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### Eight Hidden Squares

##### Stage: 2 and 3 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?

### Music to My Ears

##### Stage: 2 Challenge Level:

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

##### Stage: 2 Challenge Level:

How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.

##### Stage: 2 Challenge Level:

This practical problem challenges you to make quadrilaterals with a loop of string. You'll need some friends to help!

### Nine-pin Triangles

##### Stage: 2 Challenge Level:

How many different triangles can you make on a circular pegboard that has nine pegs?

### Ten Hidden Squares

##### Stage: 2 Challenge Level:

These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?

### Square It

##### Stage: 1, 2, 3 and 4 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.

##### Stage: 1 Challenge Level:

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

### Cut Nets

##### Stage: 2 Challenge Level:

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

### Fractional Triangles

##### Stage: 2 Challenge Level:

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

### Let Us Reflect

##### Stage: 2 Challenge Level:

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

### Counting Cards

##### Stage: 2 Challenge Level:

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?

### Skeleton Shapes

##### Stage: 1 Challenge Level:

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

### Square Corners

##### Stage: 2 Challenge Level:

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?

### A Puzzling Cube

##### Stage: 2 Challenge Level:

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?

##### Stage: 2 Challenge Level:

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

### Neighbours

##### Stage: 2 Challenge Level:

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?

### Happy Halving

##### Stage: 1 Challenge Level:

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

### One Big Triangle

##### Stage: 1 Challenge Level:

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

### A City of Towers

##### Stage: 1 Challenge Level:

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?

### Display Boards

##### Stage: 2 Challenge Level:

Design an arrangement of display boards in the school hall which fits the requirements of different people.

### Square it for Two

##### Stage: 1, 2, 3 and 4 Challenge Level:

Square It game for an adult and child. Can you come up with a way of always winning this game?

### How Would We Count?

##### Stage: 1 and 2 Challenge Level:

An activity centred around observations of dots and how we visualise number arrangement patterns.

### Take One Example

##### Stage: 1 and 2

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

### Poly Plug Pattern

##### Stage: 1 Challenge Level:

Create a pattern on the left-hand grid. How could you extend your pattern on the right-hand grid?

### 3D Stacks

##### Stage: 2 and 3 Challenge Level:

Can you find a way of representing these arrangements of balls?

### Introducing NRICH TWILGO

##### Stage: 1, 2, 3, 4 and 5 Challenge Level:

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

### Auditorium Steps

##### Stage: 2 and 3 Challenge Level:

What is the shape of wrapping paper that you would need to completely wrap this model?

### Triple Cubes

##### Stage: 1 and 2 Challenge Level:

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

### More Pebbles

##### Stage: 2 and 3 Challenge Level:

Have a go at this 3D extension to the Pebbles problem.

### Celtic Knot

##### Stage: 2 Challenge Level:

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

### 28 and It's Upward and Onward

##### Stage: 2 Challenge Level:

Can you find ways of joining cubes together so that 28 faces are visible?

### Coded Hundred Square

##### Stage: 2 Challenge Level:

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

### Holes

##### Stage: 1 and 2 Challenge Level:

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

### Hexpentas

##### Stage: 1 and 2 Challenge Level:

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

### Baravelle

##### Stage: 2, 3 and 4 Challenge Level:

What can you see? What do you notice? What questions can you ask?

### Thinking Through, and By, Visualising

##### Stage: 2, 3 and 4

This article is based on some of the ideas that emerged during the production of a book which takes visualising as its focus. We began to identify problems which helped us to take a structured view. . . .

### Air Nets

##### Stage: 2, 3, 4 and 5 Challenge Level:

Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.

### Putting Two and Two Together

##### Stage: 2 Challenge Level:

In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?

### Triangular Faces

##### Stage: 2 Challenge Level:

This problem invites you to build 3D shapes using two different triangles. Can you make the shapes from the pictures?

### Reef and Granny

##### Stage: 2 Challenge Level:

Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?

### How Many Pieces This Time?

##### Stage: 1 Challenge Level:

How many pieces of string have been used in these patterns? Can you describe how you know?

### How Many Pieces?

##### Stage: 1 Challenge Level:

How many loops of string have been used to make these patterns?

### Start Cube Drilling

##### Stage: 1 Challenge Level:

Imagine a 3 by 3 by 3 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will have holes drilled through them?

### Overlapping Again

##### Stage: 2 Challenge Level:

What shape is the overlap when you slide one of these shapes half way across another? Can you picture it in your head? Use the interactivity to check your visualisation.

### Overlaps

##### Stage: 1 Challenge Level:

What does the overlap of these two shapes look like? Try picturing it in your head and then use the interactivity to test your prediction.

### More Building with Cubes

##### Stage: 2 Challenge Level:

Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!

### Building with Cubes

##### Stage: 1 Challenge Level:

Try to picture these buildings of cubes in your head. Can you make them to check whether you had imagined them correctly?

### Painted Faces

##### Stage: 2 Challenge Level:

Imagine a 3 by 3 by 3 cube made of 9 small cubes. Each face of the large cube is painted a different colour. How many small cubes will have two painted faces? Where are they?

### Flip

##### Stage: 2 Challenge Level:

Can you picture where this letter "F" will be on the grid if you flip it in these different ways?