Poor Man's space travel

by Richard Knottenbelt

I met a youngster rummaging through a dust bin. He seemed to be interested only in large sheets of paper.

"What are you doing?"

"I am trying to get to the moon."

"Are you going to make a paper spaceship?"

"No, it's much simpler than that."

He put one piece of newspaper down and stood on it.

"I am now nearer to the moon."

He doubled the paper and stood on that. Then doubled again and stood on top of that -- there were now 4 thicknesses of paper, say a total of 4/10 of a millimetre and he carried on doubling.

After a few more doublings I began to get the idea. It is roughly 400000km to the moon. How many times must he double?

Surprisingly, the answer is only 43.

The pattern is 1, 2, 4, 8, 16,... , each term doubling the previous one. Such a sequence is called a Geometric Progression and the nth term is given by 2^n-1. The equation 2^n-1=400000 x 10⁶ x 10 is solved as follows.

Take logs on both sides (any base will do) to get

		         log (2n-1)  =   log (4 x 1012)
		       (n-1)log 2   =   log(4 x 1012)
			     n-1    =   (log(4 x 1012))/(log 2)
		             n-1    =   41.86 (approx)
		               n    =   42.86 (approx)

But n must be an integer and 42 will be too small. So n=43, i.e. 43 doublings are needed.

Geometric Progressions (GP's) often have terms which get very big like this one. For some GP's however, the terms get smaller, look at the series1, 1/2, 1/4, 1/8, 1/16,... for example.

The population explosion is often analysed by considering population increase, e.g. 3% as being added at the end of a year. The GP is made by multiplying by 1.03 to get successive terms.

``Population increases Geometrically: Food increases arithmetically. Population will therefore always outstrip food supply until famine, disease or war ensue.''

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Richard Knottenbelt is a very active and well-known maths teacher in Zimbabwe. He currently teaches with his wife at Victoria High, Masvingo.