The incredible ‘Birthday Problem'

When little numbers get surprisingly big

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 ...

The incredible ‘Birthday Problem'

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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Big Ben

On a visit to Big Ben, I was told that if you stand at the bottom of the tower with a portable radio and listen to the chimes on Radio 4 (they still transmit them live), you hear them on the radio before you hear them ‘for real’. I couldn’t believe it – but was intrigued enough to try it for myself. And you know what? It’s absolutely true. The bongs come out of the radio a fraction of a second before they reach your ears from the top of the tower. It’s something so silly, so counter-intuitive, that you have to tell people. (Well, I did.) Researching the explanation, I found that it’s because radio waves travel at the speed of light (186,000 miles per second) rather than the 700 or so miles per hour at which sound waves travel. The signal travelling down the wire from the microphone to the BBC goes at the speed of light too. Hence the radio version overtaking the real one.

I realised that this would be the perfect way to teach the principle in school physics lessons. Instead of a boring teacher droning on that ‘radio waves travel at the speed of light’, illustrate it with this beautiful and quirky little fact. The kids will remember it then. I certainly would have done if my physics teacher had taken this approach. As it was I had to wait until I heard a piece of so-called ‘trivia’ in my thirties.