How many different triangles can you make on a circular pegboard that has nine pegs?
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
What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
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?
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?
How can you put five cereal packets together to make different
shapes if you must put them face-to-face?
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 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
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.
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?
A tetromino is made up of four squares joined edge to edge. Can
this tetromino, together with 15 copies of itself, be used to cover
an eight by eight chessboard?
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?
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?
Can you help the children find the two triangles which have the
lengths of two sides numerically equal to their areas?
Zumf makes spectacles for the residents of the planet Zargon, who
have either 3 eyes or 4 eyes. How many lenses will Zumf need to
make all the different orders for 9 families?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Hover your mouse over the counters to see which ones will be
removed. Click to remover them. The winner is the last one to
remove a counter. How you can make sure you win?
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?
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!
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
What is the smallest number of coins needed to make up 12 dollars and 83 cents?
Can you find all the different triangles on these peg boards, and
find their angles?
This practical challenge invites you to investigate the different
squares you can make on a square geoboard or pegboard.
Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes 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?
Find all the different shapes that can be made by joining five
equilateral triangles edge to edge.
When intergalactic Wag Worms are born they look just like a cube.
Each year they grow another cube in any direction. Find all the
shapes that five-year-old Wag Worms can be.
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.?
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?
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
Are all the possible combinations of two shapes included in this
set of 27 cards? How do you know?
In how many ways can you stack these rods, following the rules?
These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
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 put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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 thoughtful shepherd used bales of straw to protect the area
around his lambs. Explore how you can arrange the bales.
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
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.
In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?
The Vikings communicated in writing by making simple scratches on
wood or stones called runes. Can you work out how their code works
using the table of the alphabet?