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?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
How many centimetres of rope will I need to make another mat just
like the one I have here?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
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?
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.
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?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Here are two kinds of spirals for you to explore. What do you notice?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
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?
Find out what a "fault-free" rectangle is and try to make some of
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?
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.
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
This task follows on from Build it Up and takes the ideas into three dimensions!
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 find all the ways to get 15 at the top of this triangle of numbers?
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
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.
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?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
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?
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 work out how to win this game of Nim? Does it matter if you go first or second?
This challenge asks you to imagine a snake coiling on itself.
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 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?
Find the sum of all three-digit numbers each of whose digits is
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
It starts quite simple but great opportunities for number discoveries and patterns!
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
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10