Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
It is possible to dissect any square into smaller squares. What is the minimum number of squares a 13 by 13 square can be dissected into?
In a right angled triangular field, three animals are tethered to posts at the midpoint of each side. Each rope is just long enough to allow the animal to reach two adjacent vertices. Only one animal. . . .
Can you maximise the area available to a grazing goat?
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. . . .
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.
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?
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?
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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. . . .
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
How can you make an angle of 60 degrees by folding a sheet of paper twice?
Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?
Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.
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?
You can move the 4 pieces of the jigsaw and fit them into both outlines. Explain what has happened to the missing one unit of area.
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. . . .
Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create.
Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .
A huge wheel is rolling past your window. What do you see?
ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?
A cylindrical helix is just a spiral on a cylinder, like an ordinary spring or the thread on a bolt. If I turn a left-handed helix over (top to bottom) does it become a right handed helix?
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.
Show that among the interior angles of a convex polygon there cannot be more than three acute angles.
Find all the ways to cut out a 'net' of six squares that can be folded into a cube.
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.
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?
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?
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?
Can you mentally fit the 7 SOMA pieces together to make a cube? Can you do it in more than one way?
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.
A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?
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?
What can you see? What do you notice? What questions can you ask?
Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?
You have 27 small cubes, 3 each of nine colours. Use the small cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of every colour.
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .
In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?
When dice land edge-up, we usually roll again. But what if we didn't...?
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
Have a go at this 3D extension to the Pebbles problem.