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.
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?
Can you find rectangles where the value of the area is the same as the value of the perimeter?
Can you maximise the area available to a grazing goat?
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
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".
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
If you move the tiles around, can you make squares with different coloured edges?
Which has the greatest area, a circle or a square inscribed in an isosceles, right angle triangle?
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.
If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
How many winning lines can you make in a three-dimensional version of noughts and crosses?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Different combinations of the weights available allow you to make different totals. Which totals can you make?
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?
The area of a square inscribed in a circle with a unit radius is, satisfyingly, 2. What is the area of a regular hexagon inscribed in a circle with a unit radius?
Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
How many different symmetrical shapes can you make by shading triangles or squares?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
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?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?
Explore the effect of combining enlargements.
Which set of numbers that add to 10 have the largest product?
Explore the effect of reflecting in two parallel mirror lines.
Can you describe this route to infinity? Where will the arrows take you next?
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. . . .
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?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Can all unit fractions be written as the sum of two unit fractions?
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?
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?
Here's a chance to work with large numbers...
The Egyptians expressed all fractions as the sum of different unit fractions. The Greedy Algorithm might provide us with an efficient way of doing this.
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
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?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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.
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.
How many solutions can you find to this sum? Each of the different letters stands for a different number.
An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?
A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?