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. . . .
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. . . .
In a snooker game the brown ball was on the lip of the pocket but
it could not be hit directly as the black ball was in the way. How
could it be potted by playing the white ball off a cushion?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Can you show that you can share a square pizza equally between two
people by cutting it four times using vertical, horizontal and
diagonal cuts through any point inside the square?
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?
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.
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Think of a number... follow the machine's instructions. I know what
your number is! Can you explain how I know?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
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. . . .
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. . . .
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
Can you convince me of each of the following: If a square number is
multiplied by a square number the product is ALWAYS a square
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
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?
Label this plum tree graph to make it totally magic!
Can you use the diagram to prove the AM-GM inequality?
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 know?
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?
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?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.
Show that all pentagonal numbers are one third of a triangular number.
Can you find a rule which connects consecutive triangular numbers?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
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?
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?
How to build your own magic squares.
Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number.
Cross out the numbers on the same row and column. Repeat this
process. Add up you four numbers. Why do they always add up to 34?
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
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 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. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
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?
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?”
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?
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?
32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50
x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if
Janine noticed, while studying some cube numbers, that if you take
three consecutive whole numbers and multiply them together and then
add the middle number of the three, you get the middle number. . . .
A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?