Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.
This challenge extends the Plants investigation so now four or more children are involved.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Investigate the number of faces you can see when you arrange three cubes in different ways.
An investigation that gives you the opportunity to make and justify
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 challenge encourages you to explore dividing a three-digit number by a single-digit number.
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?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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!
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
Investigate what happens when you add house numbers along a street
in different ways.
Investigate the different ways you could split up these rooms so
that you have double the number.
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.
We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Try continuing these patterns made from triangles. Can you create
your own repeating pattern?
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.
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?
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?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
How many models can you find which obey these rules?
What can you say about the child who will be first on the playground tomorrow morning at breaktime in your school?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
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?
A challenging activity focusing on finding all possible ways of stacking rods.
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.
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?
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
Is there a best way to stack cans? What do different supermarkets
do? How high can you safely stack the cans?
If the answer's 2010, what could the question be?
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?