What is the smallest cuboid that you can put in this box so that
you cannot fit another that's the same into it?
These practical challenges are all about making a 'tray' and covering it with paper.
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.
How many models can you find which obey these rules?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
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
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
If you had 36 cubes, what different cuboids could you make?
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 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?
Can you find all the different ways of lining up these Cuisenaire
Here you see the front and back views of a dodecahedron. Each
vertex has been numbered so that the numbers around each pentagonal
face add up to 65. Can you find all the missing numbers?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
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.
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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?
An investigation that gives you the opportunity to make and justify
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 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?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
How many different ways can you find of fitting five hexagons
together? How will you know you have found all the ways?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
What is the best way to shunt these carriages so that each train
can continue its journey?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
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
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?
Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?
Place eight queens on an chessboard (an 8 by 8 grid) so that none
can capture any of the others.
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?
A thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
How many triangles can you make on the 3 by 3 pegboard?
Investigate the different ways you could split up these rooms so
that you have double the number.
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?
Investigate the smallest number of moves it takes to turn these
mats upside-down if you can only turn exactly three at a time.
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?
This activity investigates how you might make squares and pentominoes from Polydron.
Can you draw a square in which the perimeter is numerically equal
to the area?
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 DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?