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!
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?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do 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 numbers?
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?
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?
Investigate the different ways you could split up these rooms so that you have double the number.
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?
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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?
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?
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?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?
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.
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 can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
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.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
In how many ways can you stack these rods, following the rules?
This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Explore one of these five pictures.
How many models can you find which obey these rules?
If the answer's 2010, what could the question be?
How many triangles can you make on the 3 by 3 pegboard?
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?
Can you find ways of joining cubes together so that 28 faces are visible?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
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?
An investigation that gives you the opportunity to make and justify predictions.
What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?
This challenge involves calculating the number of candles needed on birthday cakes. It is an opportunity to explore numbers and discover new things.
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.
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 Cambridge.
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?
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 make?
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.
Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.
How many ways can you find of tiling the square patio, using square tiles of different sizes?
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?
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?
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?