This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

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

Stop the Clock game for an adult and child. How can you make sure you always win this game?

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

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

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

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

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

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

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

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

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?

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?

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

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

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

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?

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

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

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

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

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?

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

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?

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

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

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

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

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

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.

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

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

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?

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.

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

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

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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?

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?

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 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 starts quite simple but great opportunities for number discoveries and patterns!

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?

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

This challenge is about finding the difference between numbers which have the same tens digit.

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.