Use algebra to reason why 16 and 32 are impossible to create as the
sum of consecutive numbers.
A, B & C own a half, a third and a sixth of a coin collection.
Each grab some coins, return some, then share equally what they had
put back, finishing with their own share. How rich are they?
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?
Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?
Balance the bar with the three weight on the inside.
Label this plum tree graph to make it totally magic!
How to build your own magic squares.
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 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 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?
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?
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
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?
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?
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. . . .
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?
The number 27 is special because it is three times the sum of its digits 27 = 3 (2 + 7). Find some two digit numbers that are SEVEN times the sum of their digits (seven-up 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 know?
The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21....
How many Fibonacci sequences can you find containing the number 196
as one of the terms?
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?
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. . . .
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 number... follow the machine's instructions. I know what
your number is! Can you explain how I know?
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. . . .
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
Can you make sense of these three proofs of Pythagoras' Theorem?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
Use the numbers in the box below to make the base of a top-heavy
pyramid whose top number is 200.
Manufacturers need to minimise the amount of material used to make
their product. What is the best cross-section for a gutter?
The sums of the squares of three related numbers is also a perfect
square - can you explain why?
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?
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
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. . . .
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Can you use the diagram to prove the AM-GM inequality?
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
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. . . .
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?
Can you find the area of a parallelogram defined by two vectors?
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
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. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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?
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 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. . . .
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. . . .
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.