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

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?

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

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. . . .

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

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

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)?

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

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?

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?

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

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

Can you find the area of a parallelogram defined by two vectors?

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

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?

Triangle ABC is isosceles while triangle DEF is equilateral. Find one angle in terms of the other two.

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. . . .

Two motorboats travelling up and down a lake at constant speeds leave opposite ends A and B at the same instant, passing each other, for the first time 600 metres from A, and on their return, 400. . . .

Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

A circle is inscribed in a triangle which has side lengths of 8, 15 and 17 cm. What is the radius of the circle?

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

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

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

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

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...

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 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. . . .

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

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

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?

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

In a three-dimensional version of noughts and crosses, how many winning lines can you make?

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

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

32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

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?

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?

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.

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

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

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

Take a few whole numbers away from a triangle number. If you know the mean of the remaining numbers can you find the triangle number and which numbers were removed?

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

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?

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

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?

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .