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?
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?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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 see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
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?
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?
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
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?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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 many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Can you explain the strategy for winning this game with any target?
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 asks you to imagine a snake coiling on itself.
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.
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Find out what a "fault-free" rectangle is and try to make some of your own.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
Can you find sets of sloping lines that enclose a square?
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.
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?
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.
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Here are two kinds of spirals for you to explore. What do you notice?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
It's easy to work out the areas of most squares that we meet, but what if they were tilted?
Can you tangle yourself up and reach any fraction?
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
A collection of games on the NIM theme
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.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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.