If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
The challenge here is to find as many routes as you can for a fence
to go so that this town is divided up into two halves, each with 8
You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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 many models can you find which obey these rules?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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?
An investigation that gives you the opportunity to make and justify
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
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?
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.
Ana and Ross looked in a trunk in the attic. They found old cloaks
and gowns, hats and masks. How many possible costumes could they
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99
How many ways can you do it?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
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. . . .
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 put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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.
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?
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
This challenge is to design different step arrangements, which must
go along a distance of 6 on the steps and must end up at 6 high.
Investigate the different ways you could split up these rooms so
that you have double the number.
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
In how many ways can you stack these rods, following the rules?