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?
If the answer's 2010, what could the question 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.
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?
Investigate what happens when you add house numbers along a street
in different ways.
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?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
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.
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 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?
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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!
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
Explore ways of colouring this set of triangles. Can you make
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
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?
Bernard Bagnall describes how to get more out of some favourite
Investigate these hexagons drawn from different sized equilateral
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?
An investigation that gives you the opportunity to make and justify
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?
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.
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
"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 ...?
Why does the tower look a different size in each of these pictures?
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?
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 Charlie asked his grandmother how old she is, he didn't get a
straightforward reply! Can you work out how old she is?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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?
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?
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 many tiles do we need to tile these patios?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
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
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?
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.
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.