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. . . .
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
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.
A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?
How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?
If a sum invested gains 10% each year how long before it has doubled its value?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
The sums of the squares of three related numbers is also a perfect square - can you explain why?
If the hypotenuse (base) length is 100cm and if an extra line splits the base into 36cm and 64cm parts, what were the side lengths for the original right-angled triangle?
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 describe this route to infinity? Where will the arrows take you next?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Can all unit fractions be written as the sum of two unit fractions?
Show that is it impossible to have a tetrahedron whose six edges have lengths 10, 20, 30, 40, 50 and 60 units...
A circle of radius r touches two sides of a right angled triangle, sides x and y, and has its centre on the hypotenuse. Can you prove the formula linking x, y and r?
What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?
Can you find the area of a parallelogram defined by two vectors?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
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.
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?
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.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?
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?
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?
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
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.
Can you arrange these numbers into 7 subsets, each of three numbers, so that when the numbers in each are added together, they make seven consecutive numbers?
Can you work out the dimensions of the three cubes?
There are lots of different methods to find out what the shapes are worth - how many can you find?
All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
Can you maximise the area available to a grazing goat?
Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?
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?