Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
While we were sorting some papers we found 3 strange sheets which
seemed to come from small books but there were page numbers at the
foot of each page. Did the pages come from the same book?
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?
What happens if you join every second point on this circle? How
about every third point? Try with different steps and see if you
can predict what will happen.
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
An investigation that gives you the opportunity to make and justify
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.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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!
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 different shaped boxes can you design for 36 sweets in one
layer? Can you arrange the sweets so that no sweets of the same
colour are next to each other in any direction?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Well now, what would happen if we lost all the nines in our number
system? Have a go at writing the numbers out in this way and have a
look at the multiplications table.
How many different sets of numbers with at least four members can
you find in the numbers in this box?
Investigate the numbers that come up on a die as you roll it in the
direction of north, south, east and west, without going over the
path it's already made.
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.
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Have a go at this 3D extension to the Pebbles problem.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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.
In how many ways can you stack these rods, following the rules?
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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?
If the answer's 2010, what could the question be?
In this investigation, you are challenged to make mobile phone
numbers which are easy to remember. What happens if you make a
sequence adding 2 each time?
Investigate what happens when you add house numbers along a street
in different ways.
There are ten children in Becky's group. Can you find a set of
numbers for each of them? Are there any other sets?
This activity asks you to collect information about the birds you
see in the garden. Are there patterns in the data or do the birds
seem to visit randomly?
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?
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?
In this investigation, you must try to make houses using cubes. If
the base must not spill over 4 squares and you have 7 cubes which
stand for 7 rooms, what different designs can you come up with?
Investigate the different ways you could split up these rooms so
that you have double the number.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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?
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?
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.
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
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
There are three tables in a room with blocks of chocolate on each. Where would be the best place for each child in the class to sit if they came in one at a time?
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?