Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
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.
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. . . .
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 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 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?
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.
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Can all unit fractions be written as the sum of two unit fractions?
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. . . .
Can you find sets of sloping lines that enclose a square?
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.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
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. . . .
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
Can you tangle yourself up and reach any fraction?
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?
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.
Explore the effect of combining enlargements.
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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 would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
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?
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. . . .
It starts quite simple but great opportunities for number discoveries and patterns!
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
Can you describe this route to infinity? Where will the arrows take you next?
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.
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. . . .
Explore the effect of reflecting in two intersecting mirror lines.
Can you explain the strategy for winning this game with any target?
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
Explore the effect of reflecting in two parallel mirror lines.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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.
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?