Problems

In the following grid, how many different ways are there of getting from the bottom left triangle to the bottom right triangle? You must only go from between triangles that share an edge and you can visit each triangle at most once. (You don’t have to visit all of the triangles.)

Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).

How many subsets are there of \(\{1,2,...,n\}\) (the integers from \(1\) to \(n\) inclusive) containing no consecutive digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).
For example, when \(n=3\), we have \(8\) subsets overall but only \(5\) contain no consecutive integers. The \(8\) subsets are \(\varnothing\) (the empty set), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,3\}\), \(\{1,2\}\), \(\{2,3\}\) and \(\{1,2,3\}\), but we exclude the final three of these.

A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: \(3\) points for a win, \(1\) point for a draw and \(0\) points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points.
Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team?

There is a secret gathering of a group of \(n\) aliens in a very dark room. You cannot see anyone in the room, but you hear the following questions.

1. “Is at least one of us a Goop?"

2. “Is the number of Goops amongst us an even number?"

3. “Is the number of Goops amongst us a multiple of 3?"

4. \(\dots\)

5. “Is the number of Goops amongst us a multiple of \(n\)?"

What are all the possible values of \(n\) such that this gathering can happen? Note that each of the \(n\) aliens have asked exactly one question.

You meet a group of \(n\) aliens. The first alien asks “is at least one of us a Goop?", the second alien asks “are at least two of us Goops?", the third asks “are at least three of us Goops?" and so on until the final one says “are at least \(n\) of us Goops?".

How many Goops are there?

Cut a deck of \(4\) cards. Are any of the cards in the same place as they were before?

We have a deck of \(13\) cards from Ace to King. Let Ace be the first card, \(2\) the second card and so on with King being the thirteenth card. How can you swap \(4\) and \(7\) (and leave all other cards where they are) by only switching adjacent pairs of cards?

How many permutations of 13 cards leaves the third card where it started?

Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(6\times 6\) rectangle?