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?
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
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Can you explain how this card trick works?
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?
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 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?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
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?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
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.
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
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 tangle yourself up and reach any fraction?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
A collection of games on the NIM theme
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?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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.
Can you find all the ways to get 15 at the top of this triangle of numbers?
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.
This task follows on from Build it Up and takes the ideas into three dimensions!
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
It would be nice to have a strategy for disentangling any tangled
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.
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
It starts quite simple but great opportunities for number discoveries and patterns!