Problems
We wish to lock a vault with different locks. The vault committee has \(11\) members, each of whom has keys to some of the locks, but not all of them.
What is the smallest possible number of locks that we need to lock the vault so that each group of \(6\) members can open it together with the keys they have, but no group of just \(5\) members can ever do it? Note that a lock can have multiple keys that open it and a person can have keys to more than one lock.
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 \(\angle B = \angle D = 90^{\circ}\). Show how the plane can be tiled with pentagons congruent to the given one.
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.
In the \(6\times7\) large rectangle shown below, how many rectangles are there in total formed by grid lines?
Explain why a position \(g\) is a winning position if there is a move that turns \(g\) into a losing position. On the other hand, explain why a position is a losing position if all moves turns it into a winning position.