Below is a five-by-five chess board with the central square poked out. Place a knight, bishop, rook or queen in the upper-left corner and find a path that ends at the bottom-right corner that visits every square on the board along the way without repeating. With which piece or pieces is this possible? If a bishop, rook or queen glide across squares on the way to a stop, those intervening squares count as visited. Pieces cannot land on or glide over the hole. The knight hops, so only squares it lands on count as visited, and it can also hop over the hole.
Knights can move in an L shape: two squares vertically followed by one square horizontally or one square vertically followed by two squares horizontally.
Bishops can move diagonally.
Rooks can move horizontally or vertically.
Queens can move horizontally, vertically or diagonally.
Amanda Montañez
Of these chess pieces, only the queen can find a path. Bishops only move diagonally, so they never move between squares of different colors. A bishop starting in the top left corner, therefore, could never reach the dark squares. Because every square a rook passes through during its moves gets counted as part of its path, we can treat the rook as though it only moves one square at a time. Looking at it this way, notice that it alternates between light and dark squares. Its first square is light, so its second square will be dark. Then its third will be light and fourth dark, and so on. Continuing this pattern, the 24th square that the rook visits, an even number, must be dark. The board contains 24 squares, so the rook must visit the “finish” square 24th—but it cannot, because the “finish” square is light. The knight also alternates colors, so the same argument applies to it. The queen has many viable paths, with one example below.
Amanda Montañez
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