The incredible ‘Birthday Problem'

When little numbers get surprisingly big

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How many people do you have to gather together in a room before it becomes more likely than not that at least two of them will share a birthday? This teaser is something I mention on the Central Line walk (which passes the wonderful Gresham College, where I first learned about it.) For years I've struggled to convince people that the answer - just 23 - is correct. But recently I found a way of doing it that really brings the truth of the puzzle home ...

A common answer when you ask people the question is 183, that being slightly more than half of 365, the number of possible birthdays. Or 184, if they're trying to take leap years into account. The fact that the answer is only 23 seems counter-intuitive. How can it be that with just that tiny number of people gathered in a room, the chances are that at least two of them will share a birthday?

There are any number of mathematical ways of expressing the thinking behind the answer, many of which contain fractions or decimals, most of which use the word ‘probability' and all of which leave the average layman (myself included) more or less cold. But what I love about this conundrum is that it shows how quickly a relatively small number of factors can combine to make things very complicated. And there is a way of expressing that without any complicated maths. It's to do with shapes.

Imagine you're the first person in the room. When the second person walks in, the odds that they share your birthday are 1 in 365. (Let's forget about leap years.) When the third person walks in, the odds that they share your birthday are also 1 in 365. So it's tempting to say that the chances of a birthday being shared have just doubled. But there is also the possibility that the second and third people share a birthday. So the chances have more than doubled.

You can think of the situation at this stage as being represented by a triangle. The three people are the three points of the triangle. The triangle's three sides show the three possible connections between you.

Now imagine a fourth person walks in. The situation is now a square. Its four sides show four possible connections - four possible chances of a birthday being shared. But there are also the square's two diagonals, making six possible connections in all. With five people - a pentagon - there are 10 possible connections. And so on - each new arrival into the room makes things more and more complicated. Which is another way of saying that each new arrival makes it more and more likely that a birthday will be shared. Imagine a 23-sided shape (any linguists out there who know what that's called?). Imagine all the lines linking each point of that shape to all the others. And suddenly the conundrum's answer seems all too believable.

This ‘numbers getting surprisingly big' thing can also be seen in the Rubik's cube fact mentioned elsewhere on this site. But perhaps I'll leave that for another post ...

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