In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

Which countries have the most naturally athletic populations?

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?

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?

Can you find a way to identify times tables after they have been shifted up?

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?

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

There are nasty versions of this dice game but we'll start with the nice ones...

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

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

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

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 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 work out which spinners were used to generate the frequency charts?

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

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

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Six balls of various colours are randomly shaken into a trianglular arrangement. What is the probability of having at least one red in the corner?

7 balls are shaken in a container. You win if the two blue balls touch. What is the probability of winning?

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

Can you find the values at the vertices when you know the values on the edges?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

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

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

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

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

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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?

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

There are many different methods to solve this geometrical problem - how many can you find?

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Which set of numbers that add to 10 have the largest product?

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

There are lots of different methods to find out what the shapes are worth - how many can you find?

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

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".

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

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?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

You'll need to know your number properties to win a game of Statement Snap...

Can you find any two-digit numbers that satisfy all of these statements?