Make some loops out of regular hexagons. What rules can you discover?

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

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

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. . . .

Can you find a rule which relates triangular numbers to square numbers?

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.

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

Can you find a rule which connects consecutive triangular numbers?

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

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

The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

Write down a three-digit number Change the order of the digits to get a different number Find the difference between the two three digit numbers Follow the rest of the instructions then try. . . .

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

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

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

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?

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. . . .

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?

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

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?

Think of a number Multiply it by 3 Add 6 Take away your start number Divide by 2 Take away your number. (You have finished with 3!) HOW DOES THIS WORK?

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

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

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. . . .

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

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

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?

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?

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

A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

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

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

Kyle and his teacher disagree about his test score - who is right?

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. . . .

Can you explain why a sequence of operations always gives you perfect squares?

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

Can you find rectangles where the value of the area is the same as the value of the perimeter?

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.

Choose any four consecutive even numbers. Multiply the two middle numbers together. Multiply the first and last numbers. Now subtract your second answer from the first. Try it with your own. . . .

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?

What is the total number of squares that can be made on a 5 by 5 geoboard?

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

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