Chapter 1: The Pigeonhole Principle in Action

Have you ever wondered if there are at least two individuals in your city with the same number of hairs on their heads? Excluding those who are bald, which certainly includes many individuals, a reasonable maximum estimate for the number of hairs on a human scalp is about 200,000. In an urban setting with a population exceeding this number, the answer to the question posed is undoubtedly affirmative.

This scenario illustrates the Pigeonhole Principle. This principle asserts that if you distribute m items into n containers, where m exceeds n, at least one container will contain more than one item.

For instance, imagine residing in a bustling metropolis like Tokyo, home to around 14 million residents. With only 200,000 possible hair counts, it follows that there must be at least two people with the same number of hairs. If each hair count were to be unique, the population of Tokyo would have to be 200,000 or fewer, which is clearly not the case.

Furthermore, considering that 14 million divided by 200,000 equals 70, it can be concluded that there are at least 70 individuals in Tokyo who share the same hair count. If fewer than 70 individuals had the same hair count, the total population would drop below 14 million, which contradicts the known figure.

Section 1.1: Applications of the Pigeonhole Principle

Another compelling application of this principle involves the common question of whether two individuals in a sports stadium have the same birthday. If a stadium houses 50,000 spectators, with only 366 possible birthday choices (accounting for leap years), it’s safe to say that at least a pair of attendees will share a birthday. In fact, dividing 50,000 by 366 yields approximately 136.61, indicating that at least 137 people are likely to have the same birthday.

Chapter 2: Exploring Hair Counts and Birthdays

The first video titled "How Many Humans Have the Same Number of Body Hairs? | Infinite Series" dives into this intriguing topic, exploring the mathematical implications and providing visual insights into the Pigeonhole Principle.

The second video, "The Truth About Where Hair Extensions Come From | Shady | Refinery29," sheds light on the origins of hair extensions, enhancing the discussion about hair counts and their societal implications.