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Freedom and Constraints

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 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.

Dozens

Age 7 to 14

Challenge Level Yellow star

Can you select the missing digit(s) to find the largest multiple?

How Much Can We Spend?

Age 11 to 14

Challenge Level Yellow star

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?

Largest Product

Age 11 to 14

Challenge Level Yellow star

Which set of numbers that add to 100 have the largest product?

Perimeter Expressions

Age 11 to 14

Challenge Level Yellow star

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

Cinema Problem

Age 11 to 14

Challenge Level Yellow star

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.

What Numbers Can We Make?

Age 11 to 14

Challenge Level Yellow star

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

Gabriel's Problem

Age 11 to 14

Challenge Level Yellow star

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

Odds and Evens

Age 11 to 14

Challenge Level Yellow star

Are these games fair? How can you tell?

American Billions

Age 11 to 14

Challenge Level Yellow star

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...

1 Step 2 Step

Age 11 to 14

Challenge Level Yellow star

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?