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
The number of plants in Mr McGregor's magic potting shed increases
overnight. He'd like to put the same number of plants in each of
his gardens, planting one garden each day. How can he do it?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
It starts quite simple but great opportunities for number discoveries and patterns!
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.
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?
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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. . . .
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 describe this route to infinity? Where will the arrows take you next?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
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. . . .
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. . . .
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?”
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can all unit fractions be written as the sum of two unit fractions?
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.
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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?
Choose any 3 digits and make a 6 digit number by repeating the 3
digits in the same order (e.g. 594594). Explain why whatever digits
you choose the number will always be divisible by 7, 11 and 13.
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.
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?
Can you tangle yourself up and reach any fraction?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Can you find the values at the vertices when you know the values on
It would be nice to have a strategy for disentangling any tangled
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you find sets of sloping lines that enclose a square?
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?
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Explore the effect of reflecting in two parallel mirror lines.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .