} = V_{m,n}) \\ X_{k,m,n} \land Y_{k, s} &\implies (B_{k,BOT \oplus s} = H_{m,n}) \\ X_{k,m,n} \land Y_{k, s} &\implies (B_{k,TOP \oplus s} = H_{m,n+1}) \end{align*}
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Wed, 27 Dec 2023 20:45 GMT