Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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

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

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

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?

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

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

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....?

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?

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

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

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

How many models can you find which obey these rules?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

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

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?

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

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?

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?

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?

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. . . .

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

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

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.

Can you find all the different triangles on these peg boards, and find their angles?

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?

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

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

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

How many different triangles can you make on a circular pegboard that has nine pegs?

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.

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

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.

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

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

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?

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

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

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

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?

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

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

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?