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?
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
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.
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?
What is the best way to shunt these carriages so that each train
can continue its journey?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
Use the interactivity to play two of the bells in a pattern. How do
you know when it is your turn to ring, and how do you know which
bell to ring?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
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.
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
How many models can you find which obey these rules?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
An irregular tetrahedron is composed of four different triangles.
Can such a tetrahedron be constructed where the side lengths are 4,
5, 6, 7, 8 and 9 units of length?
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?
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did he do it?
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
The planet of Vuvv has seven moons. Can you work out how long it is
between each super-eclipse?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
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?
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?
Find out what a "fault-free" rectangle is and try to make some of
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
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?
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
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.
I like to walk along the cracks of the paving stones, but not the
outside edge of the path itself. How many different routes can you
find for me to take?
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
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
How many different symmetrical shapes can you make by shading triangles or squares?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Place eight dots on this diagram, so that there are only two dots
on each straight line and only two dots on each circle.
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
This article for teachers describes several games, found on the
site, all of which have a related structure that can be used to
develop the skills of strategic planning.
Find all the different shapes that can be made by joining five
equilateral triangles edge to edge.
Mr McGregor has a magic potting shed. Overnight, the number of
plants in it doubles. He'd like to put the same number of plants in
each of three gardens, planting one garden each day. Can he do it?
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?
Can you find all the different ways of lining up these Cuisenaire
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!
If we had 16 light bars which digital numbers could we make? How
will you know you've found them all?
A magician took a suit of thirteen cards and held them in his hand
face down. Every card he revealed had the same value as the one he
had just finished spelling. How did this work?