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?

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.

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

Use your skill and judgement to match the sets of random data.

Can you recreate squares and rhombuses if you are only given a side or a diagonal?

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?

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

Which countries have the most naturally athletic populations?

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

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

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

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

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

If everyone in your class picked a number from 1 to 225, do you think any two people would pick the same number?

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

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

Can you work out the probability of winning the Mathsland National Lottery? Try our simulator to test out your ideas.

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

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?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Explore the relationships between different paper sizes.

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

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.

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

How well can you estimate 10 seconds? Investigate with our timing tool.

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

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?

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

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

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

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

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea?

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

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

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

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?

Play around with sets of five numbers and see what you can discover about different types of average...

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

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".

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

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?

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

Where should you start, if you want to finish back where you started?

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

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

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

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

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?

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