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?
Many thanks to Michael from Worth School who has sent us a
solution to this difficult problem. Michael says:
The area of triangle 1 is a half of the top left hand
Area 1 = 1/4 x 1/2 = 1/8
The area of triangle 2 is a half of a half of a half of a
Area 2 = 1/4 x 1/2 x 1/2 x 1/2 = 1/32
And so on:
Area 3 = 1/4 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 = 1/128
Area 4 = 1/4 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2= 1/512
Area 5 = 1/4 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2 x 1/2=
Area (1+2) = 1/8 + 1/32 = 5/32 = 0.15625
Area (1+2+3) = 1/8 + 1/32 + 1/128 = 21/128 = 0.1640625
Area (1+2+3+4+5) = 341/2048 = 0.16650390625
1/6 = 0.166666666
The area seems to be tending towards 1/6.
Zeno's Paradox states that motion is impossible because in order
to get from A to B, one must first pass through the mid-point C.
But to get from A to C, you must first pass through the midpoint D,
and so on passing through midpoints E, F, G, etc.
This is similar in to the triangle problem, since there are an
infinite number of triangles, most of which have tiny areas. It is
for this reason that we say that the area 'tends towards
The way to resolve these two problems is very similar. In the
triangle question we have:
1/8 + 1/32 + 1/128 + 1/512 + 1/2048 + ... = 1/6
In Zeno's Paradox, assuming that you can cover this infinite
amount of small distances, you travel:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... = 1
This means that there is not really any problem, since the sum
of the tiny distances equals 1.
A very clear explanation, thank you Michael.
Bryony, Louise and Sarah from Caldicot School in South Wales
sent in images in Word documents which show their solutions. Thank
you girls - this makes it much easier to see the fractions. Here is
Bryony's, here is
Louise's and here is