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The Mathematics of Cons – Pyramid Selling

Pyramid schemes are a very old con – but whilst illegal, still exist in various forms. Understanding the maths behind them therefore is a good way to avoid losing your savings!

The most basic version of the fraud starts with an individual making the following proposition, “pay me \$1000 to join the club, all you then need to do is recruit 6 more people to the club (paying \$1000 each) and you will have made a \$5000 profit.”

There are lots of variations – and now that most people are aware of pyramid selling, now normally revolve around multi-level-marketing (MLM).  These are often still pyramid schemes, but encourage participants to believe it is a genuine business by actually having a sales product which members have to sell.  However the main focus of the business is still the same – taking money off people who then make their money back after having signed up a set number of new recruits.

The following graphic from Consumer Fraud Reporting is a clear mathematical demonstration why these frauds only end up enriching those at the top of the pyramid:

You can see that if the requirement was to recruit 8 new members, that by the 9th level you would need to have 1 billion people already signed up.  Even with the need to recruit just 4 new members you still have rapid exponential growth which very quickly means you will run out of new potential members.  For pyramid schemes it is only those in the first 3-4 levels (the white cells) that stand any real chance of making money  – and these levels are usually filled by those in on the scam.

Ponzi schemes (like that run by Bernie Madoff) use a similar method.  A conman takes money from investors promising (say) 10% annual returns.  Lots of investors sign up.  The conman then is able to use the lump sum investments to pay the 10% annual returns.  This scam can last for years, with people thinking that they are getting a good rate of return, only to find out eventually that actually their lump sum investment has gone.

This is a good topic to look at with graphs (plotting exponential growth), interest rates, or exponential sequences – and shows why understanding maths is an important financial skill.

If you like this topic you might also like:

Benford’s Law – Using Maths to Catch Fraudsters – the surprising mathematical law that helps catch criminals.

Amanda Knox and Bad Maths in Courts – when misunderstanding mathematics can have huge consequences .

Essential resources for IB students:

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.

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All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

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P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.