How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
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 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.
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
How will you go about finding all the jigsaw pieces that have one peg and one hole?
One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?
Have a go at this 3D extension to the Pebbles problem.
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?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?
Make a cube out of straws and have a go at this practical challenge.
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?
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?
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
What is the least number of moves you can take to rearrange the bears so that no bear is next to a bear of the same colour?
This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?
What is the greatest number of squares you can make by overlapping three squares?
How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?
Can you work out what shape is made when this piece of paper is folded up using the crease pattern shown?
Can you cut up a square in the way shown and make the pieces into a triangle?
Move four sticks so there are exactly four triangles.
A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?
Move just three of the circles so that the triangle faces in the opposite direction.
What happens when you try and fit the triomino pieces into these two grids?
Can you cover the camel with these pieces?
Looking at the picture of this Jomista Mat, can you decribe what you see? Why not try and make one yourself?
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
Have a go at making a few of these shapes from paper in different sizes. What patterns can you create?
Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?
How many different triangles can you make on a circular pegboard that has nine pegs?
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 ...
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Can you describe a piece of paper clearly enough for your partner to know which piece it is?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?
Can you cut a regular hexagon into two pieces to make a parallelogram? Try cutting it into three pieces to make a rhombus!
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!
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 find ways of joining cubes together so that 28 faces are visible?