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?

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

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.

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

Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

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.

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

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.?

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

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

Can you use your powers of logic and deduction to work out the missing information in these sporty situations?

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

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

In this Sudoku, there are three coloured "islands" in the 9x9 grid. Within each "island" EVERY group of nine cells that form a 3x3 square must contain the numbers 1 through 9.

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.

Investigate the different ways you could split up these rooms so that you have double the number.

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

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

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

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?

Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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?

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?

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 second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.

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

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

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.

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

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

A few extra challenges set by some young NRICH members.

Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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