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?
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.
This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
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.
How many models can you find which obey these rules?
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.
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?
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?
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 different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
An activity making various patterns with 2 x 1 rectangular tiles.
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
These practical challenges are all about making a 'tray' and covering it with 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?
Ahmed is making rods using different numbers of cubes. Which rod is twice the length of his first rod?
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?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?
Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.
Can you create more models that follow these rules?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?
Kimie and Sebastian were making sticks from interlocking cubes and lining them up. Can they make their lines the same length? Can they make any other lines?
Can you visualise what shape this piece of paper will make when it is folded?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
These pictures show squares split into halves. Can you find other ways?
Make a chair and table out of interlocking cubes, making sure that the chair fits under the table!
What are the next three numbers in this sequence? Can you explain why are they called pyramid numbers?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
For this task, you'll need an A4 sheet and two A5 transparent sheets. Decide on a way of arranging the A5 sheets on top of the A4 sheet and explore ...
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
What do these two triangles have in common? How are they related?
Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
Is there a best way to stack cans? What do different supermarkets do? How high can you safely stack the cans?
Can you make the most extraordinary, the most amazing, the most unusual patterns/designs from these triangles which are made in a special way?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
Make a flower design using the same shape made out of different sizes of paper.
What shape is made when you fold using this crease pattern? Can you make a ring design?
This practical problem challenges you to create shapes and patterns with two different types of triangle. You could even try overlapping them.
Exploring and predicting folding, cutting and punching holes and making spirals.
You have a set of the digits from 0 – 9. Can you arrange these in the five boxes to make two-digit numbers as close to the targets as possible?
If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?