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.

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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

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

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

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

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?

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?

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?

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?

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

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

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. . . .

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

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 connects consecutive triangular numbers?

A, B & C own a half, a third and a sixth of a coin collection. Each grab some coins, return some, then share equally what they had put back, finishing with their own share. How rich are they?

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

Create some shapes by combining two or more rectangles. What can you say about the areas and perimeters of the shapes you can make?

What are the possible dimensions of a rectangular hallway if the number of tiles around the perimeter is exactly half the total number of tiles?

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

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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

This article explains how to make your own magic square to mark a special occasion with the special date of your choice on the top line.

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?

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

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

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

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

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

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. . . .

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

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?

How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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

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. . . .

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

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. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.

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

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

The Number Jumbler can always work out your chosen symbol. Can you work out how?

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?

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?

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?

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. . . .