Mr Gilderdale is playing a game with his class. What rule might he have chosen? How would you test your idea?

Investigate which numbers make these lights come on. What is the smallest number you can find that lights up all the lights?

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 complete this jigsaw of the multiplication square?

Each light in this interactivity turns on according to a rule. What happens when you enter different numbers? Can you find the smallest number that lights up all four lights?

Work out how to light up the single light. What's the rule?

Use the interactivities to complete these Venn diagrams.

Use the interactivity to sort these numbers into sets. Can you give each set a name?

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

Yasmin and Zach have some bears to share. Which numbers of bears can they share so that there are none left over?

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

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.

Imagine a wheel with different markings painted on it at regular intervals. Can you predict the colour of the 18th mark? The 100th mark?

Arrange any number of counters from these 18 on the grid to make a rectangle. What numbers of counters make rectangles? How many different rectangles can you make with each number of counters?

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....?

Incey Wincey Spider game for an adult and child. Will Incey get to the top of the drainpipe?

Play this well-known game against the computer where each player is equally likely to choose scissors, paper or rock. Why not try the variations too?

Use the interactivity to find out how many quarter turns the man must rotate through to look like each of the pictures.

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

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

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?

The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?

A game for 2 people that can be played on line or with pens and paper. Combine your knowledege of coordinates with your skills of strategic thinking.

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

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. . . .

An interactive activity for one to experiment with a tricky tessellation

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?

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!

Take it in turns to place a domino on the grid. One to be placed horizontally and the other vertically. Can you make it impossible for your opponent to play?

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

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

Move just three of the circles so that the triangle faces in the opposite direction.

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

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

Can you hang weights in the right place to make the equaliser balance?

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

Complete the squares - but be warned some are trickier than they look!

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

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

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

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

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