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.

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

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

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

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

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

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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

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

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

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

This challenge asks you to imagine a snake coiling on itself.

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

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

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'.

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?

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

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

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

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

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

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

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

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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?

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

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

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

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?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

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?

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

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Can you find sets of sloping lines that enclose a square?

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?

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

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

Watch this animation. What do you see? Can you explain why this happens?

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?

What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.

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 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?