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There are **494** NRICH Mathematical resources connected to **Visualising and representing**, you may find related items under Thinking mathematically.

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
Live 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
### Terminology

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

Problem
Primary curriculum
Secondary curriculum
### Remainders

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

Age 7 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
### Think of Two Numbers

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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
### Two Primes Make One Square

Can you make square numbers by adding two prime numbers together?

Age 7 to 11

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
### Reach 100

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Super Shapes

The value of the circle changes in each of the following problems. Can you discover its value in each problem?

Age 7 to 11

ShortChallenge 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
### Iff

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Which Scripts?

There are six numbers written in five different scripts. Can you sort out which is which?

Age 7 to 11

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
### Funny Factorisation

Using the digits 1 to 9, the number 4396 can be written as the product of two numbers. Can you find the factors?

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
### Plus Minus

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Age 14 to 16

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
### Legs Eleven

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

Age 11 to 14

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
### Counting Factors

Is there an efficient way to work out how many factors a large number has?

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
### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Age 16 to 18

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
### Catrina's Cards

How would you find out how many football cards Catrina has collected?

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
### Writing Digits

Lee was writing all the counting numbers from 1 to 20. She stopped for a rest after writing seventeen digits. What was the last number she wrote?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Biscuit Decorations

Andrew decorated 20 biscuits to take to a party. He lined them up and put icing on every second biscuit and different decorations on other biscuits. How many biscuits weren't decorated?

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

Problem
Primary curriculum
Secondary curriculum
### Round and Round the Circle

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

Age 7 to 11

Challenge Level

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

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Roll These Dice

Roll two red dice and a green dice. Add the two numbers on the red dice and take away the number on the green. What are all the different possible answers?

Age 7 to 11

Challenge Level