Hair Count and the Pigeonhole Principle

The question asks under what condition we can guarantee that at least two persons in a city have the exact same number of hair on their heads, given that each person has at least one hair.

This problem can be solved using a fundamental concept in mathematics called the Pigeonhole Principle.

Understanding the Pigeonhole Principle

The Pigeonhole Principle states that if you have more "pigeons" than "pigeonholes," and you try to put each pigeon into a pigeonhole, then at least one pigeonhole must contain more than one pigeon.

Applying Pigeonhole Principle to Hair Count

Let's define the terms in the context of our problem:

• Pigeons: The persons in the city.
• Pigeonholes: The possible number of hairs a person can have on their head.

We are told that each person has at least one hair. This means the number of hairs a person can have is 1 or more.

There is also a biological limit to the maximum number of hairs a human can have on their head. Let's call this maximum possible number of hair \(M\).

So, the possible number of hairs a person can have ranges from 1 up to \(M\). The distinct possible values for the number of hairs are:

\(1, 2, 3, \ldots, M\)

The number of these distinct possible hair counts (our pigeonholes) is \(M\).

According to the Pigeonhole Principle, if the number of pigeons (people) is greater than the number of pigeonholes (distinct possible hair counts), then at least two pigeons must occupy the same pigeonhole. In other words, if the population of the city is greater than the maximum possible number of hair \(M\), then at least two persons must have the same number of hairs.

So, the guarantee that at least two persons have the same number of hair on their heads happens when the population of the city is greater than the maximum possible number of hair on a head.

Analyzing the Options

Let's look at the given options based on our understanding:

1. is greater than the maximum possible number of hair on the head.
2. is less than the maximum possible number of hair on the head.
3. has at least one pair of identical twins.
4. is genetically homogeneous.

Option 1 directly matches our conclusion from applying the Pigeonhole Principle. If the population is greater than the maximum possible number of hair, there are more people than distinct hair counts, guaranteeing a match.

Option 2 is the opposite and would not guarantee a match. You could potentially have everyone with a different hair count if the population is less than or equal to the number of distinct hair counts.

Options 3 and 4 are not directly related to the mathematical guarantee based on population size and the range of hair counts. Identical twins might have similar hair counts, but it doesn't provide a general guarantee for the entire population based on number alone. Genetic homogeneity is also not the condition derived from the Pigeonhole Principle.

Therefore, the condition that guarantees at least two persons have exactly the same number of hair on their heads is when the population of the city is greater than the maximum possible number of hair on the head.