Answering back-of-the-envelope questions is a skill taught in many physics departments around the US. These order-of-magnitude problems are named after Enrico Fermi, who would famously ask his introductory course students a question that could not easily be looked up. This issue’s example is pawsitively impossible to find in a textbook:

How many sleeping cats are there in the US today?

Solution:

This problem is a good one because you could approach it from a few points of view. Whenever you approach a back-of-the-envelope question, you first need to decide on what you need to know to answer the question.

For this case, I want to know:

(a) How many cats live in the US?

(b) What are the odds any given cat is sleeping?

And if I multiply the answer to (a) and (b), I should get a good estimate.

I’ll start off by making an estimate about how often cats are sleeping, because it’s the easier of the two. For those of us that have had the pleasure of living with a cat, the answer may seem to be “all the time,” but that’s not really the case. I’m going to go with a safe guess of 16 hours a day. I say 16 is a safe guess because it’s in the right ballpark when you think about orders of magnitude and the range that could be the actual answer. Cats definitely sleep more than two hours a day, since most things do! It’s also a safe bet that they sleep more than four or even eight hours a day as well. On the other end of the spectrum, the most a cat can sleep in one day is 24 hours, which we know is too high. So, the actual answer is between 8 and 24, which averages to 16. Worst-case scenario, we are off by a factor of 2. That means if you were to measure an infinite number of identical cats in identical states, a so-called ensemble of cats, you’d find about 16/24, or two-thirds of them to be asleep. Thus, b) is about 2/3.

The first part is a bit more challenging to estimate than the second because we have to make some big assumptions. To answer how many cats live in the US, we need to add together the pet cat population and the stray population. I remember that there are about 320 million people in the US, and my guess is that, on average, one out of four people own one cat. That means there are about 80 million cats, or 80 Mcat (where M means mega, or 106). It’s much harder to guess how many strays live in the US, or how one would even count those cats. But, for the sake of the problem, let’s estimate there are ½ as many stray cats as pet cats. I make this guess (and it’s not a very good one) based on how many cat owners I know who got their cat through adoption. Sometimes you have to go with what you know. Thus the total number of cats is:

And since we already determined that there was a 2/3 probability a cat is asleep at any given time, we are left with 80 Mcat, or 80 million cats.

Are we right? I have no idea! I would also argue that while we might be able to count the number of pet cats, it’s very hard to know the number of stray cats. That being said, the actual answer is unlikely to be less than 40 million or more than 160 million cats, unless one of my assumptions is very wrong—where very means off by more than a factor of 2.

We may not ever be able to get the “true” answer (because surveying every cat in the country to see if it is asleep would be prohibitively expensive), but I can look up some statistics and infer a solution.

According to the ASPCA, the US has about 3.2 million cats in shelters nationwide, and there are approximately 86 million owned cats in the US.¹ Upon searching, the number of stray cats varies wildly, but some sources cite experts that claim around 70 million.² We also need to know how much cats sleep, which according to some research is about 13.2 hours.³

Thus, combining the known results:

which is surprisingly close to our back-of-the-envelope solution of 80 Mcats. (P)awesome!

Have a good idea for the next Fermi problem? sps [at] aip.org (Email us) with the subject line “Fermi problems.”

1. Pet Statistics.
2. Feral Cats.
3. S. Campbell and I. Tobler, Animal sleep: A review of sleep duration across phylogeny, Neurosci. Biobehav. Rev., Vol. 8, Issue 3, Autumn 1984, pp. 269–300.