Question

A chain-letter scheme is a famous (and usually illegal) get-rich-quick scheme. A person \(X\) receives a letter with, say, five names on it. \(X\) sends 10 to the person whose name is at the top of the list. \(X\) then deletes that name from the top of the list, adds his or her own name to the bottom of the list, and sends the letter to five "friends," all within one day. In around two weeks, \(X\) is supposed to receive 31,250. Suppose every person who receives the letter follows the instructions (including sending 10 to the person listed first!). Show that if there are only finitely many people, the scheme cannot work (in some sense of "cannot work" that you should make precise). Show that if there are countably infinitely many people, the scheme can work.

Short answer

The scheme fails with finite people but works with countably infinite people.

Step-by-step solution

Understanding the Scheme

When person X sends the letter to five friends, each friend is expected to repeat the process. This action leads to exponential growth in the number of people required to follow the scheme each day.

Analyzing Finite Case

If there are finitely many people, each level of the scheme requires 5 times more people than the previous level. After a few iterations, the required number of new participants will exceed the available people, making it impossible to sustain the scheme indefinitely with a finite population.

Understanding Countably Infinite Case

A countably infinite set means there is always a next candidate to receive the letter, emulating endless growth similar to how natural numbers (1, 2, 3, ...) continue infinitely. In this scenario, each level of the scheme can continue to find new participants.

Conclusion

In a finite population, the scheme fails when additional participants cannot be found. However, with a countably infinite population, there will always be another participant to continue the chain, theoretically allowing the scheme to work indefinitely.

Key concepts

Exponential Growth

Exponential growth is a key component of many phenomena, including chain reactions in chemistry and the spread of biological populations. In the described chain letter scheme, exponential growth occurs as each participant is instructed to recruit five new people. This growth pattern can be understood by considering how the number of participants expands: every new step multiplies the total number by five.

Mathematically, if you start with one participant, the equation to find the total number of participants after "n" steps is given by the formula:

• Initial participants = 1
• First step = 5
• Second step = 5²
• Third step = 5³
• ...and so on...

Therefore, after "n" levels, the total number of participants needed will be 5ⁿ. This rapid increase is an archetype of exponential growth, where each new stage drastically increases the number of people required to sustain the process.

Infinite Sets

Infinite sets are collections of elements that do not have a finite number of members. For example, the set of all natural numbers {1, 2, 3, ...} is infinite because there is no end to the numbers you can count. Infinite sets are divided into different kinds, including countable and uncountable infinities.

In the context of the chain letter scheme, an infinite set signifies that there is an unending pool of individuals who could potentially receive the letter. In a finite population, the process will eventually stall because there won't be enough new people to involve. But in an infinite scenario, this limitation is theoretically absent, pending other practical constraints.

Countable Infinity

Countable infinity refers to a type of infinity that can be matched one-to-one with the set of natural numbers. This means that, although infinite, the set can still be "listed" or "counted" in a sequence. A classic example is the set of integers which can be arranged as {..., -3, -2, -1, 0, 1, 2, 3, ...}.

In the exercise, an indefinite pool of participants, represented as a countably infinite set, allows the chain scheme to "work" in an idealized sense. You can continually find another participant because each person can be associated with a unique natural number in the sequence. This mathematical property ensures that the process theoretically never runs out of people in a countably infinite setting.

Mathematical Induction

Mathematical induction is a proof technique used primarily for proving statements about natural numbers. It involves two steps: the base case and the inductive step. In the base case, you prove a statement for an initial value, often zero or one. The inductive step assumes the statement holds for an arbitrary number "k" and then proves it for "k+1".

Though not directly applicable to the primary problem in the exercise, mathematical induction can illustrate why each step of spreading the chain letter continues to adhere to the same principles. By demonstrating that if the scheme works for "n" participants, it must work for "n+1" when continuous, endless growth can be assumed. This proof style solidifies understanding of processes exhibited across infinite sequences, much like the stepwise expansion in a chain letter scheme when assuming unlimited participants.