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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
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?
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 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?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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 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.
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?
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?
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?
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?
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 is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
Can you find ways of joining cubes together so that 28 faces are
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?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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,. . . .
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
How can you arrange these 10 matches in four piles so that when you
move one match from three of the piles into the fourth, you end up
with the same arrangement?
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?
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 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?
This challenge involves eight three-cube models made from
interlocking cubes. Investigate different ways of putting the
models together then compare your constructions.
This problem invites you to build 3D shapes using two different
triangles. Can you make the shapes from the pictures?
Exchange the positions of the two sets of counters in the least possible number of moves
Can you cut up a square in the way shown and make the pieces into a
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
Can you fit the tangram pieces into the outline of this goat and giraffe?
Can you visualise what shape this piece of paper will make when it is folded?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
Can you fit the tangram pieces into the outline of Mai Ling?
Here's a simple way to make a Tangram without any measuring or
Can you fit the tangram pieces into the outline of Granma T?
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 fit the tangram pieces into the outlines of the candle and sundial?
Can you fit the tangram pieces into the outlines of the workmen?
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
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 ...