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?

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?

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?

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

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

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?

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

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 many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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.

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

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

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

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

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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

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?

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

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

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

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

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?

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?

In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

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

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?

This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

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.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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?

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

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

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

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

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.

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. . . .

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

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

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.

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

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.

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?