It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
This challenge asks you to imagine a snake coiling on itself.
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?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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.
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?
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?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
How many ways can you find to do up all four buttons on my coat?
How about if I had five buttons? Six ...?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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?
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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. . . .
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
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?
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.
Can you find sets of sloping lines that enclose a square?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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?
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?
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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. . . .
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 find the values at the vertices when you know the values on
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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9,
12, 15... other squares? 8, 11, 14... other squares?
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.
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.
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.
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?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.