A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Explore the effect of reflecting in two parallel mirror lines.

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

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

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?

If you move the tiles around, can you make squares with different coloured edges?

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares?

Explore the effect of combining enlargements.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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

Can you describe this route to infinity? Where will the arrows take you next?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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

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.

How many different symmetrical shapes can you make by shading triangles or squares?

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?

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 difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Different combinations of the weights available allow you to make different totals. Which totals can you make?

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

A game for 2 or more people, based on the traditional card game Rummy. Players aim to make two `tricks', where each trick has to consist of a picture of a shape, a name that describes that shape, and. . . .

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Can all unit fractions be written as the sum of two unit fractions?

There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?

Five children went into the sweet shop after school. There were choco bars, chews, mini eggs and lollypops, all costing under 50p. Suggest a way in which Nathan could spend all his money.

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Some people offer advice on how to win at games of chance, or how to influence probability in your favour. Can you decide whether advice is good or not?

Is it always possible to combine two paints made up in the ratios 1:x and 1:y and turn them into paint made up in the ratio a:b ? Can you find an efficent way of doing this?

A decorator can buy pink paint from two manufacturers. What is the least number he would need of each type in order to produce different shades of pink.

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

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?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?

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?

Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...

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. . . .

The clues for this Sudoku are the product of the numbers in adjacent squares.

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Here is a chance to create some attractive images by rotating shapes through multiples of 90 degrees, or 30 degrees, or 72 degrees or...

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

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Do you know a quick way to check if a number is a multiple of two? How about three, four or six?

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.