During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

If we had 16 light bars which digital numbers could we make? How will you know you've found them all?

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

These rectangles have been torn. How many squares did each one have inside it before it was ripped?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

This challenge extends the Plants investigation so now four or more children are involved.

When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

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?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

An investigation that gives you the opportunity to make and justify predictions.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

Can you draw a square in which the perimeter is numerically equal to the area?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

What is the date in February 2002 where the 8 digits are palindromic if the date is written in the British way?

Can you work out some different ways to balance this equation?

What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

What happens when you round these numbers to the nearest whole number?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

How many ways can you find of tiling the square patio, using square tiles of different sizes?

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Have a go at balancing this equation. Can you find different ways of doing it?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

The pages of my calendar have got mixed up. Can you sort them out?

In this matching game, you have to decide how long different events take.

My local DIY shop calculates the price of its windows according to the area of glass and the length of frame used. Can you work out how they arrived at these prices?

What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

These practical challenges are all about making a 'tray' and covering it with paper.

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?