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?
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.
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle 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?
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?
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?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Here are two kinds of spirals for you to explore. What do you notice?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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.
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?
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.
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?
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?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Delight your friends with this cunning trick! Can you explain how
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
This challenge asks you to imagine a snake coiling on itself.
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 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?
A collection of games on the NIM theme
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
This activity involves rounding four-digit numbers to the nearest thousand.
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?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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.
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.
Can you tangle yourself up and reach any fraction?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
This task follows on from Build it Up and takes the ideas into three dimensions!
Can you find all the ways to get 15 at the top of this triangle of numbers?
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?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Find out what a "fault-free" rectangle is and try to make some of
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.
How many centimetres of rope will I need to make another mat just
like the one I have here?