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

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

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?

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

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.

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

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

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

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

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?

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.

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

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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 explain the strategy for winning this game with any target?

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

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

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

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

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

What happens when you round these numbers to the nearest whole number?

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?

What happens when you round these three-digit numbers to the nearest 100?

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

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?

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

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

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

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?

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

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

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

How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?

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?

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

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

Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the numbers on each circle add up to the same amount. Can you find the rule for giving another set of six numbers?

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

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

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 centimetres of rope will I need to make another mat just like the one I have here?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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?

Here are two kinds of spirals for you to explore. What do you notice?