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?

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

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

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?

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.

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?

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

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?

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?

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

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?

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

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.

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?

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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

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?

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

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

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

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

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

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

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

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

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?

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?

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.

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?

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

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?

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....?

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?

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

How many models can you find which obey these rules?

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

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?

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.

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

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

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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?

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?

I like to walk along the cracks of the paving stones, but not the outside edge of the path itself. How many different routes can you find for me to take?

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

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

Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?