Mathematicians often explore structures, notice patterns, find rules and make generalisations. Once they feel they understand a structure, they then push at the boundaries, change the problem, and explore the new structures that emerge.

The Freedom and Constraints pathway on wild.maths.org offers

The collection of related NRICH tasks below are ideal for teachers who want to promote creativity in the classroom. They are designed for classroom use, with accompanying Teachers' Notes and Resources.

The Freedom and Constraints pathway on wild.maths.org offers

**students**some situations which may initially seem constrained, but on closer inspection they may find there is more room to manoeuvre than they imagined.The collection of related NRICH tasks below are ideal for teachers who want to promote creativity in the classroom. They are designed for classroom use, with accompanying Teachers' Notes and Resources.

problem

### Two and Two

How many solutions can you find to this sum? Each of the different letters stands for a different number.

problem

### American Billions

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...

problem

### Cinema Problem

A cinema has 100 seats. Show how it is possible to sell exactly 100 tickets and take exactly £100 if the prices are £10 for adults, 50p for pensioners and 10p for children.

problem

### How much can we spend?

A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?

problem

### Perimeter Expressions

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

problem

### What numbers can we make?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

problem

### Gabriel's Problem

Gabriel multiplied together some numbers and then erased them. Can you figure out where each number was?

problem

### 1 Step 2 Step

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

problem

### Take Three From Five

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

problem

### Peaches today, Peaches tomorrow...

A monkey with peaches, keeps a fraction of them each day, gives the rest away, and then eats one. How long can his peaches last?

problem

### Weights

Different combinations of the weights available allow you to make different totals. Which totals can you make?

problem

### What numbers can we make now?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?