This task depends on groups working collaboratively, discussing and reasoning to agree a final product.
In how many ways can you fit two of these yellow triangles
together? Can you predict the number of ways two blue triangles can
be fitted together?
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
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.
In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?
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?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
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
How many triangles can you make on the 3 by 3 pegboard?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Can you recreate these designs? What are the basic units? What
movement is required between each unit? Some elegant use of
procedures will help - variables not essential.
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Pentagram Pylons - can you elegantly recreate them? Or, the
European flag in LOGO - what poses the greater problem?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
A merchant brings four bars of gold to a jeweller. How can the
jeweller use the scales just twice to identify the lighter, fake
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
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?
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
In a square in which the houses are evenly spaced, numbers 3 and 10
are opposite each other. What is the smallest and what is the
largest possible number of houses in the square?
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
Can you find all the different triangles on these peg boards, and
find their angles?
These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Can you put the 25 coloured tiles into the 5 x 5 square so that no
column, no row and no diagonal line have tiles of the same colour
Can you rearrange the biscuits on the plates so that the three
biscuits on each plate are all different and there is no plate with
two biscuits the same as two biscuits on another plate?
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
Take a rectangle of paper and fold it in half, and half again, to
make four smaller rectangles. How many different ways can you fold
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
In the planet system of Octa the planets are arranged in the shape
of an octahedron. How many different routes could be taken to get
from Planet A to Planet Zargon?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
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!
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
What can you say about these shapes? This problem challenges you to
create shapes with different areas and perimeters.
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?