Tommy has written 6 letters and addressed 6 envelopes. He then forgot which letter goes where and put them randomly such that no letter goes in the right envelope. In how many ways can he do this?
Annie and Hanna are preparing some Christmas baubles. They want to paint each bauble all in one colour. They have \(7\) different colours of paint and \(26\) baubles to paint. In how many ways can they do this? Two ways are considered the same if the numbers of baubles of each colour are the same. Each bauble has to be painted but not all the colours need to be used.
An \(8 \times 8\) square is divided into \(1 \times 1\) cells. It is covered with right-angled isosceles triangles (two triangles cover one cell). There are 64 black and 64 white triangles. We consider "regular" coverings - such that every two triangles having a common side are of a different colour. How many "regular" covers are there?
You are given a pentagon \(ABCDE\) such that \(AB = BC = CD = DE\), and \(\angle B = \angle D = 90^\circe\). Show how the plane can be tiled with pentagons equal to the given one.
What is logically the opposite of the statement “every \(n\) is odd or \(p<q\)"?
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?
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.
Four different digits are given. We use each of them exactly once to construct the largest possible four-digit number. We also use each of them exactly once to construct the smallest possible four-digit number which does not start with 0. If the sum of these two numbers is 10477, what are the given digits?
How many ways can the numbers \(1,1,1,1,1,2,3,\dots,9\) be listed in such a way that none of the \(1\)’s are adjacent? The number 1 appears five times and each of \(2\) to \(9\) appear exactly once.