Can all unit fractions be written as the sum of two unit fractions?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
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?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Can you find the values at the vertices when you know the values on
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.
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.
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
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
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?
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.
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
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. . . .
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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?
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. . . .
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.
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
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?
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.
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!
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
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.
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
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
It would be nice to have a strategy for disentangling any tangled
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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
Can you see how to build a harmonic triangle? Can you work out the next two rows?