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
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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 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?
Can you cover the camel with these pieces?
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?
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?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
What happens when you try and fit the triomino pieces into these
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?
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?
Use the clues to colour each square.
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?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
How many different rhythms can you make by putting two drums on the
An activity making various patterns with 2 x 1 rectangular tiles.
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. . . .
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
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?
These practical challenges are all about making a 'tray' and covering it with 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?
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 triangles can you make using sticks that are 3cm, 4cm and 5cm long?
Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
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....?
Investigate the different ways you could split up these rooms so
that you have double the number.
How many models can you find which obey these rules?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
When newspaper pages get separated at home we have to try to sort
them out and get things in the correct order. How many ways can we
arrange these pages so that the numbering may be different?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Can you find all the different ways of lining up these Cuisenaire
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?
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?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
How many trains can you make which are the same length as Matt's, using rods that are identical?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
How many different triangles can you make on a circular pegboard that has nine pegs?