"Ip dip sky blue! Who's 'it'? It's you!" Where would you position yourself so that you are 'it' if there are two players? Three players ...?
When Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
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
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.
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
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?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
Investigate the different ways these aliens count in this
challenge. You could start by thinking about how each of them would
write our number 7.
How many different sets of numbers with at least four members can
you find in the numbers in this box?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
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?
Bernard Bagnall describes how to get more out of some favourite
Explore ways of colouring this set of triangles. Can you make
There are ten children in Becky's group. Can you find a set of
numbers for each of them? Are there any other sets?
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 find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
The red ring is inside the blue ring in this picture. Can you rearrange the rings in different ways? Perhaps you can overlap them or put one outside another?
In my local town there are three supermarkets which each has a
special deal on some products. If you bought all your shopping in
one shop, where would be the cheapest?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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.
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?
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?
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!
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?
In how many ways can you stack these rods, following the rules?
How many tiles do we need to tile these patios?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
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 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.
Why does the tower look a different size in each of these pictures?
If I use 12 green tiles to represent my lawn, how many different
ways could I arrange them? How many border tiles would I need each
This problem is intended to get children to look really hard at something they will see many times in the next few months.
Take a look at these data collected by children in 1986 as part of the Domesday Project. What do they tell you? What do you think about the way they are presented?
If the answer's 2010, what could the question be?
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
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?
Investigate these hexagons drawn from different sized equilateral
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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 make these equilateral triangles fit together to cover the
paper without any gaps between them? Can you tessellate isosceles
These pictures were made by starting with a square, finding the
half-way point on each side and joining those points up. You could
investigate your own starting shape.