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
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?
Take 5 cubes of one colour and 2 of another colour. How many
different ways can you join them if the 5 must touch the table and
the 2 must not touch the table?
This problem focuses on Dienes' Logiblocs. What is the same and
what is different about these pairs of shapes? Can you describe the
shapes in the picture?
Using different numbers of sticks, how many different triangles are
you able to make? Can you make any rules about the numbers of
sticks that make the most triangles?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
Use the clues to colour each square.
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
How many different rhythms can you make by putting two drums on the
How many different triangles can you make on a circular pegboard that has nine pegs?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
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 many models can you find which obey these rules?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
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?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
Can you cover the camel with these pieces?
An activity making various patterns with 2 x 1 rectangular tiles.
These practical challenges are all about making a 'tray' and covering it with paper.
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.
How many triangles can you make on the 3 by 3 pegboard?
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
What happens when you try and fit the triomino pieces into these
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
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?
If you have three circular objects, you could arrange them so that
they are separate, touching, overlapping or inside each other. Can
you investigate all the different possibilities?
Find all the different shapes that can be made by joining five
equilateral triangles edge to edge.
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?
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?
If you had 36 cubes, what different cuboids could you make?
When intergalactic Wag Worms are born they look just like a cube.
Each year they grow another cube in any direction. Find all the
shapes that five-year-old Wag Worms can be.
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Is it possible to place 2 counters on the 3 by 3 grid so that there
is an even number of counters in every row and every column? How
about if you have 3 counters or 4 counters or....?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Let's say you can only use two different lengths - 2 units and 4
units. Using just these 2 lengths as the edges how many different
cuboids can you make?
Put 10 counters in a row. Find a way to arrange the counters into
five pairs, evenly spaced in a row, in just 5 moves, using the