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?

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

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

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

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

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.

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

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

Which countries have the most naturally athletic populations?

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

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?

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

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

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

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

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

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

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?

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

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

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

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".

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

What's the largest volume of box you can make from a square of paper?

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

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

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

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...

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

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

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

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 develop a strategy to work out the rules controlling each light?

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

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?

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.

Explore the relationships between different paper sizes.

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

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?

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

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

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?

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?

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?