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 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?
A variant on the game Alquerque
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
Move just three of the circles so that the triangle faces in the
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
This is the first article in a series which aim to provide some insight into the way spatial thinking develops in children, and draw on a range of reported research. The focus of this article is the. . . .
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,. . . .
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?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
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?
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Can you make a 3x3 cube with these shapes made from small cubes?
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?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens?
These are pictures of the sea defences at New Brighton. Can you
work out what a basic shape might be in both images of the sea wall
and work out a way they might fit together?
How many different cuboids can you make when you use four CDs or
DVDs? How about using five, then six?
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?
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
This second article in the series refers to research about levels
of development of spatial thinking and the possible influence of
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Exchange the positions of the two sets of counters in the least possible number of moves
Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.
Can you cover the camel with these pieces?
What happens when you try and fit the triomino pieces into these
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
Cut four triangles from a square as shown in the picture. How many
different shapes can you make by fitting the four triangles back
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!
This article for teachers discusses examples of problems in which
there is no obvious method but in which children can be encouraged
to think deeply about the context and extend their ability to. . . .
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?
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.
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
A game for two players. You'll need some counters.
The aim of the game is to slide the green square from the top right
hand corner to the bottom left hand corner in the least number of
If you can post the triangle with either the blue or yellow colour face up, how many ways can it be posted altogether?
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?
Which of these dice are right-handed and which are left-handed?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
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 extension of noughts and crosses in which the grid is enlarged
and the length of the winning line can to altered to 3, 4 or 5.