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?

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.

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 solutions can you find to this sum? Each of the different letters stands for a different number.

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

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?

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?

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?

Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?

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

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

Use the differences to find the solution to this Sudoku.

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

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

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. . . .

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

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

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

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

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.

The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?

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?

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

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

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

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

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

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?

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?

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.

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

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

Chris and Jo put two red and four blue ribbons in a box. They each pick a ribbon from the box without looking. Jo wins if the two ribbons are the same colour. Is the game fair?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

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?

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?

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

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

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. . . .

Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .

Explore the effect of combining enlargements.

Explore the effect of reflecting in two parallel mirror lines.

Each of the following shapes is made from arcs of a circle of radius r. What is the perimeter of a shape with 3, 4, 5 and n "nodes".

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

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

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

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

A car's milometer reads 4631 miles and the trip meter has 173.3 on it. How many more miles must the car travel before the two numbers contain the same digits in the same order?

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