Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
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 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?
Imagine a wheel with different markings painted on it at regular
intervals. Can you predict the colour of the 18th mark? The 100th
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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. . . .
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,. . . .
Investigate how the four L-shapes fit together to make an enlarged
L-shape. You could explore this idea with other shapes too.
A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .
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?
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?
What is the best way to shunt these carriages so that each train
can continue its journey?
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?
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Swap the stars with the moons, using only knights' moves (as on a
chess board). What is the smallest number of moves possible?
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?
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?
On the graph there are 28 marked points. These points all mark the
vertices (corners) of eight hidden squares. Can you find the eight
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable.
Decide which of these diagrams are traversable.
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 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 huge wheel is rolling past your window. What do you see?
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
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
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Exchange the positions of the two sets of counters in the least possible number of moves
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.
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.
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
A game for 1 person. Can you work out how the dice must be rolled from the start position to the finish? Play on line.
Which of these dice are right-handed and which are left-handed?
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?
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?
What would be the smallest number of moves needed to move a Knight
from a chess set from one corner to the opposite corner of a 99 by
99 square board?
A game for two players on a large squared space.
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.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Building up a simple Celtic knot. Try the interactivity or download
the cards or have a go on squared paper.
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
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?
When dice land edge-up, we usually roll again. But what if we
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
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 cuboid that you can put in this box so that
you cannot fit another that's the same into it?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
How many different symmetrical shapes can you make by shading triangles or squares?
Design an arrangement of display boards in the school hall which fits the requirements of different people.