Everyone in Zimbabwe knows about the infamous `savings clubs' which absorbed so many people's savings some mon for joining the club. A then receives four certificates wonths ago. Similar events in Albania led to civil war! In this article, we analyse the mathematics of such clubs.
Suppose A joins the savings crediting each of the four names on a certificate he has acquired with $50 and also crediting the `Money Maker' account with $50 as the conditiith his name at the bottom. He then (in what is called a `round' of sales) sells these to four other people, who each receive (once they have parted with club ``Money Maker'' by $250 in total) four certificates with A's in the third position and their name in the fourth. The process continues till A is at the first position, and at the next sale, his name disappears on the new certificates. He is said to have `harvested.'
Now for the analysis. Let us assume the system works perfectly, ie everyone manages to sell their certificates and all the sales in a round of sales happen at the same time. This is just to make our life easier.
For convenience let us say that A's sale of four certificates is regarded as the first. A harvests after 4 sales, so his name appears on 4 + 42 + 43 + 44 =(4/3)(44 - 1) = 340 certificates. He harvests $17000. Pretty good for an investment of just $250! No wonder people join these clubs.
Before the first sale there is only one member. After the first sale there are 4 new members. After the second sale there are 16 new members, etc. So after the nth sale there are 4n new members, 1+4+...+4n=(1/3)(4n+1 - 1) in all.
So for example, after only a dozen sales the club has had 22 million members! (Of course, people who join k times are counted k times.) And since each person paid $50 to join, the management of the club have over a billion dollars in their accounts! No wonder people form these clubs!
It takes four sales for people to harvest, so 1+4+...+4n-4=(1/3)(4n-3 - 1) people harvest.
So after 12 sales, only about 87 000 people harvest. In fact, at any time, only about ((1/3)(4n-3-1))/((1/3)(4n+1-1)), which is approximately a sixteenth. So for every 1000 who invest, 4 can be expected to harvest.
It is now quite obvious how these clubs crash. As hopefully few of you know from personal experience, a point is reached where no-one can sell their certificates any more. For everyone who wants a certificate already has one! And it is equally clear why the directors of the clubs suddenly acquire a fleet of Mercs. For one person's gain is another person's loss --- over 99% of their customers fail to harvest, and only about 80% (you can estimate this) even manage to get their money back.
Judah Makonye is a lecturer at Bondolfi Teachers' College, Masvingo, Zimbabwe
