Don’t take you hand round you head to touch your nose!

 By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

07 September 2014

A story is told that in the early 1960s, the American Space Exploration Programme spent a million dollars on research to develop a biro pen that can work in zero gravity. They figured that astronauts in space would need to do many calculations on paper (remember this was long before the pocket calculators) and so such a pen would be essential. For their part, the Soviets did not encounter this problem because they just used pencils! Apparently, this was one of the major reasons why the Soviets sent a man to space before the Americans.

I don’t know the truth about that story but it illustrates the danger of seeking a complicated solution to a simple problem. I remembered the tale after reading reactions to my comment last week about the number of telephone numbers that can fit in one prefix code. Some readers insisted said that my answer of one million was wrong because I did not take into account the mathematics of factorials.

Well, for those who have forgotten (as well as those who don’t know), the factorial of a number, say, 5, is 5 x 4 x 3 x 2 x 1 = 120. This is written mathematically as “5!”. Similarly, 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040 and so on.

Factorials are used in calculating the number of arrangements that can be made with given quantity of objects. Suppose you have two blocks; one black (B), the other white (W). How many ways can you arrange them along a straight line? The answer is simple: there are only two possible arrangements, namely, B-W and W-B.

What if a third grey (G) block was added? Here are the possible patterns: B-G-W, B-W-G, G-B-W, G-W-B, W-B-G, and W-G-B. That is, a total of six ways.

Now, when there are two objects you can get 2 (=2 x 1 = 2!) arrangements. When there are three objects, you get 6 (=3 x 2 x 1 = 3!) patterns.

With this knowledge, some readers thought that this factorial concept must somehow come into the number of telephone numbers that can be accommodated in a prefix code.

Well; that is not so. A complicated way finding out how many 6-digit numbers can be formed involves first recognising that we have ten digits to chose from. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Next we ask how many choices we have for the first place in our number. Obviously, the answer is ten.

Since repetition of digits is allowed (e.g., 0733 – 444555 is a valid telephone number), we have another ten choices for the second place; another ten for the third place and so on. Thus the total number of possible telephone numbers is 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000.

Now according to my history teacher, when Columbus went on his voyage to the Americas, he was wondering why sailors were going around Africa on their way to India. Doing that, he argued, was similar to “taking your hand around you head in order to touch your nose”.

I can say the same regarding the method I have used here to work out the number of telephone numbers accommodated in one prefix code. It is much easier to count them as I did last week.