We start with one yellow cube and build around it to make a 3x3x3
cube with red cubes. Then we build around that red cube with blue
cubes and so on. How many cubes of each colour have we used?
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
Can you make a 3x3 cube with these shapes made from small cubes?
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
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
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?
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.
Exchange the positions of the two sets of counters in the least possible number of moves
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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?
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
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?
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?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
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 dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
This article for teachers describes how modelling number properties
involving multiplication using an array of objects not only allows
children to represent their thinking with concrete materials,. . . .
Can you find ways of joining cubes together so that 28 faces are
Can you predict when you'll be clapping and when you'll be clicking
if you start this rhythm? How about when a friend begins a new
rhythm at the same time?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Paint a stripe on a cardboard roll. Can you predict what will
happen when it is rolled across a sheet of paper?
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 cut a regular hexagon into two pieces to make a
parallelogram? Try cutting it into three pieces to make a rhombus!
Can you arrange the shapes in a chain so that each one shares a
face (or faces) that are the same shape as the one that follows it?
Looking at the picture of this Jomista Mat, can you decribe what
you see? Why not try and make one yourself?
What is the total area of the four outside triangles which are
outlined in red in this arrangement of squares inside each other?
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
What shape has Harry drawn on this clock face? Can you find its
area? What is the largest number of square tiles that could cover
Can you cut up a square in the way shown and make the pieces into a
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Try this interactive strategy game for 2
This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
I've made some cubes and some cubes with holes in. This challenge
invites you to explore the difference in the number of small cubes
I've used. Can you see any patterns?
Each of the nets of nine solid shapes has been cut into two pieces.
Can you see which pieces go together?
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
On which of these shapes can you trace a path along all of its
edges, without going over any edge twice?
This problem invites you to build 3D shapes using two different
triangles. Can you make the shapes from the pictures?
Which of the following cubes can be made from these nets?
Can you fit the tangram pieces into the outline of Little Ming?
Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?
Can you fit the tangram pieces into the outline of this shape. How would you describe it?
Can you fit the tangram pieces into the outline of Granma T?
Can you fit the tangram pieces into the outline of the telescope and microscope?