How many possible necklaces can you find? And how do you know you've found them all?

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

Number problems at primary level that require careful consideration.

My coat has three buttons. How many ways can you find to do up all the buttons?

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

What could the half time scores have been in these Olympic hockey matches?

My cube has inky marks on each face. Can you find the route it has taken? What does each face look like?

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

How many trains can you make which are the same length as Matt's, using rods that are identical?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Use these head, body and leg pieces to make Robot Monsters which are different heights.

Can you see who the gold medal winner is? What about the silver medal winner and the bronze medal winner?

Find all the numbers that can be made by adding the dots on two dice.

This challenge is about finding the difference between numbers which have the same tens digit.

In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

Use the numbers and symbols to make this number sentence correct. How many different ways can you find?

Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

Find out about Magic Squares in this article written for students. Why are they magic?!

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

In this maze of hexagons, you start in the centre at 0. The next hexagon must be a multiple of 2 and the next a multiple of 5. What are the possible paths you could take?

This dice train has been made using specific rules. How many different trains can you make?

These practical challenges are all about making a 'tray' and covering it with paper.

Can you find all the different ways of lining up these Cuisenaire rods?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.