Editing Pigeonhole principle (section)

=== Hair counting ===
One can demonstrate there must be at least two people in [[London]] with the same number of hairs on their heads as follows.<ref>{{cite book|title=The Psychology of Reasoning|first=Eugenio|last=Rignano|translator-first=Winifred A.|translator-last=Holl|publisher=K. Paul, Trench, Trubner & Company, Limited|year=1923|page=72|isbn=9780415191326 |url=https://books.google.com/books?id=1i9VAAAAMAAJ&pg=PA72}}</ref><ref>To avoid a slightly messier presentation, this example only refers to people who are not bald.</ref> 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 {{math|(''m'' {{=}} 1 million}} holes). There are more than 1,000,000 people in London ({{math|''n''}} is bigger than 1&nbsp;million items). Assigning a pigeonhole to each number of hairs on a person's head, and assigning people to pigeonholes according to the number of hairs on their heads, 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, {{math|''n'' > ''m''}}). Assuming London has 9.002 million people,<ref>{{Cite web|url=http://data.london.gov.uk/dataset/londons-population|title=London's Population / Greater London Authority (GLA)|website=data.london.gov.uk}}</ref> it follows that at least ten Londoners have the same number of hairs, as having nine Londoners in each of the 1 million pigeonholes accounts for only 9 million people.

For the average case ({{math|1=''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 the 150,001st person. The principle just proves the existence of an overlap; it says nothing about the number of overlaps (which falls under the subject of [[probability distribution]]).{{Citation needed|date=February 2025}}

There is a passing, satirical, allusion in English to this version of the principle in ''A History of the Athenian Society'', prefixed to ''A Supplement to the Athenian Oracle: Being a Collection of the Remaining Questions and Answers in the Old Athenian Mercuries'' (printed for Andrew Bell, London, 1710).<ref>{{Cite web|url=https://books.google.com/books?id=JCwUAAAAQAAJ&q=mean+hairs|title = A Supplement to the Athenian Oracle: Being a Collection of the Remaining Questions and Answers in the Old Athenian Mercuries. ... To which is Prefix'd the History of the Athenian Society, ... By a Member of the Athenian Society|year = 1710}}</ref> It seems that the question ''whether there were any two persons in the World that have an equal number of hairs on their head?'' had been raised in ''The Athenian Mercury'' before 1704.<ref>{{Cite web|url=https://books.google.com/books?id=4QsUAAAAQAAJ&q=sent+quarters|title = The Athenian Oracle being an entire collection of all the valuable questions and answers|year = 1704}}</ref><ref>{{Cite web|url=https://books.google.com/books?id=GG0PAAAAQAAJ&q=town+eternity|title = The Athenian Oracle: Being an Entire Collection of All the Valuable Questions and Answers in the Old Athenian Mercuries. ... By a Member of the Athenian Society|year = 1704}}</ref>

Perhaps the first written reference to the pigeonhole principle appears in a short sentence from the French Jesuit [[Jean Leurechon]]'s 1622 work ''Selectæ Propositiones'':<ref name=leurechon/> "It is necessary that two men have the same number of hairs, [[écu]]s, or other things, as each other."<ref>{{citation|first=Jean|last=Leurechon|title=Selecæe Propositiones in Tota Sparsim Mathematica Pulcherrimæ|year=1622|publisher=Gasparem Bernardum|page=2}}</ref> The full principle was spelled out two years later, with additional examples, in another book that has often been attributed to Leurechon, but might be by one of his students.<ref name=leurechon/>