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?
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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 is the best way to shunt these carriages so that each train
can continue its journey?
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?
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
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
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?
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?
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 make a 3x3 cube with these shapes made from small cubes?
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?
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?
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
This 100 square jigsaw is written in code. It starts with 1 and
ends with 100. Can you build it up?
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?
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?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
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?
This article for teachers describes a project which explores
thepower of storytelling to convey concepts and ideas to children.
Can you find ways of joining cubes together so that 28 faces are
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?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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?
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
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?
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?
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?
An activity centred around observations of dots and how we visualise number arrangement patterns.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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.
Can you fit the tangram pieces into the outlines of the candle and sundial?
A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.
Billy's class had a robot called Fred who could draw with chalk
held underneath him. What shapes did the pupils make Fred draw?
These points all mark the vertices (corners) of ten hidden squares.
Can you find the 10 hidden squares?
This article looks at levels of geometric thinking and the types of
activities required to develop this thinking.
Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?
What happens when you turn these cogs? Investigate the differences
between turning two cogs of different sizes and two cogs which are
On which of these shapes can you trace a path along all of its
edges, without going over any edge twice?
Can you fit the tangram pieces into the outlines of the workmen?
Can you fit the tangram pieces into the outline of this sports car?