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?
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.
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.
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?
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?
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
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?
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?
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?
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 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!
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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
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 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.
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?
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
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.
Here are many ideas for you to investigate - all linked with the
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Investigate the area of 'slices' cut off this cube of cheese. What
would happen if you had different-sized block of cheese to start
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
Bernard Bagnall looks at what 'problem solving' might really mean
in the context of primary classrooms.
"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
How many tiles do we need to tile these patios?
In this investigation we are going to count the number of 1s, 2s,
3s etc in numbers. Can you predict what will happen?
How many models can you find which obey these rules?
Why does the tower look a different size in each of these pictures?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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.
In how many ways can you stack these rods, following the rules?
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?
Can you continue this pattern of triangles and begin to predict how
many sticks are used for each new "layer"?
An investigation that gives you the opportunity to make and justify
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?
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 sets of numbers with at least four members can
you find in the numbers in this box?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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
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?