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?
How many triangles can you make on the 3 by 3 pegboard?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
An activity making various patterns with 2 x 1 rectangular tiles.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
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?
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
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.
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?
If you had 36 cubes, what different cuboids could you make?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.
Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?
Can you draw a square in which the perimeter is numerically equal to the area?
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?
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.
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
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?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
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?
Find all the different shapes that can be made by joining five equilateral triangles edge to edge.
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?
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?
If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?
Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
How many models can you find which obey these rules?
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
These practical challenges are all about making a 'tray' and covering it with paper.
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 article for primary teachers suggests ways in which to help children become better at working systematically.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
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?
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?
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing 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?
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
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?
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
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?