Date modified: Mon 2023-05-15 . 09:28 AM
Date created: Sat 2023-10-07 . 09:02 PM
# Random conflict free knights. Consider a $3\times 3$ chess board, where each square has $p=1/2$ probability of having a knight or not. What is the probability that this board is free of conflict, that is to say, no knight attacking another knight? ![[---images/Random conflict free knights 2023-05-04 14.26.13.excalidraw.svg]] ///hint./// Note the center square does not matter at all. (why?) As for the remaining 8 squares, they form a closed knight tour of length 8, namely a cyclic graph of 8 vertices. Now note adjacent vertices imply two knights attacking each other. ![[---images/Random conflict free knights 2023-05-04 14.42.11.excalidraw.svg]] /// #puzzle #chess #counting #probability #graph-theory