At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.
At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening?
According to Plutarch, the Greeks found all the rectangles with
integer sides, whose areas are equal to their perimeters. Can you
find them? What rectangular boxes, with integer sides, have their
surface areas equal to their volumes?
Take an A4 piece of paper and halve it by drawing a line across
the middle, parallel to the shorter side. Each half is called an A5
Halve the bottom half by drawing a vertical line down the middle.
This creates two A6 rectangles.
Halve the right hand one by drawing a horizontal line across its
middle. This creates two A7 rectangles.
Halve the bottom one by drawing a vertical line down the middle.
This creates two A8 rectangles.
You should have something like this:
Halve the right hand A8 shape by drawing a horizontal line
across its middle. This creates two A9 rectangles.
And so on ... Keep going until the rectangles are too small to be
Now draw the diagonal of the A4 piece of paper from the top left
corner to the bottom right corner. This creates a sequence of
triangles. The first two are numbered in the diagram below but you
will have many more drawn on your sheet:
What is the total area of the first two triangles
as a fraction of the original A4 rectangle?
What is the total area of the first three triangles as a fraction
of the original A4 rectangle?
If you could you go on adding all the triangles' areas, what do you
think the total would be as a fraction of the original A4
Many thanks to Professor Michael Sewell
for providing us with this idea.
It has a connection with Zeno's paradox. Can you find out about
What is the connection between the method used to find all the
triangles' areas and the method used to explain Zeno's paradox?