Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

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

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

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?

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

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

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.

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

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?

Delight your friends with this cunning trick! Can you explain how it works?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

Can you find all the ways to get 15 at the top of this triangle of numbers?

This task follows on from Build it Up and takes the ideas into three dimensions!

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

While we were sorting some papers we found 3 strange sheets which seemed to come from small books but there were page numbers at the foot of each page. Did the pages come from the same book?

Got It game for an adult and child. How can you play so that you know you will always win?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?