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?
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?
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
Investigate what happens when you add house numbers along a street
in different ways.
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 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.
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?
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?
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 the answer's 2010, what could the question be?
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!
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?
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.
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.
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.
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?
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Explore Alex's number plumber. What questions would you like to
ask? Don't forget to keep visiting NRICH projects site for the
latest developments and questions.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
How many different sets of numbers with at least four members can
you find in the numbers in this box?
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?
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
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.
Here is your chance to investigate the number 28 using shapes,
cubes ... in fact anything at all.
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?
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?
An activity making various patterns with 2 x 1 rectangular tiles.
Explore ways of colouring this set of triangles. Can you make
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?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
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?
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 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?
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
Bernard Bagnall describes how to get more out of some favourite
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
Here are many ideas for you to investigate - all linked with the
There are ten children in Becky's group. Can you find a set of
numbers for each of them? Are there any other sets?
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?
This problem is intended to get children to look really hard at something they will see many times in the next few months.
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
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
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 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