At a certain point, you think you’ve seen all of those maddening, intentionally misleading Facebook math riddles. The first one I recall led to an unfriending on Facebook — and my first article for Slate. It discussed the history of “order of operations” and the ambiguities of math “language” (and amusingly led to just as much debate in the comments as on Facebook).
Since then, I’ve mostly skipped over them — until I saw the one about counting animals heading to the river. Have you seen it? I immediately recognized not only its familiar ambiguity, but also its potential as a great tool for demonstrating how to think critically about a medical study.
It goes like this (retaining original syntax):
“One rabbit saw six elephants while going to the river. Every elephant saw two monkeys going towards the river. Every monkey holds one parrot in their hands. How many Animals are going towards the river???”
The allegedly “correct” answer is five: Supposedly, the elephants are only observers (of the monkeys), and all of them see the two monkeys, who each have one parrot. So, one rabbit, two monkeys and two parrots equal five animals.
I did not get five. My first gut-reaction answer was 25: One rabbit plus 12 monkeys (two seen by each of six elephants) plus the 12 parrots, each carried by a monkey, equaled 25 animals. But as I went to put my answer on my friend’s post, I saw the five responses and quickly realized that the elephants could have all seen the same two monkeys.
Clearly, the question contained insufficient data to determine the answer. We have no idea how many monkeys — anywhere from two to 12 — the elephants see, so we can’t determine the total animals. Ever the nerd, I thought of my answer in terms of a quasi-confidence interval of 5-25.
I smugly left a comment on my friend’s post that the answer could be any odd number (the rabbit plus the couplings of monkeys and parrots) between five and 25. Then I showed it to my husband, who pointed out that the elephants could be going toward the river. No, the riddle didn’t say they were. But it didn’t specify they weren’t either. And you know what they say about assuming …
In one sense, this problem is a demonstration of what happens with poor, or at least imprecise, communication. But what if it’s not so much imprecise communication, but simply a lack of adequate data? Often, published medical studies include all the data available, but the data set remains incomplete. Maybe the researchers had no way of knowing whether the elephants were headed toward the river or not — much as we don’t often know whether an association involves causation, or which direction that causation might go.
In some cases, researchers use imputation to sub in values for the missing data, but imputation risks introducing bias. And most often, researchers need to draw some sort of conclusion to publish their study, despite the limitations of missing data (though, ideally, also noting those limitations in the study).
So I shared the problem myself on Facebook to see what other issues with the riddle my critically thinking friends noticed. And there are quite a few. Let’s say this problem represents a study. Here is a list of questions you should be asking the author:
- Do you know which direction the elephants are going? And are they all going the same direction? (If it’s possible that some of the elephants were going toward the river as well, that increases the possible range of animals from five to 31.)
- Is there a difference between going “to” the river versus going “toward” the river? (Go back and note the difference in language in the problem. Sometimes this is a case of being lost in translation or editing differences. Other times, there is a difference in definitions in a study.)
- Did the elephants all see the same monkeys, or did they see different monkeys? (If there were only two monkeys, why use “every” instead of “each,” as one friend noted.)
- If every monkey held one parrot “in their hands,” is that one parrot per monkey, or is it one parrot per hand of each monkey? (Now the potential range increases from five to 43 animals—at a maximum, one rabbit plus six elephants plus 12 monkeys plus 24 parrots.)
- Do the parrots count as “going towards” the river if they have no choice in the matter and are being brought along by the monkeys? This may sound like a silly or pedantic question, but given what we know of social determinants of health, we do have to consider opportunity, inequities and other factors that can influence whether someone actually ends up at a particular outcome as a result of their own actions or actions beyond their control.
- Are there any other animals, including ticks, parasites or other hangers-on, headed toward the river? The question asks about the total animals headed towards the river. There could be other animals who simply weren’t observed — just as there are sometimes confounders in a study that aren’t measured or accounted for. Further, the animals headed toward the river could have their own hitchhikers, from fleas and ticks to worms and other parasites—all part of Kingdom Animalia—headed toward the river too. Think of these critters as the hidden biases in a study that journalists and researchers have to explicitly think about and account for even if they can’t be seen or measured. They still matter.
At this point, you think I’m crazy for overthinking this to the point of ridiculousness, but in a way, this is the level of critical thinking you need to apply to every study you encounter. What data are missing? What did the researchers not take into account? What assumptions might the researchers be making? What are confounders that could influence the interpretation of the data? What are the biases that could interfere with accurate interpretation?
And finally, why is Animals capitalized? Really, it could be that no animals at all are headed to the river. Maybe they’re all waiting for the concert to start.