Can you discover whether this is a fair game?
Can you use the diagram to prove the AM-GM inequality?
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Can you find a rule which connects consecutive triangular numbers?
Can you find a rule which relates triangular numbers to square numbers?
Show that all pentagonal numbers are one third of a triangular number.
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
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. . . .
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
How many different ways can I lay 10 paving slabs, each 2 foot by 1 foot, to make a path 2 foot wide and 10 foot long from my back door into my garden, without cutting any of the paving slabs?
A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?
A huge wheel is rolling past your window. What do you see?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
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. . . .
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
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.
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
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?
The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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?
When dice land edge-up, we usually roll again. But what if we didn't...?
The reader is invited to investigate changes (or permutations) in the ringing of church bells, illustrated by braid diagrams showing the order in which the bells are rung.
Can you find a way of representing these arrangements of balls?
Can you describe this route to infinity? Where will the arrows take you next?
Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?
To avoid losing think of another very well known game where the patterns of play are similar.
Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?
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 moves.
Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
If you move the tiles around, can you make squares with different coloured edges?
A game for 2 players
A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Find the point whose sum of distances from the vertices (corners) of a given triangle is a minimum.
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . .