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
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?
What is the best way to shunt these carriages so that each train
can continue its journey?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
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?
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,. . . .
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
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?
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.
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
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?
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
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Exchange the positions of the two sets of counters in the least possible number of moves
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 many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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?
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?
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!
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
Can you find ways of joining cubes together so that 28 faces are
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?
Here are more buildings to picture in your mind's eye. Watch out -
they become quite complicated!
A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
How can you paint the faces of these eight cubes so they can be put
together to make a 2 x 2 cube that is green all over AND a 2 x 2
cube that is yellow all over?
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?
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?
A game for two players on a large squared space.
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.
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?
Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?
An extension of noughts and crosses in which the grid is enlarged
and the length of the winning line can to altered to 3, 4 or 5.
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?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?
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?
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
What happens when you turn these cogs? Investigate the differences
between turning two cogs of different sizes and two cogs which are