Evaluate \(a(4,4)\) for the function \(a(m,n)\), which is defined for integers \(m,n\ge0\) by \[\begin{align*} a(0,n)&=n+1\text{, if }n\ge0;\\ a(m,0)&=a(m-1,1)\text{, if }m>0;\\ a(m,n)&=a(m-1,a(m,n-1))\text{, if }m>0\text{, and }n>0. \end{align*}\]
Show that a knight’s tour is impossible on a \(3\times3\) grid.
Show that there is a way of placing two queens on the board such that between them they attack every square on a \(4\times4\) grid. But also show that one queen on her own cannot do it. This type of problem is called ‘queen’s domination’.
A set of chess pieces is called independent if none of them can attack each other. How many independent queens can you place on a \(4\times4\) grid?
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?
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.