Puzzle | Can 2 persons be with same number of hairs on their heads?

One day, John and Jessica were searching Google to find the maximum number of hairs on a human head. They found that the maximum number is 200,000. Then Jessica thought for a while and suddenly made a statement that there are at least two citizens of New York who have exactly the same number of hairs on their heads. Both of them knew that the population of New York is 12.3 million. However, John was still thinking about whether Jessica’s statement was correct or not.

Is Jessica’s statement 100% correct, or do we need more information to determine its truth?

Check if you were right - full answer with solution below. 

Solution:

Step 1: Arrange people by the number of hairs

Assume that all the citizens of New York are sorted according to the number of hairs on their heads.

Step 2: Create virtual bins

Imagine creating virtual bins for each possible number of hairs:

0,1,2,3,…,200,0000.

This gives a total of 200,001 bins.

Each bin represents people having the same number of hairs.

Step 3: Compare people and bins

• Total number of citizens = 12.3 million
• Total number of possible hair counts (bins) = 200,001

Since the number of people is much larger than the number of bins, at least two people must fall into the same bin.

This means at least two people have exactly the same number of hairs.

Step 4: Identify the principle used

This problem is based on the Pigeonhole Principle, which states that if more items are distributed into fewer containers, at least one container must hold more than one item.

There is no probability involved, and the conclusion is guaranteed.

Explanation:

Suppose we assume that: The total number of people in the city with at most 10,000 hairs is 2.3 million.

After removing these people from the total population:

12.3 million−2.3 million = 10 million people remaining

Now, the remaining hair-count bins are:

10,001, 10,002...................200,000

That gives only 190,000 bins.

Since 10 million people must fit into 190,000 bins, at least two people must share the same number of hairs, and that number must be greater than 10,000.

So, Jessica’s statement is absolutely correct. No additional information is required, and the result follows directly from the Pigeonhole Principle.