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?
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.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
Delight your friends with this cunning trick! Can you explain how
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?
Can you continue this pattern of triangles and begin to predict how many sticks are used for each new "layer"?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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?
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.
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Compare the numbers of particular tiles in one or all of these
three designs, inspired by the floor tiles of a church in
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Here are two kinds of spirals for you to explore. What do you notice?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Find out what a "fault-free" rectangle is and try to make some of
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
How many centimetres of rope will I need to make another mat just
like the one I have here?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
An investigation that gives you the opportunity to make and justify
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 ...?
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Can you explain the strategy for winning this game with any target?
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
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?