On your trip to London you meet a man.
“Leaving the bald people aside, I can bet a hundred bucks that there are two people living in London who have same number of hairs on their heads”.
You think that even if the man counts it would be difficult for him to find 2 people with same number of hair so you accept the bet.
The man instead of counting hair of even a single man convinces you that there are two people living in London who have same number of hairs on their heads.
So were you able to solve the riddle? Leave your answers in the comment section below.
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Answer:
This riddle is based on the pigeonhole principle, which states that if n items are put into m containers, with n > m, then at least one container must contain more than one item.
We can demonstrate there must be at least two people in London with the same number of hairs on their heads as follows.
Since a typical human head has an average of around 150,000 hairs; it is reasonable to assume (as an upper bound) that no one has more than 1,000,000 hairs on their head (m = 1 million holes).
There are more than 1,000,000 people in London (n is bigger than 1 million items).
Assigning a pigeonhole to each number of hairs on a person’s head, and assign people to pigeonholes according to the number of hairs on their head, there must be at least two people assigned to the same pigeonhole by the 1,000,001st assignment (because they have the same number of hairs on their heads) (or, n > m).
For the average case (m = 150,000) with the constraint: fewest overlaps, there will be at most one person assigned to every pigeonhole and the 150,001st person assigned to the same pigeonhole as someone else.
In the absence of this constraint, there may be empty pigeonholes because the “collision” happens before we get to the 150,001st person.
The principle just proves the existence of an overlap; it says nothing of the number of overlaps (which falls under the subject of Probability Distribution).
