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?

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?

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

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?

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

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

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?

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

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?

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

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

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

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?

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

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

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

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

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

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

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

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

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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?

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

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.

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.

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

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.

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

A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.

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

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?

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

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

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?

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

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

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

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

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

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

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

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

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?

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?