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?

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?

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

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

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

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

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

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

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

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

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.

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

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

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?

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?

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?

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

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

This activity involves rounding four-digit numbers to the nearest thousand.

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

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

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

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

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

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?

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

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?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

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?

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

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

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

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

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

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.

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

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

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

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

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

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

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?

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

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?

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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.