Two people with exactly the same number of hairs

Question

A human head has at most about 150,000 hairs. A large city, say London, has about 9 million residents. Show that at least two residents of the city have exactly the same number of hairs on their heads. Then say how large a group of people is needed before this is guaranteed, and whether the argument tells you which two people they are.

Accepted Answer

Yes, it is certain, by the pigeonhole principle. The argument. Every resident has some number of hairs between 0 and 150,000 inclusive: that is 150,001 possible values. Picture 150,001 boxes, one per value, and place each of the 9 million residents in the box for their hair count. Since 9,000,000 is far more than 150,001, some box must contain at least two people. Those two have identical hair counts. In fact, dividing 9,000,000 by 150,001 shows that some box contains at least 60 people. How many people are needed. Any group of 150,002 people is enough to force a match, because 150,002 people…

Two people with exactly the same number of hairs

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