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?
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 you’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.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?
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?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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 ...?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
Find out what a "fault-free" rectangle is and try to make some of your own.
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?
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?
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?
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.
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.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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 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?
Can you explain the strategy for winning this game with any target?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
Got It game for an adult and child. How can you play so that you know you will always win?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you find all the ways to get 15 at the top of this triangle of numbers?
Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.
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.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
Find the sum of all three-digit numbers each of whose digits is odd.
It starts quite simple but great opportunities for number discoveries and patterns!
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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.
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
An investigation that gives you the opportunity to make and justify predictions.