Problems

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?

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(6\times8\) rectangle.

Prove that one can tile the whole plane without spaces and overlaps, using any non self-intersecting quadrilaterals of the same shape. Note: quadrilaterals might not be convex.

It is impossible to completely tile the plane using only regular pentagons. However, can you identify at least three different types of pentagons (each with at least two different corresponding sides AND angles) that can be used to tile the plane in three distinct ways? Here essentially different means the tilings have different patterns.

Draw how to tile the whole plane with figures, made from squares \(1\times 1\), \(2\times 2\), \(3\times 3\), where squares are used the same number of times in the design of the figure.

Show how to cover the plane with triangles of the following shape.