Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain 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.

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

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?

Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

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

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

Find the five distinct digits N, R, I, C and H in the following nomogram

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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

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?

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

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 algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

The sums of the squares of three related numbers is also a perfect square - can you explain why?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.

Can you find a rule which connects consecutive triangular numbers?

Show that all pentagonal numbers are one third of a triangular number.

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up numbers)?

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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. . . .

A job needs three men but in fact six people do it. When it is finished they are all paid the same. How much was paid in total, and much does each man get if the money is shared as Fred suggests?

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Take any two numbers between 0 and 1. Prove that the sum of the numbers is always less than one plus their product?

Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.

Attach weights of 1, 2, 4, and 8 units to the four attachment points on the bar. Move the bar from side to side until you find a balance point. Is it possible to predict that position?

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?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

To investigate the relationship between the distance the ruler drops and the time taken, we need to do some mathematical modelling...

Can you make sense of these three proofs of Pythagoras' Theorem?

In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?