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There are **369** NRICH Mathematical resources connected to **Visualising**, you may find related items under Thinking mathematically.

Problem
Primary curriculum
Secondary curriculum
### On the Edge

If you move the tiles around, can you make squares with different coloured edges?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tangram Tangle

If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Hundred Square

A hundred square has been printed on both sides of a piece of paper. What is on the back of 100? 58? 23? 19?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cuboids

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Spider and the Fly

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shadow Play

Here are shadows of some 3D shapes. What shapes could have made them?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painted Cube

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cut Nets

Each of the nets of nine solid shapes has been cut into two pieces. Can you see which pieces go together?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Buses

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Coordinate Patterns

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Seven Squares - Group-worthy Task

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shear Magic

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odd Squares

Think of a number, square it and subtract your starting number. Is the number you're left with odd or even? How do the images help to explain this?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picturing Square Numbers

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picturing Triangular Numbers

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rolling Around

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sponge Sections

You have been given three shapes made out of sponge: a sphere, a cylinder and a cone. Your challenge is to find out how to cut them to make different shapes for printing.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fractional Triangles

Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### John's Train Is on Time

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Let Us Reflect

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Shady Symmetry

How many different symmetrical shapes can you make by shading triangles or squares?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pick's Theorem

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Counting Cards

A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Reflecting Squarely

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Age 11 to 14

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Proofs with Pictures

Some diagrammatic 'proofs' of algebraic identities and inequalities.

Age 14 to 18

Problem
Primary curriculum
Secondary curriculum
### Frogs

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triangles to Tetrahedra

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Skeleton Shapes

How many balls of modelling clay and how many straws does it take to make these skeleton shapes?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes Within Cubes

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?

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Corners

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Puzzling Cube

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Quadrilaterals

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Attractive Tablecloths

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Marbles in a Box

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Nine Colours

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?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Neighbours

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One and Three

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 metres from B. How long is the lake?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Doesn't Add Up

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Is There a Theorem?

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Picture Story

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Bendy Quad

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes Cut Into Four Pieces

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Happy Halving

Can you split each of the shapes below in half so that the two parts are exactly the same?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### One Big Triangle

Make one big triangle so the numbers that touch on the small triangles add to 10.

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A City of Towers

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Squares

What is the greatest number of squares you can make by overlapping three squares?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Age 5 to 11

Challenge Level