Just in case it is true that the Public Service revels in its paper work, here is a simple practical task for the morning tea break.

Suppose we have a large sheet of very thin paper, say 0.025 mm thick – that is, a thousand sheets will give a thickness of 2.5 centimetres, or about 1 inch.

We tear the paper in half, and put the two pieces together, one on top of the other. We tear them in half, and put the four pieces together in a pile, tear them in half and put the eight pieces together in a pile, and so on.

If we tear and put together fifty times, how high will the final stack of paper be? As high as the ceiling? As high as a ten storey building? As high as the moon?!

Hint 1

To begin with, forget the problem with the paper, and experiment with the doubling sequence:

1, 2, 4, 8, 16, 32, ...

Experiment with a calculator to get a feel of what is happening here.

Hint 2

How many terms give a magnification of 1000? How many more terms to give 1000 times this?

Solution

The consequences of repeated doubling are well known to those who work with finance as a special case of compound interest. Let us try to get a rough idea of the effect.

Consider the sequence    1, 2, 4, 8, 16, 32, ... .

Mathematically, we can write this as powers of 2:
         2⁰, 2¹, 2², 2³, 2⁴, 2⁵, ... .

Evaluating successive powers, we find that 2¹⁰ = 1024.
Similarly 2²⁰ = 2¹⁰ x 2 x ... x 2 = 2¹⁰ x 2¹⁰. This means that doubling ten times gives a multiplication of over 1000, and a further doubling ten times gives a multiplication of this result of more than 1000 – that is, over a million. It is clear that the numbers here get very large.

In terms of the original problem, the final stack of paper is 250 x 0.025 x 10⁶ km <sup>high. In this product, the first term is the multiplying factor caused by the repeated doubling, the second factor is the thickness of the paper, and the final factor converts the measurement units from millimetres to kilometres. This final height is over 28 million kilometres!

To get some idea of this immense distance, the distance of the moon from the earth is only 384 000 km, the diameter of the sun is 1.392 million km, and the mean distance of the earth from the sun is 149.6 million km.</sup>

Extensions

Suppose that at each step of the problem the paper is simply folded without tearing. A physical problem occurs here. Take a sheet of newspaper. How many times can you repeatedly fold this sheet?

What do you find?

Hint 1

Hint 2

Solution

Extensions