How many models can you find which obey these rules?

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.

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

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

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

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?

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

El Crico the cricket has to cross a square patio to get home. He can jump the length of one tile, two tiles and three tiles. Can you find a path that would get El Crico home in three jumps?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

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?

What happens when you try and fit the triomino pieces into these two grids?

Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

Tim's class collected data about all their pets. Can you put the animal names under each column in the block graph using the information?

Systematically explore the range of symmetric designs that can be created by shading parts of the motif below. Use normal square lattice paper to record your results.

Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?

I was in my car when I noticed a line of four cars on the lane next to me with number plates starting and ending with J, K, L and M. What order were they in?

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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

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?

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?

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

Alice and Brian are snails who live on a wall and can only travel along the cracks. Alice wants to go to see Brian. How far is the shortest route along the cracks? Is there more than one way to go?

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?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

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

In the planet system of Octa the planets are arranged in the shape of an octahedron. How many different routes could be taken to get from Planet A to Planet Zargon?

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?

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

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

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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?

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

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

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

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?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!

A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.