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?

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

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

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

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?

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?

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

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?

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?

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

An environment which simulates working with 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.

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....?

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?

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

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?

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

A game for 2 people that everybody knows. You can play with a friend or online. If you play correctly you never lose!

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

Can you complete this jigsaw of the multiplication square?

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

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

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

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!

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?

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

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

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

This was a problem for our birthday website. Can you use four of these pieces to form a square? How about making a square with all five pieces?

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

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?

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

Can you explain the strategy for winning this game with any target?

Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

Work out the fractions to match the cards with the same amount of money.

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

An interactive game to be played on your own or with friends. Imagine you are having a party. Each person takes it in turns to stand behind the chair where they will get the most chocolate.

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.

Find out how we can describe the "symmetries" of this triangle and investigate some combinations of rotating and flipping it.

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?

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

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

In this activity, the computer chooses a times table and shifts it. Can you work out the table and the shift each time?

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

A game for 2 players. Can be played online. One player has 1 red counter, the other has 4 blue. The red counter needs to reach the other side, and the blue needs to trap the red.