There are 179 results
Broad Topics > Thinking Mathematically > Visualising
Polygon Pictures
Age 11 to 14
Challenge Level
Can you work out how these polygon pictures were drawn, and use that to figure out their angles?
Polygon Rings
Age 11 to 14
Challenge Level
Join pentagons together edge to edge. Will they form a ring?
Seven Squares
Age 11 to 14
Challenge Level
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Partly Painted Cube
Age 14 to 16
Challenge Level
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?
Mystic Rose
Age 14 to 16
Challenge Level
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Diminishing Returns
Age 11 to 14
Challenge Level
How much of the square is coloured blue? How will the pattern continue?
Tower of Hanoi
Age 11 to 14
Challenge Level
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Christmas Chocolates
Age 11 to 14
Challenge Level
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Baravelle
Age 7 to 16
Challenge Level
What can you see? What do you notice? What questions can you ask?
Cuboid Challenge
Age 11 to 16
Challenge Level
What's the largest volume of box you can make from a square of paper?
Eight Hidden Squares
Age 7 to 14
Challenge Level
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?
Spotting the Loophole
Age 14 to 16
Challenge Level
A visualisation problem in which you search for vectors which sum to zero from a jumble of arrows. Will your eyes be quicker than algebra?
Route to Infinity
Age 11 to 14
Challenge Level
Can you describe this route to infinity? Where will the arrows take you next?
Squares in Rectangles
Age 11 to 14
Challenge Level
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?
Semi-regular Tessellations
Age 11 to 16
Challenge Level
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
Cubes Within Cubes Revisited
Age 11 to 14
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?
Square Coordinates
Age 11 to 14
Challenge Level
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?
Isosceles Triangles
Age 11 to 14
Challenge Level
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?
Efficient Cutting
Age 11 to 14
Challenge Level
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
Fence It
Age 11 to 14
Challenge Level
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Ten Hidden Squares
Age 7 to 14
Challenge Level
These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?
Square It
Age 11 to 16
Challenge Level
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.
On the Edge
Age 11 to 14
Challenge Level
If you move the tiles around, can you make squares with different coloured edges?
Cuboids
Age 11 to 14
Challenge Level
Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?
The Spider and the Fly
Age 14 to 16
Challenge Level
A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?
Painted Cube
Age 14 to 16
Challenge Level
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?
Buses
Age 11 to 14
Challenge Level
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?
Coordinate Patterns
Age 11 to 14
Challenge Level
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Seven Squares - Group-worthy Task
Age 11 to 14
Challenge Level
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?
Picturing Square Numbers
Age 11 to 14
Challenge Level
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Picturing Triangular Numbers
Age 11 to 14
Challenge Level
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Rolling Around
Age 11 to 14
Challenge Level
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?
John's Train Is on Time
Age 11 to 14
Challenge Level
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
Shady Symmetry
Age 11 to 14
Challenge Level
How many different symmetrical shapes can you make by shading triangles or squares?
Reflecting Squarely
Age 11 to 14
Challenge Level
In how many ways can you fit all three pieces together to make shapes with line symmetry?
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
Triangles to Tetrahedra
Age 11 to 14
Challenge Level
Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?
Cubes Within Cubes
Age 7 to 14
Challenge Level
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?
Marbles in a Box
Age 11 to 16
Challenge Level
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Nine Colours
Age 11 to 16
Challenge Level
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
One and Three
Age 14 to 16
Challenge Level
Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .
Doesn't Add Up
Age 14 to 16
Challenge Level
In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?
Is There a Theorem?
Age 11 to 14
Challenge Level
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?
Picture Story
Age 14 to 16
Challenge Level
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Bendy Quad
Age 14 to 16
Challenge Level
Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.
The Farmers' Field Boundary
Age 11 to 14
Challenge Level
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Constructing Triangles
Age 11 to 14
Challenge Level
Generate three random numbers to determine the side lengths of a triangle. What triangles can you draw?
NRICH is part of the family of activities in the Millennium Mathematics Project.