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?
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.
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
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Find out what a "fault-free" rectangle is and try to make some of
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
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.
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
Can you find all the ways to get 15 at the top of this triangle of 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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
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.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Are these statements relating to odd and even numbers always true, sometimes true or never true?
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
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.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
An investigation that gives you the opportunity to make and justify
Explore the effect of reflecting in two intersecting mirror lines.
Here are two kinds of spirals for you to explore. What do you notice?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How many centimetres of rope will I need to make another mat just
like the one I have here?
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?
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.
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?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?