Problems

Place eight independent queens on a standard \(8\times8\) 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 \(2\times2\times2\) cube.

Let \(s\) and \(t\) be two positive integers. Can we have \(s^2=t^2+2\)?

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.

Imagine a \(5\times6\) rectangular chocolate bar, and you want to split it between you and your \(29\) closest friends, so that each person gets one square. You repeatedly snap the chocolate bar along the grid lines until the rectangle is in \(30\) individual squares. You can’t snap more than one rectangle at a time.

The diagram shows a couple of choices for your first two snaps. For example, in the first picture, you snap along a vertical line, and then snap the left rectangle along a horizontal line.
How many snaps do you need to get the \(30\) squares?

Prove that \(n^{n+1}>(n+1)^n\) for integers \(n\ge3\).

What is the following as a single fraction? \[\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{98\times99}+\frac{1}{99\times100}.\]

Prove that \(3\) always divides \(2^{2n}-1\), where \(n\) is a positive integer.