### Cube Net

How many tours visit each vertex of a cube once and only once? How many return to the starting point?

### Binary Sequences

Show that the infinite set of finite (or terminating) binary sequences can be written as an ordered list whereas the infinite set of all infinite binary sequences cannot.

### Binomial Coefficients

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

# Groups of Sets

##### Stage: 5 Challenge Level:

The binary operation $*$ for combining sets is defined as $A*B =(A\cup B) - (A\cap B)$.

Prove that $G$, consisting of the set of all subsets of a set $S$ (including the empty set and the set $S$ itself), together with the binary operation $*$, forms a group. You may assume that the associative property is satisfied.

Consider the set of all subsets of the natural numbers and solve the equation $\{1,2,4\}*X = \{3,4\}$.