Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
Can you see how to build a harmonic triangle? Can you work out the next two rows?
How many centimetres of rope will I need to make another mat just
like the one I have here?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
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. . . .
Can all unit fractions be written as the sum of two unit fractions?
This challenge asks you to imagine a snake coiling on itself.
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.
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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.
Great Granddad is very proud of his telegram from the Queen
congratulating him on his hundredth birthday and he has friends who
are even older than he is... When was he born?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you find the values at the vertices when you know the values on
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
A collection of games on the NIM theme
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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. . . .
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. . . .
Can you find sets of sloping lines that enclose a square?
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?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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?
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.
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.
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.
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
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!
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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?
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?
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. . . .
List any 3 numbers. It is always possible to find a subset of
adjacent numbers that add up to a multiple of 3. Can you explain
why and prove it?
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.