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

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?

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

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

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

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

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?

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 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.

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?

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

How many models can you find which obey these rules?

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

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

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

Investigate the different ways you could split up these rooms so that you have double the number.

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?

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. . . .

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?

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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?

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

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

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

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?

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

Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?

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

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

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

There are seven pots of plants in a greenhouse. They have lost their labels. Perhaps you can help re-label them.

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.