Brainteaser #11 – King Eric’s Kingdom

Brainteasers and Riddles

King Eric of the kingdom of Epicus Escapicus has two daughters and two boys. The two daughters are all he expected from his children; intelligent, cunning, quick-witted and they also possess Amazonian strength. The boys, though he doesn’t like to admit, are small, unable to fight, and as thick as two short planks!

He ponders this and decides his kingdom would be better off with more girls. He then decrees to his kingdom:

“All couples must keep having children until they have a daughter!”

He ponders that this may cause massive overpopulation and makes a second decree:

“Couples must stop having children once they have a daughter!”

His subjects immediately start to follow his orders.

Once many years have passed, what’s the expected ratio of girls to boys in King Eric’s kingdom?

Firstly, the probability of having a boy or a girl is always 50:50 (in the vast, vast majority of circumstances).

Maybe you were tempted to solve this in an overly complex way. For example, each couple will half a girl half the time, who then stop immediately. Half of the remaining couples that had a boy will then have another half a chance of having a girl, who will then stop. Of course, the longer this goes on then the chance increases that couples may have loads of boys, maybe even hundreds, before they get a girl. This would be extraordinarily unlikely to happen, though.

The thing is, none of this matters. The ratio of boys to girls is always 1:1 and regardless of King Eric’s rule, his population will always level out to roughly 50:50.

If you work this out as an equation with C representing couples, it would be as follows:

C + C/2 + C/4 + C/8 + C/16. This is called a geometric series and the ratio remains constant: it will always be 1:1. Therefore, the ratio of boys to girls will be 1:1 after many years – don’t overthink it!

The only assumption you have to make here is that there are many families – the more there are, the closer the number will be to exactly 1:1, though it’ll be very close to 1:1 with just a couple thousand families or fewer.

If you’re still confused about the maths, you can read up more on it here.

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