# Great problems for new classes - September 2011, Stage 4&5

This month we have chosen a selection of problems ideal for beginning problem-solving at the start of a new school year. These tasks help to break the ice and stimulate mathematical discussions. We hope they will help new classes to develop good mathematical habits of mind.

## Problems

### Non-transitive Dice

##### Stage: 3 and 4 Challenge Level:

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

### Ideas for the Secondary Classroom

##### Stage: 3, 4 and 5

Articles about mathematics which can help to invigorate your classroom

### Changing Areas, Changing Volumes

##### Stage: 4 Challenge Level:

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

### Attractive Tablecloths

##### Stage: 4 Challenge Level:

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

### Weekly Challenges

##### Stage: 4 and 5 Challenge Level:

The NRICH Stage 5 weekly challenges are shorter problems aimed at Post-16 students or enthusiastic younger students. There are 52 of them.

### Few and Far Between?

##### Stage: 4 and 5 Challenge Level:

Can you find some Pythagorean Triples where the two smaller numbers differ by 1?

### Uniformly Unstable

##### Stage: 5 Challenge Level:

Invent shapes with different numbers of stable and unstable equilibrium points

### Graphs of Changing Areas

##### Stage: 5 Challenge Level:

Use graphs to gain insights into an area and perimeter problem, or use your knowledge of area and perimeter of rectangles to gain insight into the graphs.