Friday, April 20, 2012

The Pyramid Curse

A few months ago my daughter received a letter from a friend that invited her to join a mail game. You buy a sheet of stickers and mail it to the first address on the list. You then copy the list, removing the first address and shifting the rest, adding yours as fifths at the end. Then you mail this list with instructions to 5 of your friends. Wait a few weeks and then expect hundreds if not thousands of stickers by mail.
Sounds mathematically correct.

Such examples of pyramid behavior are everywhere:

  1. an avalanche
  2. a viral email with good jokes, scary health advisory, slides of the weird asphalt painting, gorgeous sunsets (that by-the-way never comes to your email box once)

Getting back to the Pyramid games, such as the sticker game my daughter received.  We all tried it at some point in our lives and were surprised to find out that they usually do not work. While one math reason makes them attractive, another math reason prevents them from running smoothly. What is it?

Your answers are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.


Jerome said...

Before actually answering this, I'd like to direct you to one of the most famous chain letters ever divised. It's this one.

It should be of particular interest to anyone who reads this column. The solutions are pretty interesting.

The history of these chain letters can be found here.

The way to limit the effectiveness of any chain letter is simply to ignore it. A broken link is a broken chain so to speak. For example if your name is at the bottom of the list and each of the people you send your name to decide to throw what you've sent away, you are toast.

TracyZ said...

When I see a new problem labeled "easy", I remember the last "easy" puzzle on this site (SEVEN - NINE = EIGHT; yikes!), but I'll give this one a go. :)

I think the main mathematical reason that pyramid schemes don't work, be they a chain letter or an investment, is that they assume that each generation of participants in the scheme will be able to find enough new participants to add for the next generation.

In the case of your daughter and the letter, there is the assumption that that your daughter knows five new people to mail the letter to, and that those five people each know five NEW people to mail the letter to, and so on. In this type of scheme, you quickly run out of new people to bring into it.

As you said, your daughter and her friends, and then the friends of her friends, and the friends of those friends would have to add up to 5^5 or 3,125, which seems unlikely.

Taking the scheme a few more generations:
5^6 = 15,625
5^7 = 78,125
5^8 = 390,625
5^9 = 1,953,125
5^10 = 9,765,625

9,765,625 is a number bigger than the 2010 population of 42 of the 50 US states (CA, FL, IL, MI, NY, OH, PA, and TX excepted), though the letter scheme would have collapsed well before that.

Ilya said...

At some point, new participants will inevitably mail those people who already participated at least once, because there is a finite number of people in the world. And of course those people will likely smell a rat, and not participate the second time. Structures of local social graphs and people's choices of their 5 friends will determine how quickly duplications will start happening. And if there was a way to check and not mail someone more than once, then at some point people would simple run out of candidates to mail. At least these kinds of pyramids are easy to see. But when perpetrators such as Madoff fabricate financial statements, all the math and common sense in the world might not protect you.

Anonymous said...

The formula for any type of infection, whether, computer or disease or gossip, etc. is:
Number of carriers after T periods = I times e with the exponent of (S-1)T. e is exponential growth. S is the spread factor. This is the most important number. If the spread factor is below 1, then the infection will die out.
Vice versa if the spread factor is over 1. I don't think the spread factor is over 1 since most people will be too lazy to do this or think it is a scam.


Jerome's wife said...

The best way to terminate a chain letter is to not send the one you are responsible for.

Maria said...

While one math reason makes such pyramid operations attractive, another math reason prevents them from running smoothly. You guys are right - this math reason is the "finite number of people that can play". Even assuming the ideal game scenario when everyone participates and spreads it to 5 new participants who haven't played so far, with the exponential speed of the game we quite quickly run out of the potential game players. See the convincing TracyZ's numbers. In our case these are 7-9 year old girls who are interested in stickers.

Other reasons are duplicates and only partial participation.

Check out Jerome's links on how to get a Gold Miners Fortune and the history of such chain letters.

Thank you! A puzzle point for everyone.

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