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.

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.

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?

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

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

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

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

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

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.

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?

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

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

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

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.

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

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

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.

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

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?

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?

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

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

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

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

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.?

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

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

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?

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?

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

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?

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

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

Four numbers on an intersection that need to be placed in the surrounding cells. That is all you need to know to solve this sudoku.

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?

How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

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?

Whenever a monkey has peaches, he always keeps a fraction of them each day, gives the rest away, and then eats one. How long could he make his peaches last for?

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?

Four small numbers give the clue to the contents of the four surrounding cells.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

This package contains a collection of problems from the NRICH website that could be suitable for students who have a good understanding of Factors and Multiples and who feel ready to take on some. . . .