Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
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?
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?
Can you mark 4 points on a flat surface so that there are only two different distances between them?
A package contains a set of resources designed to develop pupils' mathematical thinking. This package places a particular emphasis on “visualising” and is designed to meet the needs. . . .
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?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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 are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try. . . .
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A useful visualising exercise which offers opportunities for discussion and generalising, and which could be used for thinking about the formulae needed for generating the results on a spreadsheet.
On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
A huge wheel is rolling past your window. What do you see?
Can you describe this route to infinity? Where will the arrows take you next?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
How many different symmetrical shapes can you make by shading triangles or squares?
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?
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?
ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square.
ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm.
A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer?
Can you maximise the area available to a grazing goat?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
Is it true that any convex hexagon will tessellate if it has a pair of opposite sides that are equal, and three adjacent angles that add up to 360 degrees?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
Can you discover whether this is a fair game?
A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface. . . .
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.
What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?
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.
When dice land edge-up, we usually roll again. But what if we didn't...?
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. . . .
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?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Can you find a way of representing these arrangements of balls?
What is the shape of wrapping paper that you would need to completely wrap this model?
Every day at noon a boat leaves Le Havre for New York while another boat leaves New York for Le Havre. The ocean crossing takes seven days. How many boats will each boat cross during their journey?