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 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?

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.

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

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

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

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

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

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?

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

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

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.

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

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?

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

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

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

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

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

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

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

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

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?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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?

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.

Can you find all the different ways of lining up these Cuisenaire rods?

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

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?

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?

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

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.

Find out what a "fault-free" rectangle is and try to make some of your own.

The puzzle can be solved with the help of small clue-numbers which are either placed on the border lines between selected pairs of neighbouring squares of the grid or placed after slash marks on. . . .

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

60 pieces and a challenge. What can you make and how many of the pieces can you use creating skeleton polyhedra?

Number problems at primary level that require careful consideration.

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 Sudoku, based on differences. Using the one clue number can you find the solution?