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.

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?

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

A student in a maths class was trying to get some information from her teacher. She was given some clues and then the teacher ended by saying, "Well, how old are they?"

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

Countries from across the world competed in a sports tournament. Can you devise an efficient strategy to work out the order in which they finished?

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

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

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

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

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

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. . . .

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

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

Can you use the information to find out which cards I have used?

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

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?

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

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

The clues for this Sudoku are the product of the numbers in adjacent squares.

The letters of the word ABACUS have been arranged in the shape of a triangle. How many different ways can you find to read the word ABACUS from this triangular pattern?

A few extra challenges set by some young NRICH members.

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.?

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?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

An extra constraint means this Sudoku requires you to think in diagonals as well as horizontal and vertical lines and boxes of nine.

The puzzle can be solved by finding the values of the unknown digits (all indicated by asterisks) in the squares of the $9\times9$ grid.

This is a variation of sudoku which contains a set of special clue-numbers. Each set of 4 small digits stands for the numbers in the four cells of the grid adjacent to this set.

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.

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.

Bellringers have a special way to write down the patterns they ring. Learn about these patterns and draw some of your own.

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

A challenging activity focusing on finding all possible ways of stacking rods.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

Two sudokus in one. Challenge yourself to make the necessary connections.

Each of the main diagonals of this sudoku must contain the numbers 1 to 9 and each rectangle width the numbers 1 to 4.

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

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

Rather than using the numbers 1-9, this sudoku uses the nine different letters used to make the words "Advent Calendar".

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?

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.