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.
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?
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?
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Use the animation to help you work out how many lines are needed to draw mystic roses of different 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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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
Find out what a "fault-free" rectangle is and try to make some of
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
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?
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 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?
Can you find all the ways to get 15 at the top of this triangle of numbers?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle 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?
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.
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This task follows on from Build it Up and takes the ideas into three dimensions!
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
An investigation that gives you the opportunity to make and justify
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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.
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
What happens when you round these three-digit numbers to the nearest 100?
What happens when you round these numbers to the nearest whole number?
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 two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Got It game for an adult and child. How can you play so that you know you will always win?
It starts quite simple but great opportunities for number discoveries and patterns!
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?