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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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 can you put five cereal packets together to make different shapes if you must put them face-to-face?
How many models can you find which obey these rules?
How many triangles can you make on the 3 by 3 pegboard?
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.
If you had 36 cubes, what different cuboids could you make?
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?
A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
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?
Find all the different shapes that can be made by joining five equilateral triangles edge to edge.
These rectangles have been torn. How many squares did each one have inside it before it was ripped?
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.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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.
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.
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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.
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?
Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?
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.
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?
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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
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?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?
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?
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?
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?
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?