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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
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.
Let's suppose that you are going to have a magazine which has 16
pages of A5 size. Can you find some different ways to make these
pages? Investigate the pattern for each if you number the pages.
Can you find ways of joining cubes together so that 28 faces are
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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?
Have a go at this 3D extension to the Pebbles problem.
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?
The ancient Egyptians were said to make right-angled triangles
using a rope with twelve equal sections divided by knots. What
other triangles could you make if you had a rope like this?
Investigate and explain the patterns that you see from recording
just the units digits of numbers in the times tables.
An investigation that gives you the opportunity to make and justify
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.
In this investigation, we look at Pascal's Triangle in a slightly different way - rotated and with the top line of ones taken off.
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?
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?
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 what happens when you add house numbers along a street
in different ways.
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.
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.
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?
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.
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?
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?
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Can you find out how the 6-triangle shape is transformed in these
tessellations? Will the tessellations go on for ever? Why or why
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
A group of children are discussing the height of a tall tree. How would you go about finding out its height?
How many tiles do we need to tile these patios?
In this investigation, you are challenged to make mobile phone
numbers which are easy to remember. What happens if you make a
sequence adding 2 each time?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
A challenging activity focusing on finding all possible ways of stacking rods.
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?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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.
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
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?
In how many ways can you stack these rods, following the rules?
Why does the tower look a different size in each of these pictures?
How many models can you find which obey these rules?
Explore one of these five pictures.