# Too many pigeons for the holes?

While learning for my discrete mathematics exam, I stumbled across the pigeonhole principle. A pigeonhole is where pigeons like to sit. I started liking it, when I discovered, that it is so obvious in the metaphor but, at least for me, took a while to become able to really apply it. The pigeon metaphor goes like this:

If there are more pigeons than pigeonholes, at least one hole will host at least two pigeons.

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Obvious, but

How many people in Sydney have the same amount of hair on their head?

Lets estimate the maximum amount of hair: 150K. In Sydney live around 5.3M people. Preliminary conclusion: At least 2 people in Sydney have the exact same amount of hair.

The pigeonhole principle goes further: It also tells us, how many people (at least) in Sydney have the same amount of hair. Just divide the pigeons by the pigeonholes and ceil the number.

5.3M / 150K ~= 35,3 ceiled to 36.

Conclusion: At least 36 people in Sydney have the exact same amount of hair.

I believe there are many real world situations, in which the pigeonhole principle makes a lot of sense, and I cannot wait to stumble across some of them. To be honest: For this example, it would be more effective to count (or estimate) the amount of bald people, assuming they are the biggest group with the same amount of hair.

Homework: At my wedding we had 125 guests. There were 6 different main courses and 3 different side dishes, and each guest took one main course and one side dish on their plate. How many people at least had the exact same meal?

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