In most my articles I explain how maths can be applied to real world problems. Time to look at the real world applied to maths!
Pigeon keeping doesn’t seem to be popular in the UK, which is another thing I like about this country (you wouldn’t believe how messy it can get). This hobby gave a name to one of my favourite theorems, the pigeonhole principle. It’s a very powerful and also very intuitive – yes, bear with me, even if you think you “just can’t get maths”!
By Pigeons-in-holes.jpg by en:User:BenFrantzDale; this image by en:User:McKay [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)%5D, via Wikimedia Commons
Domestic pigeons live in pigeonholes. What happens if we have more birds than holes? Then at least one of them must contain two birds. Now you can start showing off – you just learned a very important theorem. Pigeonhole principle states precisely that if n items are put into m containers, with n > m > 0, then at least one container must contain more than one item. So if 10 pigeons will try to sleep in 9 pigeonholes, at least two of them cannot enjoy the luxury of a single room.
Why do we care about pigeons so much? This theorem has surprisingly many applications. Let’s take a look at a few of them.
- Take any book in English, open it at a random page and underline a sequence of 27 words. Now look at their first letters. I bet at least two of them start with the same letter, don’t they? That’s because English alphabet consists of 26 letters (“pigeonholes”) and you just picked 27 words (“pigeons”).
- In London we can find at least two people with the same number of hairs on their heads (even if we exclude bold citizens). We can have only several hundred thousand hairs (“pigeonholes”), while London has a few million residents (“pigeons”).
- Next time you go to Royal Albert Hall, you can bet your friends that there are at least two visitors with the same first and last initials. According to Wikipedia, this concert venue can seat up to 5,272 people (“pigeons”). To be on a safe side, I’ll assume that there are 40 letters for each initial, just because as a proper Pole I don’t want to forget about letters from different alphabets. This gives us 40×40=1600 possible initial combinations (“pigeonholes”). If the show is boring, go around and check id’s, maybe you’ll be lucky to find the initial twins!
- At least 2,733 attendees of London New Year’s Day Parade must share their birthday. Every year around one million people gather in London to celebrate the New Year. We have 366 possible birthdays, so on average there are 1,000,000/366=2,732.24 people per one date. But the maximum must be at least the average (think about it!), so there must be a birthday shared by at least 2733 spectators.
These are just fun facts easily proven by the pigeonhole principle. However, applications of this powerful theorem are ubiquitous in many areas of mathematics and other sciences. So next time you play musical chairs, you can count it as one of your 5-a-day maths exercises – because not only maths is fun, but also fun is maths!