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?
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?
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.
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?
If it takes four men one day to build a wall, how long does it take
60,000 men to build a similar wall?
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?
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.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
Can you maximise the area available to a grazing goat?
There are four children in a family, two girls, Kate and Sally, and
two boys, Tom and Ben. How old are the children?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
Explore the effect of reflecting in two parallel mirror lines.
Explore the effect of combining enlargements.
Here are four tiles. They can be arranged in a 2 by 2 square so that this large square has a green edge. If the tiles are moved around, we can make a 2 by 2 square with a blue edge... Now try to. . . .
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Sissa cleverly asked the King for a reward that sounded quite modest but turned out to be rather large...
What is the smallest number with exactly 14 divisors?
How many different symmetrical shapes can you make by shading triangles or 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?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
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?
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G =
F and A-H represent the numbers from 0 to 7 Find the values of A,
B, C, D, E, F and H.
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
How many solutions can you find to this sum? Each of the different letters stands for a different number.
A car's milometer reads 4631 miles and the trip meter has 173.3 on
it. How many more miles must the car travel before the two numbers
contain the same digits in the same order?
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.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
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?
A jigsaw where pieces only go together if the fractions are
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
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?
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?
Can all unit fractions be written as the sum of two unit fractions?
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
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.
Here's a chance to work with large numbers...
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?