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

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?

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

An activity making various patterns with 2 x 1 rectangular tiles.

How many trapeziums, of various sizes, are hidden in this picture?

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

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

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

Can you fill in the empty boxes in the grid with the right shape and colour?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

This activity investigates how you might make squares and pentominoes from Polydron.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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

Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

What is the best way to shunt these carriages so that each train can continue its journey?

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

Seven friends went to a fun fair with lots of scary rides. They decided to pair up for rides until each friend had ridden once with each of the others. What was the total number rides?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

Number problems for lower primary that will get you thinking.

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

In how many ways can you stack these rods, following the rules?

How many models can you find which obey these rules?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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 will you go about finding all the jigsaw pieces that have one peg and one hole?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

If you put three beads onto a tens/ones abacus you could make the numbers 3, 30, 12 or 21. What numbers can be made with six beads?

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Can you draw a square in which the perimeter is numerically equal to the area?