# Great problems for new classes - September 2011, All Stages

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

### Poly Plug Pattern

##### Stage: 1 Challenge Level:

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

### All Change

##### Stage: 1 Challenge Level:

There are three versions of this challenge. The idea is to change the colour of all the spots on the grid. Can you do it in fewer throws of the dice?

### Poly Plug Rectangles

##### Stage: 1 Challenge Level:

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

### Plants

##### Stage: 1 and 2 Challenge Level:

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

### Spot Thirteen

##### Stage: 2 Challenge Level:

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

### Fifteen Cards

##### Stage: 2 Challenge Level:

Can you use the information to find out which cards I have used?

### American Billions

##### Stage: 3 Challenge Level:

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

### Summing Consecutive Numbers

##### Stage: 3 Challenge Level:

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Changing Areas, Changing Perimeters

##### Stage: 3 Challenge Level:

How can you change the area of a shape but keep its perimeter the same? How can you change the perimeter but keep the area the same?

### Non-transitive Dice

##### Stage: 3 Challenge Level:

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

### Changing Areas, Changing Volumes

##### Stage: 3 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 to gain insights into the graphs...