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

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?

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

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

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?

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?

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?

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. . . .

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?

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

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

Is there an efficient way to work out how many factors a large number has?

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

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

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. . . .

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

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

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?

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

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

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

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

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

Two ladders are propped up against facing walls. The end of the first ladder is 10 metres above the foot of the first wall. The end of the second ladder is 5 metres above the foot of the second. . . .

Your school has been left a million pounds in the will of an ex- pupil. What model of investment and spending would you use in order to ensure the best return on the money?

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

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?

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

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

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

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

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

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

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

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using the digits 1 to 9 each once and only once. The number 4396 can be written as just such a product. Can. . . .

Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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?

Ben passed a third of his counters to Jack, Jack passed a quarter of his counters to Emma and Emma passed a fifth of her counters to Ben. After this they all had the same number of counters.

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?

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

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?

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

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