Math CentralQuandaries & Queries


Question from Lesley, a teacher:

In the mail you receive a chain letter with a list of ten names on it. The instructions tell you to send $2 to the person named on the top of the list, cross his name out, and place your name on the bottom. You then have to send ten copies of this letter to friends with the same instructions. If no one breaks the chain how much money should you receive?

Hi Lesley.

I wouldn't recommend giving this question to your students. What's described would be illegal here in Canada and most other countries; it's called a "pyramid scheme". If you do give this question to your class it deserves a discussion about pyramid schemes and why they are illegal.

You'll never get any money! Just the people near the beginning of the chain will receive any money, and since the first person wrote down ten names (aliases or accomplices), only they get paid.

Let's say you and 9 others received the letter and were the first non-criminals to get it. We'll call the names on the list C1 to C10 (criminals 1 through 10).

You send $2 off to C1. Now you cross off C1 and write LESLEY at the bottom and following the rule, you send it to 10 friends (you won't have them for long!). Let's call them V1-V10 (victims 1 through 10).

C1 got $20 from you and the other nine people who got the same letter as you. Each of them made 10 copies with their own names on it, but all 90 of them plus your ten friends sent $2 to C2. So C2 collects $200. And C3 gets $2,000, C4 gets $20,000...all the way down to C10, who receives $20 trillion dollars from the ten trillion people involved.

Oh wait...there aren't a ten trillion people on the planet! So you see nothing.

Let's say there were though...

Eventually you receive money but only from the 10th generation below you. The number of people involved (all this assumes that no one is contacted twice; also rather unrealistic) under you is 10 trillion people. So the answer is $20 trillion.

But this is a makebelieve world. In reality, you'd be a victim to be swindled and never see a cent.

On the other hand, you may earn a decent prison sentence if you get involved!

Even this is not the way it really works. Criminals would get more money by having more people get the first copy, so why would they just make 1 copy and send it to you - why not send it to a thousand people? As well, later people would be tempted to make more than 10 copies so that their own name appears on more lists and they will get more money.

Stephen La Rocque and Harley Weston

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