Problems

Show how five queens can dominate a standard $8\times 8$ chessboard. That is, each square is attacked by some queen.

How many independent queens can you place on a $5\times 5$ 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 $8\times 8$ chessboard. That is, every square either contains a bishop, or is attacked by some bishop.

Place eight independent queens on a standard $8\times 8$ chessboard.

Show how to swap the two pairs of knights on the following strangely-shaped grid. That is, the knights make one move at a time, and you’re trying to get the black nights to where the white knights are, and the white knights to where the black knights are.

Let $n$ be a positive integer. Prove that it’s impossible to have a closed knight’s tour on a $4\times n$ grid.

Find an open knight’s tour on a $2\times 2\times 2$ cube.

Four football teams play in a tournament. There’s the Ulams ($U$), the Vandermondes ($V$), the Wittgensteins ($W$) and the Xenos ($X$). Each team plays every other team exactly once, and matches can end in a draw.
If a game ends in a draw, then both teams get $1$ point. Otherwise, the winning team gets $3$ points and the losing team gets $0$ points. At the end of the tournament, the teams have the following points totals: $U$ has $7$, $V$ has $4$, $W$ has $3$ and $X$ has $2$.

Work out the results of each match, including showing that there’s no other way the results could have played out.