Initially, on each cell of a \(1 \times n\) board a checker is placed. The first move allows you to move any checker onto an adjacent cell (one of the two, if the checker is not on the edge), so that a column of two pieces is formed. Then one can move each column in any direction by as many cells as there are checkers in it (within the board); if the column is on a non-empty cell, it is placed on a column standing there and unites with it. Prove that in \(n - 1\) moves you can collect all of the checkers on one square.
Show that a knight’s tour is impossible on a \(3\times3\) grid.
Show that two queens together can attack every square on a \(4\times4\) grid, but one queen on her own cannot do it. This type of problem is called ‘queen’s domination’.
How many queens can you place on a \(4\times4\) grid so that none of them attack each other?
Show an knight’s tour on a \(5\times6\) chessboard. That is, a path where a knight starts at one square, and then visits every square exactly once, making only moves legal to a knight.
Show how five queens can dominate a standard \(8\times8\) chessboard. That is, each square is attacked by some queen.
How many independent queens can you place on a \(5\times5\) grid? That is, so none of them attack each other.
How many ways can you place \(8\) rooks independently on a chessboard? That is, so that none of them attack each other.
Why are there no closed knight’s tours on an \(n\times n\) grid when \(n\) is odd? A knight’s tour is closed if you can get to the first square from the last square by a knight’s move.
Show how to place fourteen dominating bishops on a standard \(8\times8\) chessboard. That is, every square either contains a bishop, or is attacked by some bishop.