Can you find the values at the vertices when you know the values on the edges?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

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. . . .

An article which gives an account of some properties of magic squares.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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 an efficient method to work out how many handshakes there would be if hundreds of people met?

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. . . .

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

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?

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 diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = nĀ² Use the diagram to show that any odd number is the difference of two squares.

Can you find sets of sloping lines that enclose a square?

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?

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

It would be nice to have a strategy for disentangling any tangled ropes...

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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?

Can all unit fractions be written as the sum of two unit fractions?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Can you describe this route to infinity? Where will the arrows take you next?

It starts quite simple but great opportunities for number discoveries and patterns!

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

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?

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?”

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. . . .

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

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

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 three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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.

The diagram shows a 5 by 5 geoboard with 25 pins set out in a square array. Squares are made by stretching rubber bands round specific pins. What is the total number of squares that can be made on a. . . .

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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. . . .

These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.