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?
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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
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.
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
My cousin was 24 years old on Friday April 5th in 1974. On what day
of the week was she born?
Can you order pictures of the development of a frog from frogspawn
and of a bean seed growing into a plant?
Alice's mum needs to go to each child's house just once and then
back home again. How many different routes are there? Use the
information to find out how long each road is on the route she
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?
On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?
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?
On a digital clock showing 24 hour time, over a whole day, how many
times does a 5 appear? Is it the same number for a 12 hour clock
over a whole day?
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Can you work out how to balance this equaliser? You can put more
than one weight on a hook.
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
This problem focuses on Dienes' Logiblocs. What is the same and
what is different about these pairs of shapes? Can you describe the
shapes in the picture?
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
An investigation that gives you the opportunity to make and justify
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
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?
Place the numbers 1 to 6 in the circles so that each number is the
difference between the two numbers just below it.
During the third hour after midnight the hands on a clock point in
the same direction (so one hand is over the top of the other). At
what time, to the nearest second, does this happen?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.
Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?
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
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
Can you fill in the empty boxes in the grid with the right shape
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
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?
10 space travellers are waiting to board their spaceships. There
are two rows of seats in the waiting room. Using the rules, where
are they all sitting? Can you find all the possible ways?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
Ben and his mum are planting garlic. Use the interactivity to help
you find out how many cloves of garlic they might have had.
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
Is it possible to place 2 counters on the 3 by 3 grid so that there
is an even number of counters in every row and every column? How
about if you have 3 counters or 4 counters or....?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
This activity investigates how you might make squares and pentominoes from Polydron.
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?