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 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?
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
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 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?
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?
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?
Lolla bought a balloon at the circus. She gave the clown six coins
to pay for it. What could Lolla have paid for the balloon?
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.
Investigate what happens when you add house numbers along a street
in different ways.
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.
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?
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.
An investigation that gives you the opportunity to make and justify
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
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?
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.
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
If the answer's 2010, what could the question be?
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?
Ben has five coins in his pocket. How much money might he have?
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
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?
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.
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?
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?
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.
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possibilities that could come up?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
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.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
Investigate the different ways you could split up these rooms so
that you have double the number.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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?
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 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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
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?
What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
An activity making various patterns with 2 x 1 rectangular tiles.
Make new patterns from simple turning instructions. You can have a go using pencil and paper or with a floor robot.