Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
A circle is inscribed in a triangle which has side lengths of 8, 15
and 17 cm. What is the radius of the circle?
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
Find the missing angle between the two secants to the circle when
the two angles at the centre subtended by the arcs created by the
intersections of the secants and the circle are 50 and 120 degrees.
Fifteen students had to travel 60 miles. They could use a car, which could only carry 5 students. As the car left with the first 5 (at 40 miles per hour), the remaining 10 commenced hiking along the. . . .
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
My train left London between 6 a.m. and 7 a.m. and arrived in Paris
between 9 a.m. and 10 a.m. At the start and end of the journey the
hands on my watch were in exactly the same positions but the. . . .
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. . . .
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
A cyclist and a runner start off simultaneously around a race track each going at a constant speed. The cyclist goes all the way around and then catches up with the runner. He then instantly turns. . . .
Can you make sense of these three proofs of Pythagoras' Theorem?
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
In a three-dimensional version of noughts and crosses, how many winning lines can you make?
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Jo made a cube from some smaller cubes, painted some of the faces
of the large cube, and then took it apart again. 45 small cubes had
no paint on them at all. How many small cubes did Jo use?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
Triangle ABC is isosceles while triangle DEF is equilateral. Find
one angle in terms of the other two.
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Polygons drawn on square dotty paper have dots on their perimeter
(p) and often internal (i) ones as well. Find a relationship
between p, i and the area of the polygons.
Choose any two numbers. Call them a and b. Work out the arithmetic mean and the geometric mean. Which is bigger? Repeat for other pairs of numbers. What do you notice?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
A moveable screen slides along a mirrored corridor towards a
centrally placed light source. A ray of light from that source is
directed towards a wall of the corridor, which it strikes at 45
degrees. . . .
Take a few whole numbers away from a triangle number. If you know
the mean of the remaining numbers can you find the triangle number
and which numbers were removed?
Imagine starting with one yellow cube and covering it all over with
a single layer of red cubes, and then covering that cube with a
layer of blue cubes. How many red and blue cubes would you need?
How to build your own magic squares.
Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.
Think of a number and follow my instructions. Tell me your answer,
and I'll tell you what you started with! Can you explain how I
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Crosses can be drawn on number grids of various sizes. What do you
notice when you add opposite ends?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
Think of a number... follow the machine's instructions. I know what
your number is! Can you explain how I know?
Attach weights of 1, 2, 4, and 8 units to the four attachment
points on the bar. Move the bar from side to side until you find a
balance point. Is it possible to predict that position?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
If you take two tests and get a marks out of a maximum b in the first and c marks out of d in the second, does the mediant (a+c)/(b+d)lie between the results for the two tests separately.
A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Find b where 3723(base 10) = 123(base b).
A job needs three men but in fact six people do it. When it is
finished they are all paid the same. How much was paid in total,
and much does each man get if the money is shared as Fred suggests?
Jo has three numbers which she adds together in pairs. When she
does this she has three different totals: 11, 17 and 22 What are
the three numbers Jo had to start with?”
Many numbers can be expressed as the difference of two perfect
squares. What do you notice about the numbers you CANNOT make?
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 little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .