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?
Problem solving is at the heart of the NRICH site. All the problems
give learners opportunities to learn, develop or use mathematical
concepts and skills. Read here for more information.
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.
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?
A cheap and simple toy with lots of mathematics. Can you interpret
the images that are produced? Can you predict the pattern that will
be produced using different wheels?
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
Find your way through the grid starting at 2 and following these
operations. What number do you end on?
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?
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?
Can you find ways of joining cubes together so that 28 faces are
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?
What happens when you try and fit the triomino pieces into these
What can you see? What do you notice? What questions can you ask?
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?
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 see why 2 by 2 could be 5? Can you predict what 2 by 10
Use the three triangles to fill these outline shapes. Perhaps you can create some of your own shapes for a friend to fill?
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?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
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?
Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?
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
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?
A game for 2 people. Take turns joining two dots, until your opponent is unable to move.
Can you cover the camel with these pieces?
Move just three of the circles so that the triangle faces in the
This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .
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?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?
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?
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?
Square It game for an adult and child. Can you come up with a way of always winning this game?
What is the least number of moves you can take to rearrange the
bears so that no bear is next to a bear of the same colour?
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?
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 triangles can you make on a circular pegboard that has nine pegs?
How many DIFFERENT quadrilaterals can be made by joining the dots
on the 8-point circle?
Exchange the positions of the two sets of counters in the least possible number of moves
Mathematics is the study of patterns. Studying pattern is an
opportunity to observe, hypothesise, experiment, discover and
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?
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?
Can you find a way of representing these arrangements of balls?
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?
Lyndon Baker describes how the Mobius strip and Euler's law can
introduce pupils to the idea of topology.
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.