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.
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?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
Take a counter and surround it by a ring of other counters that
MUST touch two others. How many are needed?
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?
Explore the effect of reflecting in two intersecting mirror lines.
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?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Explore the effect of reflecting in two parallel mirror lines.
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?
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.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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 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?
Can you explain how this card trick works?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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.
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 dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Here are two kinds of spirals for you to explore. 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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
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.
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?
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
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?
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?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?