There are to be 6 homes built on a new development site. They could
be semi-detached, detached or terraced houses. How many different
combinations of these can you find?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
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!
These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
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?
Lolla bought a balloon at the circus. She gave the clown six coins
to pay for it. What could Lolla have paid for the balloon?
48 is called an abundant number because it is less than the sum of its factors (without itself). Can you find some more abundant numbers?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Investigate the different ways you could split up these rooms so
that you have double the number.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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 triangles can you make on the 3 by 3 pegboard?
Ben has five coins in his pocket. How much money might he have?
I like to walk along the cracks of the paving stones, but not the
outside edge of the path itself. How many different routes can you
find for me to take?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
In how many ways can you stack these rods, following the rules?
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.
Place this "worm" on the 100 square and find the total of the four
squares it covers. Keeping its head in the same place, what other
totals can you make?
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?
We can arrange dots in a similar way to the 5 on a dice and they
usually sit quite well into a rectangular shape. How many
altogether in this 3 by 5? What happens for other sizes?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?
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
How many models can you find which obey these rules?
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.
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
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?
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
If the answer's 2010, what could the question be?
Try continuing these patterns made from triangles. Can you create your own repeating pattern?
Complete these two jigsaws then put one on top of the other. What
happens when you add the 'touching' numbers? What happens when you
change the position of the jigsaws?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
An investigation that gives you the opportunity to make and justify
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.