![Pinned Squares](/sites/default/files/styles/medium/public/thumbnails/content-99-07-six4-icon.gif?itok=Izbfoz0u)
Creating and manipulating expressions and formulae
![Pinned Squares](/sites/default/files/styles/medium/public/thumbnails/content-99-07-six4-icon.gif?itok=Izbfoz0u)
![Lower Bound](/sites/default/files/styles/medium/public/thumbnails/content-99-07-six2-icon.jpg?itok=jMF87hfR)
problem
Lower Bound
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
![Pareq Calc](/sites/default/files/styles/medium/public/thumbnails/content-99-06-six5-icon.jpg?itok=KjlEgv2i)
problem
Pareq Calc
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 lines are 1 unit and 2 units.
![How big?](/sites/default/files/styles/medium/public/thumbnails/content-99-06-six3-icon.jpg?itok=zjvwiWlo)
problem
How big?
If the sides of the triangle in the diagram are 3, 4 and 5, what is
the area of the shaded square?
![The Pillar of Chios](/sites/default/files/styles/medium/public/thumbnails/content-99-05-six6-icon.jpg?itok=Tr95BTpT)
problem
The Pillar of Chios
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
![One and three](/sites/default/files/styles/medium/public/thumbnails/content-99-05-six4-icon.jpg?itok=lZ0sSWI1)
problem
One and three
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 metres from B. How long is the lake?
![Mindreader](/sites/default/files/styles/medium/public/thumbnails/content-99-04-six5-icon.jpg?itok=HySsEe3g)
problem
Mindreader
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 the last two digits of her answer. Now you can really amaze her by giving the whole answer and the three consecutive numbers used at the start.
![Hand Swap](/sites/default/files/styles/medium/public/thumbnails/content-99-03-six3-icon.jpg?itok=aOJIkHQa)
problem
Hand Swap
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 minute
hand and hour hand had swopped places. What time did the train
leave London and how long did the journey take?
![Janine's Conjecture](/sites/default/files/styles/medium/public/thumbnails/content-98-07-six3-icon.jpg?itok=Wrx0XlQy)
problem
Janine's Conjecture
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. Does
this always work? Can you prove or disprove this conjecture?
![Boxed In](/sites/default/files/styles/medium/public/thumbnails/content-98-05-six1-icon.jpg?itok=IyOmaKi1)
problem
Boxed In
A box has faces with areas 3, 12 and 25 square centimetres. What is
the volume of the box?