A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?
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?
Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?
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. . . .
Explore when it is possible to construct a circle which just
touches all four sides of a quadrilateral.
Do you know a quick way to check if a number is a multiple of two? How about three, four or six?
How many different symmetrical shapes can you make by shading triangles or squares?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
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 find rectangles where the value of the area is the same as the value of the perimeter?
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?
Take any four digit number. Move the first digit to the 'back of
the queue' and move the rest along. Now add your two numbers. What
properties do your answers always have?
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?
Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?
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?
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?
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
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?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
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 napkin is folded so that a corner coincides with the midpoint of
an opposite edge . Investigate the three triangles formed .
Which set of numbers that add to 10 have the largest product?
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
The clues for this Sudoku are the product of the numbers in adjacent squares.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
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...
The number 2.525252525252.... can be written as a fraction. What is
the sum of the denominator and numerator?
What is the greatest volume you can get for a rectangular (cuboid)
parcel if the maximum combined length and girth are 2 metres?
Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?
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?
Is there an efficient way to work out how many factors a large number has?
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?
Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...
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?
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
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. . . .
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?
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.
What is the same and what is different about these circle
questions? What connections 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?
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
The diagonals of a trapezium divide it into four parts. Can you
create a trapezium where three of those parts are equal in area?
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. . . .
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Explore the effect of reflecting in two parallel mirror lines.
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".
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?
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?