Problems

In chess, knights can move one square in one direction and two squares in a perpendicular direction. This is often seen as an ‘L’ shape on a regular chessboard. A closed knight’s tour is a path where the knight visits every square on the board exactly once, and can get to the first square from the last square.

This is a closed knight’s tour on a $6\times 6$ chessboard.

Can you draw a closed knight’s tour on a $3\times 3$ torus? That is, a $3\times 3$ square with both pairs of opposite sides identified in the same direction, like the diagram below.

You may remember the game Nim. We will now play a slightly modified version, called Thrim. In Thrim, there are two piles of stones (or any objects of your choosing), one of size $1$ and the other of size $5$.
Whoever takes the last stone wins. The players take it in turns to remove stones - they can only remove stones from one pile at a time, and they can remove at most $3$ stones at a time.
Does the player going first or the player going second have a winning strategy?

A circle is inscribed in a triangle (that is, the circle touches the sides of the triangle on the inside). Let the radius of the circle be $r$ and the perimeter of the triangle be $p$. Prove that the area of the triangle is $\frac{pr}{2}$.

We meet a group of people, all of whom are either knights or liars. Knights always tell the truth and liars always lie. Prove that it’s impossible for someone to say “I’m a liar".

We’re told that Leonhard and Carl are knights or liars (the two of them could be the same or one of each). They have the following conversation.

Leonhard says “If $49$ is a prime number, then I am a knight."

Carl says “Leonhard is a liar".
Prove that Carl is a liar.

Let $p$ be a prime number greater than $3$. Prove that $p^2-1$ is divisible by $12$.

Split the numbers from $1$ to $9$ into three triplets such that the sum of the three numbers in each triplet is prime. For example, if you split them into $124$, $356$ and $789$, then the triplet $124$ is correct, since $1+2+4=7$ is prime. But the other two triples are incorrect, since $3+5+6=14$ and $7+8+9=24$ are not prime.

A family is going on a big holiday, visiting Austria, Bulgaria, Cyprus, Denmark and Estonia. They want to go to Estonia before Bulgaria. How many ways can they visit the five countries, subject to this constraint?

Let $p$, $q$ and $r$ be distinct primes at least $5$. Can $p^2+q^2+r^2$ be prime? If yes, then give an example. If no, then prove it.

How many subsets of $\{1,2,...,n\}$ (that is, the integers from $1$ to $n$) have an even product? For the purposes of this question, take the product of the numbers in the empty set to be $1$.