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?
Delight your friends with this cunning trick! Can you explain how
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
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.
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
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.
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Can you find the values at the vertices when you know the values on
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?
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A collection of games on the NIM theme
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.
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Find some examples of pairs of numbers such that their sum is a
factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and
16 is a factor of 48.
The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
Consider all two digit numbers (10, 11, . . . ,99). In writing down
all these numbers, which digits occur least often, and which occur
most often ? What about three digit numbers, four digit numbers. . . .
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?
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 see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Got It game for an adult and child. How can you play so that you know you will always win?
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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.
It starts quite simple but great opportunities for number discoveries and patterns!
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?