The Boots of Abba and Babba

Nov 09, 2012

RAY: In the little Asian country of Tuvaniska, there are the two towns of Abba and Babba, a few miles apart on either side of a mountain. The preferred footwear in this region is the combat boot.

There are 20,000 people who live in Abba. And 1 percent of them have one foot only, so they wear, what? One boot. Half of the remaining people in the town wear no boots at all. They go barefoot.

The rest of the people wear two boots. Now you can figure out how many boots there are in Abba.

In Babba, 20 percent of the people have one foot. Of the remaining people, half of them wear no boots and half of them wear two boots -- and there are 20 thousand boots worn in Babba.

Here’s the question. What's the population of Babba? 
RAY: Here’s how you solve it. Let's go to Abba for a second, where one percent of the population is one-booted. Half of the remaining population wear no boots. And obviously, the other half of the remaining population wear the conventional two boots.

If you go and if you figure that out, you'll find out that there are 20,000 boots worn in Abba. And it turns out that it doesn't make any difference what percentage of the population has one leg.

And so, when you go to Babba, even though 20 percent of the population has one foot, because they have one foot, that's also equivalent to being, like, half the population, is it not? So every person that has one boot is just like the other part of the population where half of the people wear no boots.

So, the average is that half the people wear a boot. Therefore, one boot per person comes out to 20,000. So, if there are 20,000 boots worn in Babba, the population must be 20,000.

TOM: That was obfuscated! That was beautiful. The winner this week is Debra Garcia from Tucson, Arizona. Congratulations, Debra! 

Get the Car Talk Newsletter