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.
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?
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
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.
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
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 pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
For this challenge, you'll need to play Got It! Can you explain the
strategy for winning this game with any target?
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?
Can you find the values at the vertices when you know the values on
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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. . . .
It would be nice to have a strategy for disentangling any tangled
Can you tangle yourself up and reach any fraction?
Can you find sets of sloping lines that enclose a square?
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. . . .
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. . . .
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
It starts quite simple but great opportunities for number discoveries and patterns!
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Explore the effect of combining enlargements.
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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. . . .
Triangle numbers can be represented by a triangular array of
squares. What do you notice about the sum of identical triangle
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.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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?
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 see how to build a harmonic triangle? Can you work out the
next two rows?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
The Egyptians expressed all fractions as the sum of different unit
fractions. Here is a chance to explore how they could have written
Can all unit fractions be written as the sum of two unit fractions?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
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
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?