How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
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.
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?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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.
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 find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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.
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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
This activity involves rounding four-digit numbers to the nearest thousand.
Can you describe this route to infinity? Where will the arrows take you next?
This challenge asks you to imagine a snake coiling on itself.
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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 centimetres of rope will I need to make another mat just
like the one I have here?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
Can you explain how this card trick works?
Here are two kinds of spirals for you to explore. What do you notice?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
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?
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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
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?
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?
Delight your friends with this cunning trick! Can you explain how
We can show that (x + 1)² = x² + 2x + 1 by considering
the area of an (x + 1) by (x + 1) square. Show in a similar way
that (x + 2)² = x² + 4x + 4
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Find the sum of all three-digit numbers each of whose digits is
Find out what a "fault-free" rectangle is and try to make some of
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
Can you tangle yourself up and reach any fraction?