Birthdays and paradoxes

I was sitting down, thinking of an amusing article to write, and I didn’t quite feel up to writing. It’s been a long weekend, and besides, it’s a special day for me today.

32 years old

So I thought, “What the heck”.

Then I thought I should at least leave you with something. So I’ll introduce you to the concept of the birthday paradox. Basically, given a group of 23 people, what’s the percentage chance that any 2 persons in that group share the same birthday? (hint: it’s higher than you think. Here’s a less clinical explanation of the paradox than the one from Wikipedia.).

The barber paradox

Suppose there’s a barber who shaves only those who do not shave themselves. The question is, does the barber shave himself?

That was the question I posed in a previous article. The hint to the answer was actually given in the article: “What if your original assumptions were wrong?”

So let’s begin, with two fictitious men, John and Ricky. John has this unfortunate streak of accidental cuts, so he doesn’t shave himself if he could help it. Ricky has this inexplicable fear of people wielding sharp objects around his face (ever watched Sweeney Todd?), so he shaves his own beard.

It’s actually a mathematical question on logic. So forming the statements, we have:

If John does not shave himself, the barber shaves John.
If Ricky shaves himself, the barber does not shave Ricky.

Here’s the interesting part. Let’s substitute “John” and “Ricky” with “the barber”.

If the barber does not shave himself, the barber shaves the barber.
If the barber shaves himself, the barber does not shave the barber.

Either way, the statements don’t make logical sense, each contradicting itself and creating a paradox. So what went wrong?

The statements were correctly formed. It’s our original assumption that’s wrong. What’s our original assumption? That

there’s a barber who shaves only those who do not shave themselves

So the correct answer is, there’s no such barber.