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?
Imagine a pyramid which is built in square layers of small cubes. If we number the cubes from the top, starting with 1, can you picture which cubes are directly below this first cube?
Move just three of the circles so that the triangle faces in the
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
What happens when you try and fit the triomino pieces into these
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?
Can you cover the camel with these pieces?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
This article for teachers describes how modelling number properties
involving multiplication using an array of objects not only allows
children to represent their thinking with concrete materials,. . . .
Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.
What is the best way to shunt these carriages so that each train
can continue its journey?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
We start with one yellow cube and build around it to make a 3x3x3
cube with red cubes. Then we build around that red cube with blue
cubes and so on. How many cubes of each colour have we used?
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?
A game for two players. You'll need some counters.
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?
Exchange the positions of the two sets of counters in the least possible number of moves
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?
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?
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?
What does the overlap of these two shapes look like? Try picturing
it in your head and then use the interactivity to test your
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
A game for 1 or 2 people. Use the interactive version, or play with friends. Try to round up as many counters as possible.
If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
An activity centred around observations of dots and how we visualise number arrangement patterns.
In this town, houses are built with one room for each person. There
are some families of seven people living in the town. In how many
different ways can they build their houses?
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?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
Can you predict when you'll be clapping and when you'll be clicking
if you start this rhythm? How about when a friend begins a new
rhythm at the same time?
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?
A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.
Which of these dice are right-handed and which are left-handed?
I found these clocks in the Arts Centre at the University of
Warwick intriguing - do they really need four clocks and what times
would be ambiguous with only two or three of them?
What can you see? What do you notice? What questions can you ask?
Can you fit the tangram pieces into the outlines of the watering can and man in a boat?
Can you fit the tangram pieces into the outlines of the candle and sundial?
Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?
This second article in the series refers to research about levels
of development of spatial thinking and the possible influence of
Here are shadows of some 3D shapes. What shapes could have made
A shape and space game for 2,3 or 4 players. Be the last person to be able to place a pentomino piece on the playing board. Play with card, or on the computer.
Make a cube out of straws and have a go at this practical
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?
Investigate the number of paths you can take from one vertex to
another in these 3D shapes. Is it possible to take an odd number
and an even number of paths to the same vertex?