Prove that two non-bald residents of NYC have exactly the same number of hairs. In New York City there are two non-bald people who have the same number of hairs ( the human head can contain up to several hundred thousands with maximum of about 500,000) How can I prove the pigeonhole principle of the two non-bald people in New York City ? The pigeonhole principle: if k is a positive integer and k+1 or more objects are placed into k boxes then there is at least one box containing two or more of the objects.

Let's say the max number of hairs is 500,000 (which i think is rreeaallyy high..) so the claim is true for any city with 500,000+ (non bald) people. (there should be an assumption that there are more then 500,000 non bald people). proof: try to give each person a different number of hairs, 1,2,3... After 500,000 people, you've run out of different numbers. meaning the 500,001th person will have to get a number of hairs which is equal to some other person's number of hairs. These two are in the same pigeonhole