# Sets and Groups - April 2005, All Stages

## Problems

### The Set of Numbers

##### Stage: 1 Challenge Level:

Can you place the numbers from 1 to 10 in the grid?

### Posting Triangles

##### Stage: 1 Challenge Level:

If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?

### Sorting Symmetries

##### Stage: 2 Challenge Level:

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

### Butterfly Cards

##### Stage: 2 Challenge Level:

Four children were sharing a set of twenty-four butterfly cards. Are there any cards they all want? Are there any that none of them want?

### Sets of Four Numbers

##### Stage: 2 Challenge Level:

There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

### Consecutive Seven

##### Stage: 3 Challenge Level:

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

### Arithmagons

##### Stage: 3 Challenge Level:

Can you find the values at the vertices when you know the values on the edges?

### Shuffle Shriek

##### Stage: 3 Challenge Level:

Can you find all the 4-ball shuffles?

### Stars

##### Stage: 3 Challenge Level:

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

### Spot the Card

##### Stage: 4 Challenge Level:

It is possible to identify a particular card out of a pack of 15 with the use of some mathematical reasoning. What is this reasoning and can it be applied to other numbers of cards?

### Sheffuls

##### Stage: 4 Challenge Level:

Discover a handy way to describe reorderings and solve our anagram in the process.

### Groups of Sets

##### Stage: 5 Challenge Level:

The binary operation * for combining sets is defined as the union of two sets minus their intersection. Prove the set of all subsets of a set S together with the binary operation * forms a group.

### What's a Group?

##### Stage: 5 Challenge Level:

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

### Binary Sequences

##### Stage: 5 Challenge Level:

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.