Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
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?
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?
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.
Can you find all the different triangles on these peg boards, and
find their angles?
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?
A mathematician goes into a supermarket and buys four items. Using
a calculator she multiplies the cost instead of adding them. How
can her answer be the same as the total at the till?
How many different triangles can you make on a circular pegboard that has nine pegs?
This activity investigates how you might make squares and pentominoes from Polydron.
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?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Can you draw a square in which the perimeter is numerically equal
to the area?
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?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?
How many triangles can you make on the 3 by 3 pegboard?
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
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?
What is the smallest number of tiles needed to tile this patio? Can
you investigate patios of different sizes?
Investigate all the different squares you can make on this 5 by 5
grid by making your starting side go from the bottom left hand
point. Can you find out the areas of all these squares?
How many ways can you find of tiling the square patio, using square
tiles of different sizes?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
These practical challenges are all about making a 'tray' and covering it with paper.
This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.
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
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Can you shunt the trucks so that the Cattle truck and the Sheep
truck change places and the Engine is back on the main line?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Put 10 counters in a row. Find a way to arrange the counters into
five pairs, evenly spaced in a row, in just 5 moves, using the
What is the best way to shunt these carriages so that each train
can continue its journey?
Pentagram Pylons - can you elegantly recreate them? Or, the
European flag in LOGO - what poses the greater problem?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
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?
An investigation that gives you the opportunity to make and justify
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A Sudoku with clues as ratios or fractions.
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?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
You have 4 red and 5 blue counters. How many ways can they be
placed on a 3 by 3 grid so that all the rows columns and diagonals
have an even number of red counters?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
Use the interactivity to listen to the bells ringing a pattern. Now
it's your turn! Play one of the bells yourself. How do you know
when it is your turn to ring?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Can you find all the different ways of lining up these Cuisenaire
Remember that you want someone following behind you to see where
you went. Can yo work out how these patterns were created and
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