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?

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

Can you find a rule which connects consecutive triangular numbers?

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?

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

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?

How good are you at finding the formula for a number pattern ?

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?

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.

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

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

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?

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?

Show that all pentagonal numbers are one third of a triangular 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

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

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

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

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

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

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. . . .

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?

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

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

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?

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?

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}.

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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.

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?

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

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?

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

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

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

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.

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. . . .

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

I added together the first 'n' positive integers and found that my answer was a 3 digit number in which all the digits were the same...

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

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

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?

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

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?

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