Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.

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

Starting with the number 180, take away 9 again and again, joining up the dots as you go. Watch out - don't join all the dots!

Try out the lottery that is played in a far-away land. What is the chance of winning?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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

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

If you have only four weights, where could you place them in order to balance this equaliser?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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

Here is a chance to play a version of the classic Countdown Game.

An environment which simulates working with Cuisenaire rods.

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

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....?

Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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

Can you find all the different triangles on these peg boards, and find their angles?

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Our 2008 Advent Calendar has a 'Making Maths' activity for every day in the run-up to Christmas.

NRICH December 2006 advent calendar - a new tangram for each day in the run-up to Christmas.

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Can you complete this jigsaw of the multiplication square?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Can you find a reliable strategy for choosing coordinates that will locate the robber in the minimum number of guesses?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

Can you fit the tangram pieces into the outline of Granma T?

Can you locate the lost giraffe? Input coordinates to help you search and find the giraffe in the fewest guesses.

Train game for an adult and child. Who will be the first to make the train?

Ahmed has some wooden planks to use for three sides of a rabbit run against the shed. What quadrilaterals would he be able to make with the planks of different lengths?

An interactive game for 1 person. You are given a rectangle with 50 squares on it. Roll the dice to get a percentage between 2 and 100. How many squares is this? Keep going until you get 100. . . .

Exchange the positions of the two sets of counters in the least possible number of moves

These formulae are often quoted, but rarely proved. In this article, we derive the formulae for the volumes of a square-based pyramid and a cone, using relatively simple mathematical concepts.